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					         Optimal Derivative Strategies with Discrete
                       Rebalancing


               Nicole Branger∗           Beate Breuer‡           Christian Schlag§




                               This version: March 1, 2007




   ∗
                     u            a                           a    u               a
     Finance Center M¨nster, Westf¨lische Wilhelms-Universit¨t M¨nster, Universit¨tsstr. 14-16, 48143
  u
M¨nster, Germany, E-mail: Nicole.Branger@wiwi.uni-muenster.de
   ‡
     Graduate Program ’Finance and Monetary Economics’, Goethe University, Mertonstr. 17, Uni-
Postfach 77, D-60054 Frankfurt am Main, Germany, E-mail: breuer@finance.uni-frankfurt.de.
   §
     Finance Department, Goethe University, Mertonstr. 17, Uni-Postfach 77, D-60054 Frankfurt am Main,
Germany, E-mail: schlag@finance.uni-frankfurt.de.
       Optimal Derivative Strategies with Discrete Rebalancing

                 Nicole Branger∗          Beate Breuer‡          Christian Schlag§



                                    This version: March 1, 2007



                                             Abstract

       Optimal asset allocation strategies are often derived in continuous time models,
       but have to be implemented in discrete time. It has been shown that in models
       with stochastic volatility or jumps, an investor who just uses the continuous time
       strategy in discrete time has to trade at least daily to profit from having access
       to derivatives. In this paper, we determine the optimal investment strategy when
       discrete rebalancing is explicitly taken into account. We find that the investor buys
       a more conservative portfolio and reduces extreme positions in the derivatives com-
       pared to the continuous time case. In particular, his exposure to volatility risk is
       significantly reduced. We show that even with monthly rebalancing, the investor
       profits from trading derivatives and can realize up to 63% of the utility gain from
       option trading in continuous time. He also profits from having access to a variance
       contract, since the more stable exposures of the variance contract to the risk factors
       over time allow him to take larger positions in volatility and jump risk.

       Keywords: Asset Allocation, Discrete Trading, Use of Derivatives, Variance Con-
       tract

       JEL: G11




   ∗
                     u            a                           a    u               a
     Finance Center M¨nster, Westf¨lische Wilhelms-Universit¨t M¨nster, Universit¨tsstr. 14-16, 48143
  u
M¨nster, Germany, E-mail: Nicole.Branger@wiwi.uni-muenster.de
   ‡
     Graduate Program ’Finance and Monetary Economics’, Goethe University, Mertonstr. 17, Uni-
Postfach 77, D-60054 Frankfurt am Main, Germany, E-mail: breuer@finance.uni-frankfurt.de.
   §
     Finance Department, Goethe University, Mertonstr. 17, Uni-Postfach 77, D-60054 Frankfurt am Main,
Germany, E-mail: schlag@finance.uni-frankfurt.de.
1     Introduction
There is a huge literature on dynamic asset allocation in continuous time models, starting
with Merton (1971). His basic approach has been extended in several dimensions. Liu
(2006) gives closed-form solutions for affine models, which include the case of stochastic
volatility. Liu, Longstaff, and Pan (2003) consider a model with stochastic volatility and
contemporaneous jumps in the stock price and volatility where only the stock and the
money market account are traded. Liu and Pan (2003) add derivatives to the set of traded
assets to complete the market, but assume that jumps can occur in the stock price only.
Branger, Schlag, and Schneider (2006) extend their approach to include also jumps in
volatility.
     All these models assume that the portfolio can be rebalanced continuously. In reality,
however, the investor can only trade at discrete points in time, and the more often the
portfolio is rebalanced, the larger the transaction costs, especially for retail investors. An
ad-hoc solution to this problem is to implement the continuous time strategy in discrete
time. The performance of these naive discretization strategies is analyzed by Branger,
Breuer, and Schlag (2006). They show that even yearly rebalancing is nearly perfect if the
investor only trades the stock and the money market account. With options, however, the
picture changes completely: discrete trading can cause huge utility losses, and the investor
has to trade at least daily to benefit from having access to derivatives.
     In this paper, we solve for the optimal investment strategy in discrete time when
the investor uses derivatives. With discrete trading, the market is – technically speaking –
incomplete, and the choice of derivatives matters. We consider the cases where the investor
trades, in addition to stock and money market account, one or two options, and we also
consider the case where one of these options is replaced by a variance contract. There is
no analytical solution for the optimal investment strategy in discrete time, and we use
backward induction combined with numerical optimization and a Monte Carlo simulation
to solve the portfolio planning problem.
     We are not the first to consider discrete trading strategies involving options. Brandt
and Santa-Clara (2006) show how to approximate dynamic investment strategies by static
strategies in timing and conditioning portfolios. They use quadratic utility and extend the
basic Markowitz approach to portfolio selection to multiple periods and the use of condi-
tioning information. Although they argue that their method works for derivatives as well,
they use only stocks, bonds and the money market account in their numerical examples.
In our study, we focus on the use of derivatives, and we do not rely on approximating
strategies, but rather determine the truly optimal strategy in discrete time.
     Another strand of the literature analyzes the empirical performance of option strate-
gies to assess whether options are ’too good’, which would indicate a mispricing of options.


                                              1
Doran and Hamernick (2006) explore the historical performance of several standard op-
tion investment strategies using S&P 500 index options. They consider the average returns
and Sharpe ratios of various option investment strategies, where options constitute only
a small part of the overall portfolio. ITM and ATM protective put strategies, OTM and
ATM covered call strategies and short OTM and ATM put and call positions clearly out-
perform the S&P 500 in case of an option maturity of one month. With a maturity of one
year, various synthetic stock and long call strategies also outperform the S&P 500. The
performance is measured by the Sharpe ratio. Goetzmann, Ingersoll, Spiegel, and Welch
(2002), however, argue that the Sharpe ratio may be misleading in case of option payoffs.
Driessen and Maenhout (2004) investigate the optimal position in OTM puts or ATM
straddles of standard expected utility maximizers and non-expected-utility investors us-
ing historical data on the S&P 500 index and index options. They find that nearly all
types of investors optimally take short positions in puts and straddles. This result is ro-
bust with respect to transaction costs, margin requirements and Peso problems. While
these papers analyze whether investments in options have been ’too good’ in the past, we
are interested in the effect of discrete trading. We therefore use a simulation setup where
options are priced correctly by assumption, and we study to which degree the investor
can benefit from derivatives, even if he cannot trade continuously.
     A recent paper that analyzes investment in the variance contract is Egloff, Leippold,
and Wu (2006). They determine the optimal investment in variance contracts in a model
with two diffusive variance factors but no jumps and estimate their model using data on
the S&P 500 and S&P 500 variance swap rates. The utility improvements resulting in
the model are substantial,and this result is also confirmed by empirical analysis of the
strategy for the period from January 2003 to January 2006. However, one has to keep
in mind that the variance swap rate declined almost monotonically over this period, so
that large gains from shortening variance contracts are not surprising. Different from their
approach, we consider a model with jumps in returns and volatility, and we take discrete
trading into account when searching for the optimal strategy.
     We find that optimal strategies in discrete time are significantly better than naive
discretization approaches. The performance of an investment strategy is measured by the
fraction of the overall potential utility gain due to derivatives that is realized by this
strategy. The maximal utility gain due to derivatives is defined as the difference between
the expected utilities of two investors having access to different sets of securities. While
the first investor can trade only the stock and the money market account, the second
investor has access to a complete market where he implements a dynamic continuous
time trading strategy. Even with one option only, an investor who rebalances his portfolio
only once a month can realize about 56% of this maximal utility gain, and up to 63%
when using two derivatives. The best results are achieved with longer-term options, for
which the sensitivities with respect to the risk factors are much more stable over time


                                            2
than for short-term options. The choice of the strike price is important when the investor
can only use one call, but does not matter much if he can use two options. We also replace
one of the call options by a variance contract, for which the payoff is equal to the realized
variance over some time period. With the variance contract as the only derivative, the
investor can realize 59% of the maximal utility gain, three percentage points more than
with the optimal choice of options. If he can use the variance contract and a call option,
his utility gain increases to 63%, again three percentage points more than in the case with
two options.
     Compared to the optimal strategy in continuous time, the investor buys a more
conservative portfolio and reduces extreme positions in derivatives significantly. As a
consequence, his exposure to the risk factors is reduced, too. This is in particular true
for volatility risk, where the exposure shrinks to just over one third of the exposure from
the continuous time model. For discrete trading, the optimal exposure with respect to
the risk factors depends on volatility, while it is independent of volatility for continuous
trading. When volatility is high and rebalancing thus becomes more important due to
more unstable exposures, the investor chooses a more conservative portfolio if he can only
trade at discrete points in time. Furthermore the optimal stock position changes from a
short position in continuous time, which would be problematic from a general equilibrium
point of view anyway, to a long position in discrete time. When a variance contract is
available for investment, the investor can achieve much more constant factor exposures
over all volatility levels. As a consequence, utility gains are higher with the variance
contracts than with options only. The exposure to volatility, however, is still only about
half as large as with continuous trading.
     We check the robustness of our results with respect to the risk aversion of the investor,
the restriction of no borrowing, and the introduction of margin requirements. For a higher
risk aversion, the investor gains less in absolute terms from having access to derivatives,
but he can again realize more than half of the maximal utility gain from trading derivatives
in continuous time. We also impose the restriction that the investor cannot take a short
position in the money market account. This restriction is binding, and with one option,
the investor can only realize about one third of the utility gain, so that the borrowing
constraint is also economically important. With two options, however, there is hardly any
difference between the utility gains he can realize with and without borrowing constraint.
Finally, the introduction of margin requirements leads only to very small utility losses.
    The paper proceeds as follows. In Section 2, we shortly describe the market model
and the pricing of options. The numerical optimization methodology and the performance
measure are explained in Section 3. Section 4 describes the results in detail. Robustness
checks are discussed in Section 5, and Section 6 concludes.




                                              3
2      Model Setup
We consider a model with stochastic volatility, jumps in the stock price, and jumps in
volatility. In the context of option pricing, this SVCJ model is studied in, e.g., Duffie, Pan,
and Singleton (2000) or Broadie, Chernov, and Johannes (2005). Liu, Longstaff, and Pan
(2003) and Branger, Schlag, and Schneider (2006) analyze its implications for portfolio
planning.
     The dynamics of the stock price St and the local variance Vt are given by the following
stochastic differential equations
                                                                                (1)
 dSt =       r + η1 Vt + µX (λP − λQ )Vt St dt +                        Vt St dBt + St− µX (dNt − λP Vt dt)         (1)
                                                          (1)                     (2)
    dVt = κP (¯P − Vt )dt + σV
              v                                   Vt ρdBt +             1 − ρ2 dBt       + µY (dNt − λP Vt dt).     (2)

The Poisson process Nt has a jump intensity of λP Vt . µX and µY are the return and the
variance jump size, respectively, where we assume that both jump sizes are deterministic.
  √           √                                            (1)        (2)
η1 Vt and η2 Vt are the risk premia for the diffusions Bt and Bt , respectively. The
investor thus receives on average a compensation of η1 Vt and η2 Vt for taking on one unit
   √      (1)      √      (2)
of Vt dBt and Vt dBt , respectively. The difference µX (λP − λQ )Vt represents the
instantaneous compensation for jump risk.
     Given the market prices of risk for the two diffusion processes and for the jump risk
factor, the dynamics of the stock price and the volatility process under the risk-neutral
measure Q are
                                            (1)
     dSt = rSt dt +              Vt St dBt + St− µX (dNt − λQ Vt dt)                                                (3)
                                                            (1)                     (2)
     dVt = κQ (¯Q − Vt )dt + σV
               v                                   Vt   ρ dBt       +     1−   ρ2 dBt         + µY (dNt − λQ Vt dt), (4)

where

                         κQ = κP + σV ρη1 +                         1 − ρ2 η2 + λP − λQ µY
                    κQ v Q = κP v P .
                       ¯        ¯
             (1)           (2)
Under Q, Bt and Bt               are standard Brownian motions, and Nt is a Poisson process with
intensity λQ Vt .
     The investor can trade in the stock and the money market account. Furthermore, he
                                                                      (i)
has access to derivatives. The dynamics of the price of a derivative Ot = g (i) (t, St , Vt )
are given by
                     (i)              (i)
                                       (i)     (i)                                                  (1)
                   dOt     = rOt dt + gs St + gv σV ρ                        η1 Vt dt +        Vt dBt
                                      (i)                                               (2)
                                   + gv σV          1 − ρ2 η2 Vt dt +          Vt dBt
                                   + ∆g (i) · (λP − λQ )Vt dt + dNt − λP Vt dt ,


                                                                4
                                                             (1)        (2)
where the exposure to the risk factors dBt , dBt                                and dNt follows from the sensitivities

                    (i)             ∂g (i) (s, v)
                   gs =
                                         ∂s             (St ,Vt )
                                          (i)
                    (i)             ∂g (s, v)
                   gv =
                                       ∂v               (St ,Vt )
                      (i)               (i)
                 ∆g           = g             ((1 + µX )St− , Vt− + µY ) − g (i) (St− , Vt− ).

To calculate option prices and sensitivities in the SVCJ model, we use Fourier inversion
as e.g. explained in Duffie, Pan, and Singleton (2000). The resulting differential equations
are solved numerically.
                            (i)   (i)
      The sensitivities gs , gv and ∆g (i) depend on the maturity of the option, the current
stock price, and the current volatility. When one of these variables changes, the sensi-
tivities change as well. If a portfolio is not rebalanced continuously, its exposure with
respect to the risk factors will not remain constant until the next rebalancing date, but
can change rather dramatically. As Branger, Breuer, and Schlag (2006) have shown, this
non-stability can cause significant utility losses in case of discrete trading.
    Besides options, we also consider a variance contract, which is e.g. analyzed in Bon-
darenko (2004). When discrete returns are used to calculate the realized variance, its
payoff at time T is given by
                                                   T          2             T                 T
                                                       dSt
                   RV (0, T ) =                                     =           Vu du +           µ2 dNu .
                                                                                                   x
                                               0       St−              0                 0

The variance contract is special in that its exposure to volatility risk and jump risk does
not depend on the current level of the stock price or volatility, but only on its time
to maturity. We thus expect the exposure of a portfolio involving the variance contract
instead of an option to be more stable over time. On the other hand, the relation between
the exposure to volatility risk and jump risk is fixed for the variance contract, whereas
it depends on the moneyness for options. With call options, the investor is thus able to
choose the optimal mixture of these two exposures. This turns out to be important in the
case with one derivative only.



3     Numerical Methodology and Performance Measure-
      ment

3.1    Portfolio Planning Problem

The investor derives utility from terminal wealth only. We assume a CRRA-utility function
with risk aversion γ, i.e. U (W ) = W 1−γ /(1 − γ). The investor maximizes his expected


                                                                  5
utility, and his indirect utility function is defined as

                   J(w, v, t) =      max           Et U (WT ) Wt = w, Vt = v .
                                  {Φ[t,T ] ∈At }

Φ[t,T ] denotes his investment strategy over the time period between t and T . At is the set
of admissible strategies, which depends on whether the investor can trade continuously
or only at discrete points in time and on the assets he can trade. We assume that the
stock and the money market account are always available for trading. Furthermore, the
investor might be able to include one or several contingent claims in his portfolio, where
we focus on options and on the variance contract.


3.1.1     Continuous Time

In case of continuous trading, the solutions to the portfolio planning problem have been
studied extensively in the literature. Liu, Longstaff, and Pan (2003) analyze the case where
only the stock and the money market account are traded. With stochastic volatility and
deterministic jump sizes and either only jumps in the stock price or simultaneous jumps
in the stock price and the volatility, two derivatives are needed to complete the market.
This case is studied in Liu and Pan (2003) for a model without jumps in volatility, and in
Branger, Schlag, and Schneider (2006) for a model which includes jumps in volatility. Since
the market is complete, the investor can achieve any desired exposure to the risk factors,
and it turns out to be useful to work with these exposures instead of asset positions. The
dynamics of the investor’s wealth can thus be written as
                        B1                                (1)      B2                     (2)
        dWt = rWt dt + θt Wt η1 Vt dt +              Vt dBt     + θt Wt η2 Vt dt +   Vt dBt
                    N
                 + θt Wt− dNt − λQ Vt dt .                                                      (5)

where θB1 describes the share of the investor’s wealth invested in the B1-diffusion, θB2
the position in the B2-diffusion, and θN the share of the investors wealth invested in jump
risk. The corresponding positions in the assets follow from the sensitivities of the assets
with respect to the risk factors.


3.1.2     Discrete Time

If the investor can only trade at discrete points in time, the set of admissible strategies
is reduced to those which are constant between the rebalancing dates, and the market
becomes incomplete. With a naive discretization strategy, the investor just applies the
continuous time strategy in discrete time, while the alternative is to determine the optimal
strategy in discrete time.
    Branger, Breuer, and Schlag (2006) find that even with naive discretization, the loss
due to discrete trading can almost be neglected in the incomplete market where only the


                                                      6
stock and the money market account are traded. However, if the investor also has access to
derivatives, utility losses can become very large. Even with as high as weekly rebalancing
frequency, there is still a significant risk for the investor to end up with negative wealth
and thus a realized utility of minus infinity. In these cases, he is better off if he does not
use derivatives at all. To profit from having access to derivatives, the investor has to trade
at least daily if he relies on naive discretization.
     We denote the points in time where the investor adjusts his portfolio (plus the end
of the investment horizon) by 0 = t0 < t1 < t2 < . . . < tn = T . The dynamics of wealth
are
                                            K                                      K
                             St             Ok
    Wti = Wti−1        φti−1 i +      ψti−1 kti +
                                       k
                                                                     1 − φti−1 −          k
                                                                                         ψti−1   er(ti −ti−1 ) , (6)
                            Sti−1 k=1      Oti−1                                   k=1

                   k
where φti−1 and ψti−1 denote the weight of the stock and k−th derivative (k = 1, . . . , K)
at time t, repectively.
     The optimal portfolio of the investor in discrete time solves

          Jdisc (w, v, ti ) =               max                    Eti U (WT ) Wti = w, Vti = v
                                       k
                                {φtj ,ψt ,k=1,...,K,j=i,...,n−1}
                                        j


where we optimize over the portfolio composition from time ti to time tn−1 . From the
Bellman principle, we know that the optimal portfolio can be found by backward induction

     Jdisc (w, v, ti ) =            max           Eti Jdisc (Wti+1 , Vti+1 , ti+1 ) Wti = w, Vti = v , (7)
                                  k
                           {φti ,ψt ,k=1,...,K}
                                    i


where the dynamics of wealth are given by Equation (6) and where the dynamics of V
follow from the model specification.


3.1.3   Implementation

For discrete trading, there is no closed form solution for the portfolio planning problem,
and we now describe our optimization algorithm in more detail.
     We discretize the state space, which is given by time, wealth level, and local variance.
The discretization in time direction is determined by the pre-specified trading dates. For
the state variable variance V , we use a grid which is more narrowly spaced for smaller
variance levels with 10 equally spaced intervals between 0.003 and 0.12 and 10 equally
spaced intervals between 0.12 and 0.76. For variance levels in between the grid points,
conditional utility levels and optimal portfolio positions are interpolated linearly. When
the variance increases above or falls below the level of the largest respectively smallest grid
point, the utility levels and optimal portfolio weights for the respective grid points on the
boundary are used. Finally, note that in the CRRA case Jdisc (w, v, t) = w1−γ Jdisc (1, v, t),


                                                          7
so that it is sufficient to determine the optimal indirect utility function and the optimal
portfolio weights for w = 1.
     The portfolio planning problem is solved by backward induction, i.e. starting at time
tn ≡ T for which J is known. For every grid point characterized by a point in time ti
(i = n − 1, n − 2, . . . , 1, 0) and a variance level v , we then have to find the indirect utility
function and the optimal portfolio weights, that is the stock position φ, the position in the
first derivative ψ1 and, if included, the position in the second derivative ψ2 . The numerical
optimization uses a sequential quadratic programming method. The optimizations are
each carried out with two different starting values, once with the optimal allocation from
the previous variance grid point and once with the stock position initialized to one and
the derivative positions initialized to zero. Expected utility for a given candidate portfolio
composition is calculated by Monte-Carlo simulation, where we simulate the variance and
the wealth level at time ti+1 and then rely on the already known indirect utility at time
ti+1 (which has to be interpolated), using Equation (7). The Monte-Carlo simulation is
based on 500,000 paths, and we use the same random numbers for each of these Monte-
Carlo simulations. To gain efficiency, we tabulate the prices of the options at the end of
each period when they are sold using the above-mentioned variance grid and a grid for the
moneyness (stock price over strike price). Prices between the grid points are interpolated
linearly. We use the parameter values shown in Table 1 and adapted from Liu, Longstaff,
and Pan (2003).


3.2    Performance Measurement

Our objective is to assess the performance of optimal trading strategies in continuous
time. Furthermore, we want to identify which derivatives the investor should trade and
how much he profits from more frequent trading. To do so, we need a measure for the
improvement of the investor’s expected utility compared to some reference strategy. We
rely on the annualized percentage difference in certainty equivalent wealth between the
strategy we want to evaluate and a reference strategy. This measure has been used, e.g.,
in Liu and Pan (2003).
     The first benchmark we consider is the optimal strategy with continuous trading in
a complete market. This strategy gives the maximal expected utility the investor can
achieve in this market. Its certainty equivalent wealth Wcont,w is defined implicitly via
                                                          1−γ
                                                         Wcont,w
                                 Jcont,w (0, W0 , V0 ) =         ,
                                                         1−γ
and the certainty equivalent wealth Wdisc,w of the discrete strategy with derivatives is
defined by
                                                       1−γ
                                                      Wdisc,w
                              Jdisc,w (0, W0 , V0 ) =         .
                                                      1−γ


                                                8
The improvement the investor could achieve by continuous trading instead of discrete
trading is then given by the annualized percentage difference in certainty equivalent wealth

                                          ln(Wcont,w /Wdisc,w )
                         Rdisc→cont,w =                         · 100.
                                                   T
This number will be positive since the discrete time strategies cannot be better than the
continuous time strategies.
     The second benchmark we consider is a discrete strategy where the investor trades
the stock and the money market account only. The certainty equivalent wealth for this
case without derivatives is denoted by Wdisc,w/o . The rebalancing frequency is the same
for both strategies (with and without derivatives), and the portfolio improvement due to
access to derivatives is measured by

                                          ln(Wdisc,w /Wdisc,w/o )
                        Rdisc,w/o→w =                             · 100.
                                                    T
This number will also be positive since the investor cannot be worse off with additional
investment possibilities. However, this does not hold for a naive discretization strategy.
For this strategy, the improvement due to additional assets could be more than offset
by the utility loss due to using a strategy that is optimal with continuous trading, but
sub-optimal with discrete trading. Branger, Breuer, and Schlag (2006) show that this is
indeed the case even for rebalancing frequencies of one week.
     The sum Rdisc,w/o→w +Rdisc→cont,w gives the utility gain when going from the optimal
discrete time strategy with the stock and the money market account only to the optimal
continuous time strategy in a complete market. This term only depends on the rebalancing
frequency and decreases the more often the portfolio is rebalanced, but it does not depend
on the choice of derivatives used to complete the market. Comparing Rdisc,w/o→w to this
sum then shows how much of the utility gain due to derivatives can be realized with
discrete trading.


3.3    Margin Requirements

An investor is not only faced with the problem that he can only trade at discrete points in
time, but he may also face margin requirements which limit the positions he can take in
risky assets and in particular in derivatives. The margin requirements used in this study
are adapted from Interactivebrokers.com (2006). In case of differences between day and
overnight requirements, overnight requirements are used.
     We use relative margin requirements, i.e. we impose restrictions on the portfolio
weights. φ denotes the relative investment in the stock, and ψ is the investment in a
specific call option. Furthermore, we assume that margin requirements are only imposed


                                                9
at rebalancing times. As a consequence, they are independent of the stock price, and we
can normalize the stock price to one. Monitoring the margin in continuous time, as it is
the case in practice, would introduce path-dependence into the problem and would make
it computationally intractable.
    For long stock positions, the required margin is 50% of the stock price:

                                      M R(S) = 0.5φ.

Margin requirements for short stock positions are not imposed since short stock positions
are not optimal in our study except for one marginal case.
    For a long position in a call, the margin requirement is given by the call price, i.e.

                                      M R(LC) = ψ.

The margin requirement for a short position depends on whether it is naked or covered.
With a portfolio weight of ψ for the option, the number of options per unit of wealth is
given by −ψ , where O is the call price. For ψ < 0, the number of covered calls is then
          O
given by min −ψ , φ , and the number of naked calls is
               O

                       −ψ           −ψ                 −ψ
                          − min        ,φ      = max      − φ, 0 .
                       O            O                  O

For the covered call, the margin requirement is equal to the option price if the option is
in the money and zero otherwise:
                                       −ψ
                                min    O
                                          ,φ   · O ITM-call
                 M R(CC) =
                                0                  OTM-call, ATM-call

Margin requirements for short naked calls are more complicated. They are given by 100%
of the option market value plus the maximum of 15% of the underlying market value
minus the out-of-the-money amount and 10% of the underlying market value:

                          −ψ
      M R(N C) = max         − φ, 0 [O + max {0.15 − max(Strike − 1.0, 0), 0.1}] .
                          O

The overall value of the portfolio then has to be larger than the sum of the margin
requirements:
                  M R(S) + M R(N C) + M R(CC) + M R(LC) ≤ 1.0.
In case of two options, the margin requirements for the second option are calculated in
the same way.




                                            10
4     Results

4.1    Call Options as Traded Derivatives

We first consider the case where the investor uses only one call option and rebalances his
portfolio once a month. The results for a risk aversion of γ = 3 are given in Table 2, and
the results for a higher risk aversion (γ = 8) are qualitatively similar. We analyze how
much of the maximal utility gain the investor can realize, and what the optimal options
both in terms of moneyness and in terms of time to maturity are.
     The highest utility gains are obtained for call options with a time to maturity of 6
months and a strike of 104% of the initial stock price and for call options with a time
to maturity of one year and a strike of 116% of the initial stock price. For these options,
Rdisc,w/o→w is equal to 1.85%, i.e. the investor’s utility gain corresponds to an annualized
excess return of 1.85% compared to the optimal strategy without options, which is about
56% of the maximal gain Rdisc,w/o→w + Rdisc→cont,w . With a time to maturity of three
months to maturity, the maximal utility gain is reduced to 1.58%. Options with one
month to maturity give a maximal gain of 1.09%, which is about 33% of the potential
gain.
     The bad performance of options with only one month to maturity can be explained
intuitively. Options allow the investor to achieve an exposure to volatility risk (which he
cannot obtain with the stock), and to unbundle the fixed relation between diffusion risk
and jump risk embedded in the stock. With a rebalancing frequency of one month and a
time to maturity of the option of one month, the investor holds the option until expiry.
At the end of this one month, the exposure to volatility risk vanishes completely (and
approaches zero before), and the actual exposure of the portfolio to volatility risk will
thus move away from the optimal one rather fast, which deteriorates the performance.
For an option with longer time to maturity, the volatility exposure will be much more
stable over time.
     The optimal strike of the call is the larger the longer the time to maturity. While for
a time to maturity of 3 months, an ATM-call is optimal, the investor should use an OTM-
call with a moneyness (defined as strike price divided by stock price) of 104% for 6 months
to maturity and a deep OTM-call with a moneyness of 116% of the initial stock price for
one year to maturity. The intuition that the short call short serves two purposes. First
the investor wants a negative exposure to volatility risk due to the negative risk premium,
and, second, the option makes it possible to obtain a better exposure to volatility and
jump risk than offered by the stock alone. In our example, the investor needs additional
negative exposure to jump risk. To achieve these two goals, the investor has to choose a
call with the appropriate ratio of jump risk exposure to B2-exposure after hedging for



                                            11
B1-exposure, and this ratio depends on the time to maturity.
     For the optimal strike, the investor sells more calls short than he owns stocks. Thus
his payoff is not monotonically increasing in the stock price. He trades potential gains
in case of extremely large stock price increases for a higher return is case of a bad or
moderate performance of the stock.
     The results for the case when the investor can include two options in his portfolio
are given in Table 3, where we have assumed that one of the options is the ATM-call.
The maximum utility gain is hardly larger than with one option. The largest value for
Rdisc,w/o→w of about 1.99% is again obtained using calls with a time to maturity of six
months or one year, and is smaller for shorter times to maturity of the options. The strike
price for the second option is less important than in the case where only one option is
used, and utility differences between different strike prices are very small.


4.2    Rebalancing Frequency

Table 4 shows the results for rebalancing intervals from 3 months to two weeks. We
consider the case with one call, and due to the results presented above, we use a time to
maturity of one year and moneyness values from 100% to 120%.
     The utility gain is the larger the more often the investor rebalances his portfolio. It
increases from an extra annualized excess return of 1.34% in case of quarterly rebalancing
to 1.51% in case of bimonthly rebalancing, 1.85% in case of monthly rebalancing and
2.19% in case of biweekly rebalancing. The higher the rebalancing frequency, the higher
the optimal strike used by the investor. Furthermore, the optimal portfolio becomes more
extreme the higher the rebalancing frequency. When he has the possibility to rebalance
his portfolio more often, the investor worries less about the stability of his exposure to
the risk factors.


4.3    Factor Exposures

To interpret the findings, we take a look at the exposure of the strategies to the different
risk factors. In all tables, we give the exposures to the three different risk factors at the
beginning of the first period where the volatility is equal to its long-run mean. θB1 denotes
the diffusive stock price risk, θB2 the diffusive volatility risk, and θN the jump risk. These
exposures are compared to two benchmarks. The first benchmark is given by a continuous
trading strategy in an incomplete market where only the stock and the money market
account are used. In this case, the optimal exposures at t = 0 are θB1 = 0.81, θB2 = 0
and θN = −0.20. The second benchmark is given by the optimal exposure in a complete




                                            12
market, where we have θB1 = 0.51, θB2 = −0.73, and θN = −0.42 (cf. Branger, Breuer,
and Schlag (2006), Figure 4, 1st row).
     In case of discrete trading, the investor weights several objectives against each other.
First, he wants to get as close as possible to the optimal exposure in continuous time.
Second, he wants the exposure to be stable over time. And third, he is particularly afraid
of ending up with very low levels of terminal wealth. The last two points will lead to a
more conservative portfolio and to less leverage.
     With two options, the investor could always set up a portfolio for which the initial
exposure is equal to the optimal exposure in continuous time. The results in Table 3,
however, show that he chooses a portfolio with an initial exposure that is smaller than
the optimal one in absolute terms. The differences are particulary large for the exposure
to volatility diffusion risk, where the initial exposure is less than 40% of the optimal
one with continuous rebalancing, while the jump risk exposure and exposure to stock
diffusion risk are both quite close to the optimal one from the continuous time model.
Since the exposure to volatility risk is achieved by option positions, these are in absolute
terms also much smaller for discrete trading than for continuous trading. This can also be
seen in Table 5 which shows the optimal portfolio decompositions for continuous trading
(upper panel) and discrete trading (lower panel). The smaller positions in options reduce
the effect of changes in option sensitivities on the portfolio exposure over time and lead
to a portfolio exposure that is more stable over time. Put together, the investor thus
foregoes parts of the risk premium earned on volatility risk in order to obtain a more
stable portfolio. If he rebalances more often, stability becomes less of an issue.
     The optimal position in the stock differs significantly between continuous and discrete
trading. Table 5 shows that the investor takes a short position in the stock if he can trade
continuously, but a long position if he can only rebalance monthly. To get the intuition,
assume that the optimal portfolio is put together step by step. First, the investor uses a
position in the stock to achieve the desired B1-exposure. This also gives him a certain
jump exposure. Second, the B2-exposure and the remaining jump exposure (which are
both negative in our example) are achieved by a portfolio of two options which is hedged
by a position in the stock so that it has zero B1-exposure. As can be seen from Table 5,
this induces a long position in the call with the lower strike price and a short position
in the call with the higher strike price. For a moderate volatility exposure as in case of
discrete trading, the hedging position in the stock is positive, while for a large volatility
exposure in case of continuous trading, this stock position is negative. The stock position
from the first step is positive. Put together, this gives a long position in the stock for
discrete trading and a short position for continuous trading.
     The objective to achieve a stable exposure also explains why the investor prefers
options with a maturity of at least six months, for which the sensitivities vary much less


                                             13
than for shorter-term contracts.
     With one option, the investor cannot realize every desired factor exposure, since the
market is incomplete. The choice of the strike price therefore becomes important. From
a comparison of Tables 2 and 3, it can be seen that with just one option those strikes
are best where the investor can realize an initial factor exposure very close to the initial
factor exposure with two options.
      Until now we have only looked at the factor exposures realized at the beginning of
the first period with an initial volatility equal to the long-run mean. Now we take a look
at the optimal factor exposures conditional on the remaining number of time periods and
the spot volatility. Figure 1 shows these exposures for the case where the investor uses
two options, both with a time to maturity of 1 year. The first option is an ATM-call, the
second one is an ITM-call with a strike price equal to 89% of the stock price at the time
when the portfolio is bought (the choice with the best results). The shorter the remaining
time horizon, the smaller (in absolute terms) the optimal exposure to B1-risk and to jump
risk, which can be attributed to a hedging demand which vanishes over time. The optimal
exposure to B2-risk, on the other hand, hardly depends on the time horizon.
      The dependence of the optimal factor exposures on the current volatility is much more
pronounced, which is quite surprising given that the optimal exposure in continuous time
does not depend on volatility at all. The demand for B1 exposure increases in volatility,
and for very high levels of volatility, it even exceeds the optimal exposure in continuous
time. The demand for B2 and jump exposure, on the other hand, decreases in volatility
in absolute terms and approaches the optimal exposure from a continuous time model for
low volatility levels. To get the intuition, note that the investor is not only interested in
being as close as possible to the optimal exposure from a continuous time model. He also
wants to achieve a stable portfolio exposure and to reduce the risk of ending up with a
very low or even negative terminal wealth. For the jump component, the crucial point is
that the intensity of jumps is proportional to the level of variance. The risk of multiple
jumps until the next rebalancing date thus increases in variance. Since a second or third
jump might drive the investor into bankruptcy, he reduces his jump exposure if volatility
is high. For high volatility, the exposure of the portfolio to volatility risk is less stable over
time, and the reaction of the investor is again to reduce his overall investment into risky
assets. The reduced exposure to volatility risk and jump risk reduces the expected excess
return on the portfolio. This induces him to increase the position in the stock slightly,
which offers a stable risk exposure, which explains why the exposure to B1 increases in
volatility.




                                               14
4.4    Value of Volatility Timing

Rebalancing in our setup can be seen as a two-stage process. First allocations that drifted
away from initial allocations are brought back to the initial allocations and second these
allocations are adjusted for the difference in optimal positions due to the difference be-
tween current and initial volatility. The question that arises is which share of the volatility
gain is due to the first step and which share is due to the second. To answer this ques-
tion, we carried out only the first step, i. e. we rebalanced the portfolio just back to the
initial allocations which are optimal for a variance equal to the average variance. We
analyzed the value of volatility timing for options with a time-to-maturity of 6 months
and a strike of 108%. For monthly rebalancing the value of adjusting the portfolio to the
current volatility amounts to 12% of the Rdisc,w/o→w , for bimonthly rebalancing to 19%
and for rebalancing every 3 months to 22% of Rdisc,w/o→w . This means the relative value
of volatility timing becomes larger if the portfolio is rebalanced less often. The reason
is that with less frequent rebalancing the total utility gain due to rebalancing decreases,
which makes volatility timing more important.


4.5    Distribution of Terminal Wealth

Next, we have a look at the distribution of terminal wealth, where we consider the case
with one option, but different times to maturity and strike prices. Figure 2 gives the
first four moments as well as the minimum terminal wealth and the utility improvement
Rdisc,w/o→w as a function of the strike price. To plot all numbers in one graph, we have
standardized each moment as well as the minimum and the performance measure by its
mean in the respective maturity group.
     The optimal strike is always indicated by the maximum of the standardized Rdisc,w/o→w .
The investor prefers portfolios with a high mean, and he accepts a certain level of standard
deviation. The dependence on skewness and kurtosis as well as on the minimum wealth
level is more involved. The smaller skewness and kurtosis and the larger the minimum,
the smaller the risk of ending up with very low terminal wealth levels, and the more
conservative the portfolio. However, the risk premium earned on the portfolio decreases,
too. The optimal trade-off between risk and return then depends on the risk aversion of
the investor.


4.6    No-Leverage Constraints

As shown above, the optimal portfolios can be quite extreme. To see whether our results
depend strongly on leverage, we have conducted a robustness check by restricting the
investor from taking a short position in the money market account.


                                              15
     In the case with one option, this no-leverage constraint reduces the optimal Rdisc,w/o→w
from about 1.85% to about 1.17%. There is a strong tendency for the results to improve
with an increasing strike price. Looking at the factor exposures we see that the investor
can now realize about the same stock price diffusive exposure as in the unconstrained case,
but much smaller volatility and jump exposures. He is not able to increase his volatility
exposure further in absolute terms, since a larger short position in the call would also lower
his B1 exposure which now cannot be compensated by leveraging the stock position. For
this reason, he prefers short calls with high strikes which yield a high B2 exposures with
the smallest reduction in the B1 exposure.
     If the investor is allowed to invest into two options, he will go long one option and
short the other. Then, the leverage is quite small, and the restriction causes practically
no utility losses, but he can still realize an Rdisc,w/o→w of 1.99%.


4.7    Margin Requirements

The results for the case with margin requirements are given in Table 6. With one option,
the optimal Rdisc,w/o→w is reduced from 1.85% to 1.83%. The optimal strike drops from
116% to 112%. In terms of the optimal investment strategy both the stock investment and
the short position in the call are reduced. In terms of factor exposures this corresponds
to an increase in the stock price diffusion exposure and a reduction in the volatility and
jump exposure.
     With two options, the optimal Rdisc,w/o→w is 1.83% compared to 1.99% without
margin requirements and thus practically the same as with one option. The optimal
strike of the second call increases from 89% to 113%. This is due to the fact that for
a stock investment we require only a margin of 50 % while for the in-the-money call
the whole purchase price is deducted from the margin account. This, in turn, practically
leads to a complete loss of the utility gains from the second option. Put together, margin
requirements do not have a strong influence on the investor’s utility gain.


4.8    Variance Contracts

Up to now, the investor has had access to call options only. Now we assume that also the
variance contract is available. The question is which of these two contracts is the better
choice for the investor. The results are given in Table 7.
     In the first case, the investor is not allowed to trade options. The optimal Rdisc,w/o→w
in this case is 1.98%. The utility gain is thus significantly higher than with one call option,
where the investor could realize an Rdisc,w/o→w of 1.85%. Like for call options, the investor
is best off if he uses claims with long times to maturity. The reason for the superiority of


                                             16
the variance contract is that its factor exposures do not change when volatility changes.
Therefore the risk of a deviation of realized factor exposures from optimal factor exposures
is smaller. The investor will then take a larger position and thus earn a big part of the
risk premia even when the volatility is high.
     In the case with a call option with the same time to maturity as the variance contract,
the best results are achieved for a maturity of 6 months and a moneyness of 112% and
for a time to maturity of 1 year and a moneyness of 116% with an Rdisc,w/o→w of 2.11%
each. This is about 63% of the utility gain due to derivatives in a continuous time model.
This utility gain is again higher than with the optimal choice of two call options. For the
optimal portfolio, the investor takes a short position in the variance contract, while the
position in the OTM call changes from a short position for low volatility levels to a long
position for high volatility levels.
     Figure 3 shows the factor exposure of the optimal portfolios as a function of the
remaining investment horizon and volatility. It confirms that with the variance contract,
the factor exposures are much closer to the optimal factor exposures from the continuous
time model. However, the volatility exposure is still about only half as large as in the
continuous time model. Furthermore, the dependence of the optimal exposure on volatility
is much less pronounced than with two calls.
     Figure 4 shows the prices and sensitivities of variance contracts for different times
to maturity. We see that, different from calls, the vega and the price change in case of
a jump are constant across volatility levels (delta is zero). The price itself is quadratic
in volatility (linear in variance). Variance contracts with longer maturities have larger
sensitivities and higher prices. The reason that they perform better in the asset allocation
problem is that their sensitivities change less over time than those of variance contracts
with shorter maturities.



5     Robustness Checks

5.1    Transaction Costs

Our optimization setup requires that the whole option position is turned over at every
rebalancing date and replaced by a position of options with the original maturity and a
defined strike price. Therefore it is only possible to consider transaction costs under the
somewhat extreme assumption that the investor actually liquidates all assets in the port-
folio, before setting up the new position. Since the turnover in the stock position is small
relative to the turnover in options and since the investor makes only small adjustments to
the stock position, and we can only apply transaction costs to the whole stock position or
none at all, we set transaction costs on stocks equal to zero. Transaction costs on options


                                            17
are set to numbers between 0.5% and 3.0% of the option price for a one-way transaction.
For comparison, Bakshi, Cao, and Chen (1997) consider transaction costs of 6% per round
trip for OTM puts and 4% per round trip for ATM options. We calculate the results for
different rebalancing frequencies to see which rebalancing frequency is optimal given a
certain level of transaction costs.
      The results are given in Table 2. Already moderate transaction costs of only 0.5%
reduce the achievable Rdisc,w/o→w from 1.827% to 1.072%. However monthly rebalancing
is still better than less frequent rebalancing. With transaction costs of 1.0% respectively
2.0%, rebalancing every 2 months is optimal with an Rdisc,w/o→w of 0.804% respectively
0.520%. With transaction costs of 3.0% it is only worthwhile to rebalance every 3 months
yielding an Rdisc,w/o→w of 0.373%. The optimal strike of the option increases with the
amount of transaction costs, since farther out of the money calls are cheaper and provide
a higher exposure to volatility risk per unit of cost. Since our calculations however over-
state the influence of transaction costs, further research is necessary to determine optimal
derivative positions under more realistic assumption on portfolio turnover.


5.2    Parameter Uncertainty

In this section we want to analyze how sensitive allocations are to the mis-measurement
of parameters. We assume that the investor can infer the Q-measure parameters perfectly
from the data, since there are many option prices at hand to calibrate an option pricing
model. We assume that there is no uncertainty about the general model for option prices.
The investor is uncertain about the size of the risk premia η1 , η2 and λP −λQ . He expresses
his uncertainty by assuming that risk premia are normally distributed with a certain mean
and variance derived from empirical studies. The standard deviations are 0.92 for η1 , 1.10
for η2 and 0.28 for λP .
     We vary the assumed mean of the respective risk premium µ, and see how this
affects the investor’s optimal allocations and the utility gain when the performance of
the obtained strategy is evaluated using the true parameters. The results are given in
Table 9. For comparison in the first lines results for the case where he can only invest
in stock and money market account are given. Assume that the investor is uncertain
about the risk premium for B1-diffusive risk. If the mean is estimated correctly there
is practically no utility loss. The same can be observed with options for all three risk
premia. Even overestimation by half a standard deviation causes only moderate utility
losses. The largest utility losses are observed if η1 is uncertain with Rdisc,w/o→w falling from
1.786% to 1.698%. The losses are larger if the mean is over- or underestimated by a full
standard deviation. For η1 , Rdisc,w/o→w then falls to 1.447% respectively 1.497%. If η2 is
mis-estimated the utility losses are much smaller although the standard deviation is larger



                                              18
and the parameters are of similar size. Rdisc,w/o→w falls just to 1.735% respectively 1.772%.
This is due to the fact that under discrete trading the optimal position in volatility risk
does not increase linearly in |η2 |. The optimal position in volatility risk is restricted by the
stability of the factor exposures and therefore does not change much if the risk premium
is changed. All in all the optimal positions are not very sensitive to mis-measurement of
the risk premia.



6     Conclusion
An investor will in general profit from having access to derivatives. If he uses naive dis-
cretization strategies, however, he should rebalance his portfolio at least daily. Optimal
trading strategies in discrete time, on the other hand, give acceptable results even for
monthly rebalancing. We have shown that an investor can realize up to 63% of the max-
imal gain from derivative investment in continuous time. And although two derivatives
are needed in the continuous-time model to complete the market, the investor can realize
almost the whole utility gain with one derivative – given that he chooses the optimal
strike of the option or uses the variance contract.
     This result enlarges the group of investors that can profit from trading derivatives
significantly. It also justifies why retail investor might want to have access to derivatives.
Since variance contracts yield better results than options, investors would benefit from
the introduction of these contracts where they do not exist yet.
     In general, the investor chooses a more conservative portfolio if he can only trade
at discrete points in time. He foregoes some risk premia, in particular on volatility risk,
to avoid extreme positions which would make the portfolio exposure more unstable over
time. The best choice for the investor is to short long-term OTM calls or the variance
contract.
     The strategy of shortening long-term OTM calls together with a stock investment
is packaged as ’discount certificates’, which are thus a good and simple way for retail
investors to profit from trading derivatives. Hedge funds, that have larger resources and
face lower transaction costs, can realize large utility gains by following more sophisticated
strategies and rebalancing more often. Nevertheless, they should also reduce the positions
that are optimal in a continuous time model to take discrete trading into account.




                                               19
References
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  Pricing Models, Journal of Finance 52, 2003–2049.

Bondarenko, O., 2004, Market Price of Variance Risk and Performance of Hedge Funds,
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Brandt, M., and P. Santa-Clara, 2006, Dynamic Portfolio Selection by Augmenting the
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Branger, N., B. Breuer, and C. Schlag, 2006, Discrete-Time Implementation of
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Branger, N., C. Schlag, and E. Schneider, 2006, Optimal Portfolios When Volatility can
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Broadie, M., M. Chernov, and M. Johannes, 2005, Model Specification and Risk Premi-
  ums: Evidence From Futures Options, Working Paper.

Doran, James S., and Robert Hamernick, 2006, Is There Money to be Made Investing in
 Options? A Historical Perspective, Working Paper.

Driessen, J., and P. Maenhout, 2004, A Portfolio Perspective on Option Pricing Anomalies,
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Duffie, D., J. Pan, and K. Singleton, 2000, Transform Analysis and Asset Pricing for
 Affine Jump Diffusions, Econometrica 68, 1343–1376.

Egloff, D., M. Leippold, and L. Wu, 2006, Variance Risk Dynamics, Variance Risk Premia
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Goetzmann, W., J. Ingersoll, M. Spiegel, and I. Welch, 2002, Sharpening Sharpe Ratios,
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Interactivebrokers.com, 2006, Description of Margin Requirements for Interactivebro-
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Liu, J., 2006, Portfolio Selection in Stochastic Environments, Review of Financial Studies.

Liu, J., F.A. Longstaff, and J. Pan, 2003, Dynamic Asset Allocation with Event Risk,
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Liu, J., and J. Pan, 2003, Dynamic Derivative Strategies, Journal of Financial Economics
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                                            20
Merton, R. C., 1971, Optimum Consumption and Portfolio Rules in a Continuous-Time
 Model, Journal of Economic Theory 3, 373–413.




                                       21
                                             SVCJ
                                     κP        5.3
                                     vP
                                     ¯     0.02172
                                     λP    1.84156
                                     σV    0.22478
                                       ρ    − 0.57
                                     µX     − 0.25
                                     µY    0.22578
                                      η1   1.38117
                                      η2     − 2.0
                                     λQ    7.36623
                                     κQ    3.50630
                                     vQ
                                     ¯     0.03283

                           Table 1: Calibrated Parameters

The table shows the calibrated parameters for the SVCJ model.




                                           22
                                                        B2       N
           St. 1 (φ )0 (ψ1 )0      (# calls) θB10      θ0       θ0 Rdisc→cont,w Rdisc,w/o→w
                                          time to maturity: 1 month
           1.00   1.282 -0.025      (-1.194) 0.629 -0.081 -0.297         2.529         0.811
           0.96   1.461 -0.055      (-1.114) 0.528 -0.050 -0.312         2.295         1.045
           0.92   1.615 -0.096      (-1.115) 0.552 -0.025 -0.312         2.254         1.086
                                         time to maturity: 3 months
           1.08   1.328   -0.031    (-3.772) 0.599 -0.230 -0.313         2.130         1.210
           1.04   1.502   -0.053    (-2.630) 0.505 -0.220 -0.336         1.788         1.552
           1.00   1.573   -0.077    (-1.908) 0.454 -0.165 -0.332         1.759         1.581
           0.96   1.658   -0.109    (-1.592) 0.451 -0.114 -0.326         1.875         1.466
           0.92   1.815   -0.156    (-1.539) 0.478 -0.079 -0.327         1.982         1.358
                                         time to maturity: 6 months
           1.20   1.237   -0.029    (-9.921) 0.624 -0.252 -0.308         2.253         1.087
           1.16   1.387   -0.043    (-6.525) 0.574 -0.277 -0.331         1.932         1.409
           1.12   1.455   -0.059    (-4.348) 0.493 -0.266 -0.332         1.660         1.681
           1.08   1.626   -0.084    (-3.384) 0.448 -0.258 -0.352         1.514         1.827
           1.04   1.711   -0.109    (-2.631) 0.421 -0.219 -0.351         1.494         1.846
           1.00   1.728   -0.133    (-2.093) 0.416 -0.169 -0.335         1.585         1.755
           0.96   1.889   -0.179    (-1.968) 0.428 -0.140 -0.341         1.671         1.669
           0.92   1.915   -0.215    (-1.765) 0.453 -0.102 -0.322         1.831         1.509
                                           time to maturity: 1 year
           1.20   1.624   -0.101    (-5.043) 0.432 -0.253 -0.344         1.536         1.805
           1.16   1.698   -0.118    (-3.954) 0.443 -0.231 -0.348         1.486         1.854
           1.12   1.609   -0.124    (-2.895) 0.430 -0.186 -0.319         1.506         1.834
           1.08   1.849   -0.169    (-2.860) 0.415 -0.192 -0.347         1.523         1.817
           1.04   1.849   -0.191    (-2.413) 0.417 -0.160 -0.331         1.599         1.742
           1.00   2.012   -0.239    (-2.340) 0.426 -0.147 -0.338         1.827         1.513
           0.96   2.033   -0.272    (-2.118) 0.443 -0.120 -0.322         1.753         1.587
           0.92   2.171   -0.334    (-2.125) 0.448 -0.105 -0.317         1.827         1.513

Table 2: Optimal Investment Strategies for Stock, Money Market Account and 1 Call
Option with Varying Strike and Time to Maturity in Case of Monthly Rebalancing

The table shows the optimal asset allocation of an investor with a 6 months investment horizon,
who can invest into the stock, the money market account and a call option with a time-to-
maturity of 1, 3, 6, and 12 months. The optimal allocation and the expected utility are found
by backward induction searching over 500,000 simulated stock price paths. Rdisc→cont,w is the
annualized excess return of the optimal continuous time strategy in a complete market over
the optimal discrete time strategy using the available options. Rdisc,w/o→w gives the annualized
excess return of the optimal discrete time strategy with derivatives over the return achievable
                                                                          B1    B2       N
with discrete trading and with stock and money market account only. θ0 , θ0 , and θ0 denote
the exposure to stock diffusion risk, volatility diffusion risk, and jump risk, respectively, at time
t = 0.




                                                    23
                                                         B1      B2       N
 St. 1 (φ )0     (ψ1 )0   (# calls) (ψ2 )0 (# calls) θ0         θ0       θ0 Rdisc→cont,w Rdisc,w/o→w
                                         time to maturity: 1 month
 1.01   1.310    0.032      (2.107) -0.063 (-3.036) 0.607 -0.063      -0.298       2.459       0.881
 0.97   1.493    -0.061    (-1.459) 0.007     (0.347) 0.541 -0.053    -0.322       2.312       1.028
 0.93   1.537    -0.069    (-0.896) -0.007 (-0.330) 0.517 -0.046      -0.312       2.249       1.092
 0.89   1.624    -0.093    (-0.807) -0.011 (-0.510) 0.557 -0.046      -0.308       2.275       1.065
 0.85   1.628    -0.107    (-0.692) -0.015 (-0.720) 0.553 -0.056      -0.295       2.381       0.959
 0.81   1.747    -0.139    (-0.718) -0.017 (-0.834) 0.582 -0.061      -0.298       2.443       0.897
                                        time to maturity: 3 months
 1.09   1.574    0.001      (0.097) -0.079 (-1.953) 0.444 -0.163      -0.331       1.572       1.769
 1.05   1.562    -0.011    (-0.671) -0.059 (-1.462) 0.486 -0.179      -0.335       1.735       1.605
 1.01   1.459    -0.180    (-5.177) 0.124     (3.068) 0.484 -0.189    -0.322       1.703       1.637
 0.97   1.591    0.041      (0.680) -0.112 (-2.767) 0.460 -0.187      -0.342       1.679       1.662
 0.93   1.711    0.003      (0.040) -0.089 (-2.196) 0.457 -0.187      -0.360       1.662       1.678
 0.89   1.402    0.075      (0.586) -0.100 (-2.478) 0.488 -0.191      -0.331       1.706       1.634
 0.85   1.374    0.089      (0.541) -0.099 (-2.432) 0.463 -0.195      -0.335       1.720       1.621
 0.81   1.453    0.074      (0.364) -0.093 (-2.285) 0.466 -0.190      -0.346       1.702       1.639
                                        time to maturity: 6 months
 1.09   1.596    -0.063    (-2.897) -0.026 (-0.404) 0.432 -0.244      -0.341       1.532       1.808
 1.05   1.553    -0.179    (-4.857) 0.113     (1.778) 0.462 -0.257    -0.347       1.465       1.876
 1.01   1.390    -1.575   (-27.274) 1.554 (24.366) 0.459 -0.274       -0.340       1.420       1.921
 0.97   1.516    -0.576     (6.882) -0.612 (-9.591) 0.429 -0.266      -0.358       1.391       1.950
 0.93   1.578    0.307      (2.701) -0.339 (-5.324) 0.426 -0.264      -0.374       1.354       1.987
 0.89   1.142    0.416      (2.838) -0.322 (-5.052) 0.453 -0.271      -0.353       1.381       1.959
 0.85   1.009    0.437      (2.406) -0.280 (-4.394) 0.454 -0.266      -0.358       1.377       1.963
 0.81   0.735    0.552      (2.533) -0.272 (-4.266) 0.442 -0.273      -0.365       1.370       1.970
                                          time to maturity: 1 year
 1.09   1.537    -0.414    (-7.566) 0.379     (3.703) 0.430 -0.273    -0.363       1.403       1.937
 1.05   1.316    -0.991   (-13.432) 1.024 (10.015) 0.436 -0.271       -0.353       1.409       1.932
 1.01   1.247    -6.401   (-66.586) 6.479 (63.382) 0.418 -0.280       -0.363       1.371       1.970
 0.97   1.092    2.696     (22.168) -2.553 (-24.981) 0.427 -0.273     -0.362       1.364       1.977
 0.93   0.829    1.589     (10.610) -1.337 (-13.077) 0.422 -0.276     -0.362       1.358       1.983
 0.89   0.749    1.217      (6.752) -0.912 (-8.921) 0.423 -0.264      -0.367       1.351       1.989
 0.85   0.598    1.117      (5.261) -0.730 (-7.138) 0.417 -0.256      -0.370       1.354       1.987
 0.81   -0.029   1.354      (5.510) -0.692 (-6.774) 0.414 -0.260      -0.366       1.369       1.972

Table 3: Optimal Investment Strategies for Stock, Money Market Account and 2 Call
Options with Varying Strike and Time-to-Maturity in Case of Monthly Rebalancing

The table shows the optimal asset allocation of an investor with a 6 months investment horizon,
who can invest into the stock, the money market account and two call options with the same
time-to-maturity (of 1, 3, 6, or 12 months) but different moneyness levels (defined as strike
price over spot price). The moneyness of the first call is given in Column 1, the second call is
at-the-money. The optimal allocation and the expected utility are found by backward induction
searching over 500,000 simulated stock price paths. Rdisc→cont,w is the annualized excess return
of the optimal continuous time strategy in a complete market over the optimal discrete time
strategy using the available options. Rdisc,w/o→w gives the annualized excess return of the optimal
discrete time strategy with derivatives over the return achievable with discrete trading and with
                                           B1   B2        N
stock and money market account only. θ0 , θ0 , and θ0 denote the exposure to stock diffusion
risk, volatility diffusion risk, and jump risk, respectively, at time t = 0.




                                                    24
                                                     B1       B2       N
     H. Time   Strike   (φ )0   (ψ )0 (# calls)     θ0       θ0       θ0 Rdisc→cont,w Rdisc,w/o→w
     3 mo.     1.20     1.350   -0.069 (-3.447)   0.536   -0.173   -0.296      2.071         1.270
     3 mo.     1.16     1.290   -0.075 (-2.499)   0.497   -0.146   -0.274      2.065         1.276
     3 mo.     1.12     1.434   -0.101 (-2.356)   0.474   -0.152   -0.291      1.999         1.341
     3 mo.     1.08     1.502   -0.122 (-2.060)   0.469   -0.138   -0.292      2.004         1.336
     3 mo.     1.04     1.546   -0.145 (-1.830)   0.460   -0.122   -0.287      2.037         1.308
     3 mo.     1.00     1.488   -0.152 (-1.490)   0.478   -0.093   -0.267      2.108         1.233
     2 mo.     1.20     1.486   -0.085 (-4.207)   0.491   -0.211   -0.320      1.838         1.502
     2 mo.     1.16     1.481   -0.097 (-3.238)   0.454   -0.189   -0.308      1.853         1.488
     2 mo.     1.12     1.544   -0.115 (-2.675)   0.454   -0.172   -0.309      1.828         1.513
     2 mo.     1.08     1.553   -0.130 (-2.197)   0.452   -0.148   -0.299      1.852         1.488
     2 mo.     1.04     1.589   -0.151 (-1.905)   0.458   -0.127   -0.293      1.882         1.458
     2 mo.     1.00     1.703   -0.185 (-1.812)   0.475   -0.114   -0.298      1.909         1.431
     1 mo.     1.20     1.624   -0.101 (-5.043)   0.432   -0.253   -0.344      1.536         1.805
     1 mo.     1.16     1.698   -0.118 (-3.954)   0.443   -0.231   -0.348      1.486         1.854
     1 mo.     1.12     1.609   -0.124 (-2.895)   0.430   -0.186   -0.319      1.506         1.834
     1 mo.     1.08     1.849   -0.169 (-2.860)   0.415   -0.192   -0.347      1.523         1.817
     1 mo.     1.04     1.849   -0.191 (-2.413)   0.417   -0.160   -0.331      1.599         1.742
     1 mo.     1.00     2.012   -0.239 (-2.340)   0.426   -0.147   -0.338      1.827         1.513
     2 we.     1.20     1.747   -0.114 (-5.684)   0.403   -0.285   -0.367      1.239         2.102
     2 we.     1.16     1.887   -0.138 (-4.620)   0.421   -0.270   -0.383      1.150         2.190
     2 we.     1.12     2.010   -0.169 (-3.952)   0.400   -0.254   -0.389      1.153         2.188
     2 we.     1.08     2.083   -0.200 (-3.382)   0.387   -0.227   -0.384      1.191         2.150
     2 we.     1.04     1.991   -0.210 (-2.662)   0.411   -0.177   -0.352      1.257         2.084
     2 we.     1.00     2.171   -0.266 (-2.599)   0.409   -0.163   -0.359      1.339         2.002

Table 4: Optimal Investment Strategies in Stock, Money Market Account and 1 Call
Option with Varying Strike and a Time-to-Maturity of 1 Year for Rebalancing Intervals
Between 3 Months and 2 Weeks

The table shows the optimal asset allocation of an investor with a 6 months investment horizon,
who can invest into the stock, the money market account and one call option. The optimal
allocation and the expected utility are found by backward induction searching over 500,000 sim-
ulated stock price paths. Rdisc→cont,w is the annualized excess return of the optimal continuous
time strategy in a complete market over the optimal discrete time strategy using the available
options. H. time is the time span during which the investor does not rebalance. Rdisc,w/o→w
gives the annualized excess return of the optimal discrete time strategy with derivatives over
the return achievable with discrete trading and with stock and money market account only. θ0 , B1
 B2 , and θ N denote the exposure to stock diffusion risk, volatility diffusion risk, and jump risk,
θ0         0
respectively, at time t = 0.




                                                   25
St. 1    (φ )0   (ψ1 )0     (ψ2 )0 θB1 (φ) θB2 (φ) θN (φ) θB1 (ψ1 ) θB2 (ψ1 ) θN (ψ1 ) θB1 (ψ2 ) θB2 (ψ2 ) θN (ψ2 )
                                             Continuous Rebalancing
1.09     -0.680 -2.042      2.867 -0.680       0.0 0.170 -17.827      -2.490    1.394 19.013        1.758 -1.982
1.05     -1.212 -4.646      5.684 -1.212       0.0 0.303 -35.972      -4.218    3.208 37.690        3.486 -3.929
1.01     -1.903 -30.019    31.336 -1.903       0.0 0.476 -205.391 -19.950 20.767 207.800 19.218 -21.661
0.97     -2.809 13.227    -11.536 -2.809       0.0 0.702   79.814      6.343 -9.095 -76.498        -7.075    7.974
0.93     -4.006   7.665    -5.074 -4.006       0.0 1.002   40.788      2.623 -5.202 -36.276        -3.355    3.782
0.89     -5.599   6.743    -3.860 -5.599       0.0 1.400   31.699      1.635 -4.486 -25.594        -2.367    2.668
0.85     -7.732   6.975    -3.144 -7.732       0.0 1.933   29.084      1.196 -4.525 -20.847        -1.928    2.173
0.81    -10.593   7.908    -2.760 -10.593      0.0 2.648   29.399      0.960 -4.974 -18.301        -1.693    1.908
                                               Monthly Rebalancing
1.09     1.369   -0.468     0.474   1.369      0.0 -0.342   -4.082    -0.571    0.319     3.143     0.291 -0.328
1.05     1.417   -0.950     0.960   1.417      0.0 -0.354   -7.352    -0.862    0.655     6.365     0.589 -0.663
1.01     1.138   -6.840     6.954   1.138      0.0 -0.285 -46.790     -4.547    4.728 46.103        4.266 -4.803
0.97     0.968    2.972    -2.787   0.968      0.0 -0.242  17.933      1.426 -2.043 -18.476        -1.709    1.925
0.93     0.719    1.753    -1.449   0.719      0.0 -0.180    9.327     0.600 -1.189      -9.609    -0.889    1.001
0.89     0.812    1.210    -0.918   0.812      0.0 -0.203    5.690     0.294 -0.805      -6.086    -0.563    0.634
0.85     0.723    1.054    -0.711   0.723      0.0 -0.181    4.395     0.181 -0.684      -4.711    -0.436    0.491
0.81     0.099    1.267    -0.662   0.099      0.0 -0.025    4.711     0.154 -0.797 -4. 391        -0.406    0.457

         Table 5: Time 0 Asset Position for Continuous and Monthly Rebalancing

The table shows the positions in the different assets and their contributions to the risk factor
positions at time 0 for continuous and monthly rebalancing. The time to maturity of the options
is 1 year, the holding period is 6 months. The strike of the first call option is given in Column
1, the second option is an ATM call.




                                                   26
                                                         B1      B2      N
  St. 1 (φ )0 (ψ1 )0      (# calls) (ψ2 )0 (# calls) θ0        θ0       θ0 Rdisc→cont,w Rdisc,w/o→w
                                     1 Option, time to maturity: 1 year
  1.20   1.424   -0.077    (-3.824)                   0.520 -0.192 -0.309        1.635         1.706
  1.16   1.550   -0.098    (-3.282)                   0.509 -0.192 -0.324        1.541         1.799
  1.12   1.654   -0.123    (-2.866)                   0.486 -0.185 -0.331        1.513         1.828
  1.08   1.733   -0.152    (-2.571)                   0.444 -0.173 -0.329        1.550         1.791
  1.04   1.679   -0.164    (-2.032)                   0.445 -0.138 -0.306        1.609         1.732
  1.00   1.907   -0.214    (-2.092)                   0.489 -0.131 -0.329        1.649         1.692
  0.96   1.634   -0.183    (-1.426)                   0.563 -0.081 -0.283        1.942         1.398
  0.92   1.607   -0.197    (-1.250)                   0.594 -0.062 -0.269        2.113         1.227
                                    2 Options, time to maturity: 1 year
  1.17   1.394   -0.072    (-3.961) 0.000    (0.000) 0.530 -0.189 -0.306         1.684         1.656
  1.17   1.155   -0.105    (-3.874) 0.077    (0.751) 0.517 -0.172 -0.275         1.608         1.733
  1.13   1.629   -0.116    (-2.960) 0.000    (0.000) 0.492 -0.187 -0.330         1.513         1.827
  1.09   1.488   -0.154    (-2.819) 0.050    (0.491) 0.474 -0.157 -0.301         1.563         1.777
  1.05   1.812   -0.174    (-2.354) 0.000    (0.000) 0.468 -0.158 -0.333         1.573         1.767
  1.01   1.880   -0.205    (-2.135) 0.000    (0.000) 0.476 -0.136 -0.328         1.642         1.699
  0.97   1.883   0.000      (0.000) -0.216 (-2.115) 0.450 -0.133 -0.322          1.680         1.660
  0.93   1.680   0.046      (0.309) -0.218 (-2.131) 0.482 -0.118 -0.301          1.688         1.653
  0.89   1.906   0.000      (0.000) -0.214 (-2.092) 0.489 -0.131 -0.329          1.646         1.695
  0.85   1.896   0.000      (0.000) -0.215 (-2.102) 0.471 -0.132 -0.326          1.653         1.688
  0.81   1.884   0.002      (0.008) -0.215 (-2.107) 0.463 -0.132 -0.323          1.672         1.669

Table 6: Optimal Investment Strategies for Stock, Money Market Account and 1 or 2 Call
Options with Varying Strike in Case of Monthly Rebalancing with Margin Requirements

The table shows the optimal asset allocation of an investor with a 6 months investment horizon,
who can invest into the stock, the money market account and an option with a time-to-maturity
of 12 months. The optimal allocation and the expected utility are found by backward induction
searching over 500,000 simulated stock price paths. Rdisc→cont,w is the annualized excess return of
the optimal continuous time strategy with options over the optimal discrete time strategy using
the available options. Rdisc,w/o→w gives the annualized excess return of the optimal discrete
time strategy with derivatives over the return achievable with discrete trading and with stock
                                    B1    B2       N
and money market account only. θ0 , θ0 , and θ0 denote the exposure to stock diffusion risk,
volatility diffusion risk, and jump risk, respectively, at time t = 0.




                                                   27
                                                                        B1      B2        N
St. 2 ttm       (φ )0    (ψ1 )0   (# contracts) (ψ2 )0    (# calls) θ0         θ0        θ0 Rdisc→cont,w Rdisc,w/o→w
                                                        1 Derivative
       3 mo 0.351        -0.025        (-2.708)                      0.434   -0.120 -0.402         1.501       1.840
       6 mo 0.324        -0.047        (-2.401)                      0.427   -0.149 -0.409         1.377       1.963
       1 ye 0.304        -0.088        (-2.139)                      0.410   -0.152 -0.389         1.362       1.979
                                                       2 Derivatives
1.08   3   mo   0.583    -0.011        (-1.308)   -0.022 (-2.452) 0.406      -0.189   -0.423       1.355       1.986
1.04   3   mo   0.792    -0.019        (-2.119)   -0.024 (-1.183) 0.409      -0.193   -0.426       1.308       2.032
1.00   3   mo   0.848    -0.013        (-1.403)   -0.031 (-0.766) 0.442      -0.128   -0.350       1.431       1.909
0.96   3   mo   0.426    -0.023        (-2.534)   -0.005 (-0.072) 0.449      -0.118   -0.396       1.475       1.865
0.92   3   mo   0.142    -0.028        (-3.046)   0.025    (-0.766) 0.449    -0.122   -0.409       1.483       1.858
1.16   6   mo   0.780    -0.025        (-1.259)   -0.022 (-3.251) 0.429      -0.216   -0.359       1.299       2.042
1.12   6   mo   0.748    -0.034        (-1.706)   -0.026 (-1.930) 0.394      -0.224   -0.406       1.227       2.113
1.08   6   mo   0.823    -0.030        (-1.505)   -0.036 (-1.429) 0.390      -0.202   -0.389       1.242       2.099
1.04   6   mo   0.832    -0.032        (-1.606)   -0.045 (-1.073) 0.375      -0.189   -0.397       1.250       2.090
1.00   6   mo   0.676    -0.036        (-1.823)   -0.036 (-0.568) 0.398      -0.159   -0.392       1.321       2.020
0.96   6   mo   0.299    -0.047        (-2.387)   0.001     (0.014) 0.412    -0.147   -0.402       1.357       1.983
0.92   6   mo   0.694    -0.030        (-1.548)   -0.045 (-0.366) 0.458      -0.117   -0.353       1.415       1.925
1.20   1   ye   0.823    -0.058        (-1.394)   -0.043 (-2.162) 0.381      -0.207   -0.383       1.248       2.092
1.16   1   ye   0.846    -0.062        (-1.500)   -0.050 (-1.678) 0.387      -0.205   -0.398       1.230       2.110
1.12   1   ye   0.863    -0.058        (-1.395)   -0.055 (-1.281) 0.410      -0.182   -0.383       1.258       2.082
1.08   1   ye   1.153    -0.034        (-0.827)   -0.093 (-1.569) 0.407      -0.164   -0.346       1.319       2.021
1.04   1   ye   0.513    -0.061        (-1.476)   -0.019 (-0.246) 0.440      -0.121   -0.331       1.355       1.986
1.00   1   ye   0.450    -0.087        (-2.095)   -0.022 (-0.216) 0.407      -0.162   -0.404       1.298       2.043
0.96   1   ye   0.212    -0.100        (-2.425)   0.013     (0.099) 0.406    -0.167   -0.416       1.308       2.033
0.92   1   ye   -0.093   -0.103        (-2.498)   0.079     (0.500) 0.435    -0.153   -0.395       1.335       2.006

Table 7: Optimal Investment Strategies for Stock, Money Market Account, 1 Variance
Contract and in Case of 2 Derivatives 1 Call Option for Monthly Rebalancing

The table shows the optimal asset allocation for an investor with a 6 months investment horizon,
who can invest into the stock, the money market account, a variance contract and a call (with
varying strike prices) which have the same time-to-maturity. The optimal allocation and the
expected utility are found by backward induction searching over 500,000 simulated stock price
paths. Rdisc→cont,w is the annualized excess return of the optimal continuous time strategy in a
complete market over the optimal discrete time strategy using the available options. Rdisc,w/o→w
gives the annualized excess return of the optimal discrete time strategy with derivatives over
the return achievable with discrete trading with stock and money market account only. θ0 ,     B1
 B2 , and θ N denote the exposure to stock diffusion risk, volatility diffusion risk, and jump risk,
θ0         0
respectively, at time t = 0.




                                                         28
  Hold. per. Opt. Str. (φ )0 (ψ1 )0 (# calls) (θB1 )0 (θB2 )0 (θN )0 Rdisc→cont,w Rdisc,w/o→w
                                No Transaction Costs
  1 mo       1.08       1.626 -0.084 (-3.384) 0.448 -0.258 -0.352          1.514         1.827
  2 mo.      1.12       1.544 -0.115 (-2.675) 0.454 -0.172 -0.309          1.828         1.513
  3 mo.      1.12       1.434 -0.101 (-2.356) 0.474 -0.152 -0.291          1.999         1.341
             Transaction Costs on Options: 0.5% one-way per Rebalancing
  1 mo       1.12       1.385 -0.049 (-3.606) 0.587 -0.220 -0.320          2.268         1.072
  2 mo       1.08       1.317 -0.053 (-2.146) 0.569 -0.164 -0.295          2.294         1.047
  3 mo       1.04       1.278 -0.061 (-1.471) 0.557 -0.123 -0.277          2.438         0.902
  6 mo       1.00       1.155 -0.056 (-0.872) 0.608 -0.070 -0.249          2.692         0.649
             Transaction Costs on Options: 1.0% one-way per Rebalancing
  1 mo       1.16       1.200 -0.028 (-4.184) 0.679 -0.177 -0.290          2.687         0.653
  2 mo       1.12       1.199 -0.034 (-2.529) 0.639 -0.154 -0.281          2.536         0.804
  3 mo       1.08       1.186 -0.041 (-1.639) 0.616 -0.125 -0.270          2.614         0.726
  6 mo       1.04       1.081 -0.037 (-0.882) 0.649 -0.073 -0.245          2.788         0.552
             Transaction Costs on Options: 2.0% one-way per Rebalancing
  2 mo       1.16       1.062 -0.019 (-2.797) 0.714 -0.119 -0.259          2.821         0.520
  3 mo       1.12       1.068 -0.023 (-1.730) 0.685 -0.106 -0.254          2.841         0.500
  6 mo       1.08       0.997 -0.021 (-0.856) 0.699 -0.065 -0.236          2.914         0.427
             Transaction Costs on Options: 3.0% one-way per Rebalancing
  3 mo       1.16       0.981 -0.013 (-2.016) 0.730 -0.085 -0.240          2.967         0.373
  6 mo       1.08       0.968 -0.018 (-0.710) 0.721 -0.054 -0.231          2.992         0.349

Table 8: Optimal Investment Strategies for Stock, Money Market Account and 1 Call Option
with Transaction Costs

The table shows the optimal investment strategies, the optimal strike used and the realized
utility gains for one-way transaction costs on the option position between 0.5 and 3 %.
Rebalancing intervals between 1 and 6 months are investigated. The options employed
have an initial time-to-maturity of 6 months. In every category only the results for the
option with the best strike are given. The optimal allocation and the expected utility
are found by backward induction searching over 500,000 simulated stock price paths.
Rdisc→cont,w is the annualized excess return of the optimal continuous time strategy in
a complete market over the optimal discrete time strategy using the available options.
Rdisc,w/o→w gives the annualized excess return of the optimal discrete time strategy with
derivatives over the return achievable with discrete trading and with stock and money
                       B1  B2       N
market account only. θ0 , θ0 , and θ0 denote the exposure to stock diffusion risk, volatility
diffusion risk, and jump risk, respectively, at time t = 0.




                                              29
                                                                B1      B2
        par.                    strike φ0    ψ0      (# calls) θ0      θ0        N
                                                                                θ0 Rtrue
                                                                                    w/o→w
        Stock and Money Market Account Only
        no –                           0.829 –                 0.83    0.0    -0.21      0.0
        η1   µ = µtrue                 0.827 –       –         0.83    0.0    -0.21      0.0
        η1   µ = µtrue + 0.5std        0.957 –       –         0.96    0.0    -0.24   -0.080
        η1   µ = µtrue + 1.0std        1.083 –       –         1.08    0.0    -0.27   -0.305
        With Options
        no                      108    1.52 -0.075   (-3.02)   0.47   -0.23   -0.33   1.786
        no                      104    1.57 -0.095   (-2.29)   0.45   -0.19   -0.33   1.762
        no                      100    1.61 -0.118   (-1.86)   0.45   -0.15   -0.32   1.689
        no                      96     1.66 -0.146   (-1.61)   0.47   -0.11   -0.31   1.585
        η1   µ = µtrue          108    1.51 -0.075   (-3.00)   0.47   -0.23   -0.33   1.785
        η1   µ = µtrue          104    1.56 -0.095   (-2.28)   0.45   -0.19   -0.32   1.762
        η1   µ = µtrue          100    1.61 -0.118   (-1.85)   0.45   -0.15   -0.32   1.688
        η1   µ = µtrue          96     1.66 -0.146   (-1.60)   0.47   -0.11   -0.31   1.584
        η2   µ = µtrue          108    1.52 -0.075   (-3.02)   0.47   -0.23   -0.33   1.786
        η2   µ = µtrue          104    1.57 -0.095   (-2.29)   0.45   -0.19   -0.33   1.763
        η2   µ = µtrue          100    1.61 -0.118   (-1.86)   0.45   -0.15   -0.32   1.690
        η2   µ = µtrue          96     1.66 -0.146   (-1.61)   0.47   -0.11   -0.31   1.587
        λP µ = µtrue            108    1.51 -0.075   (-3.01)   0.47   -0.23   -0.33   1.784
        λP µ = µtrue            104    1.57 -0.095   (-2.28)   0.45   -0.19   -0.33   1.762
        λP µ = µtrue            100    1.61 -0.118   (-1.85)   0.45   -0.15   -0.32   1.674
        λP µ = µtrue            96     1.66 -0.146   (-1.61)   0.47   -0.11   -0.31   1.584
        all µ = µtrue           108    1.51 -0.074   (-2.99)   0.47   -0.23   -0.33   1.785
        all µ = µtrue           104    1.56 -0.094   (-2.27)   0.45   -0.19   -0.32   1.760
        all µ = µtrue           100    1.61 -0.118   (-1.84)   0.45   -0.15   -0.32   1.678
        all µ = µtrue           96     1.65 -0.145   (-1.60)   0.47   -0.11   -0.31   1.583
        η1   µ = µtrue + 0.5std 108    1.53 -0.066   (-2.65)   0.60   -0.20   -0.34   1.698
        η1   µ = µtrue + 1.0std 108    1.54 -0.057   (-2.28)   0.74   -0.17   -0.35   1.447
        η1   µ = µtrue − 0.5std 108    1.50 -0.083   (-3.35)   0.33   -0.26   -0.32   1.717
        η1   µ = µtrue − 1.0std 108    1.48 -0.092   (-3.69)   0.20   -0.28   -0.31   1.497
        η2   µ = µtrue + 0.5std 108    1.49 -0.071   (-2.84)   0.50   -0.22   -0.33   1.770
        η2   µ = µtrue + 1.0std 108    1.46 -0.066   (-2.64)   0.54   -0.20   -0.32   1.735
        η2   µ = µtrue − 1.0std 108    1.56 -0.083   (-3.34)   0.39   -0.26   -0.34   1.772
        η2   µ = µtrue − 0.5std 108    1.54 -0.079   (-3.19)   0.43   -0.24   -0.33   1.786
        λP µ = µtrue + 0.5std 108      1.49 -0.073   (-2.95)   0.46   -0.22   -0.32   1.771
        λP µ = µtrue + 1.0std 104      1.52 -0.092   (-2.21)   0.44   -0.18   -0.32   1.745
        λP µ = µtrue + 2.0std 108      1.39 -0.068   (-2.72)   0.44   -0.21   -0.30   1.711
        λP µ = µtrue − 0.5std 108      1.53 -0.076   (-3.05)   0.47   -0.23   -0.33   1.786
        λP µ = µtrue − 1.0std 108      1.55 -0.077   (-3.09)   0.48   -0.24   -0.34   1.774
        λP µ = µtrue − 2.0std 108      1.58 -0.078   (-3.14)   0.49   -0.24   -0.34   1.567

Table 9: Influence of Parameter Mis-Measurement on Optimal Allocations and Utility
Gains

This table shows how optimal positions are influenced if the investor employs a subjective
distribution of the risk premium, which may be centered at the correct value or a half or one
standard deviation above or below. For comparison in the first lines the respective results are
given for the case where the investor can only invest in stock and money market account.




                                             30
                                                         B1 Exposure



                        0.6


                        0.4


                        0.2


                          0
                              0.75                                                           5
                                          0.5                                          4
                                                  0.25                     3
                                    Volatility                       2 Remaining Periods
                                                           0    1

                                                         B2 Exposure

                              0

                        −0.1

                        −0.2

                        −0.3

                        −0.4

                        −0.5
                                  0.75                                                       5
                                           0.5                                         4
                                                  0.25                         3
                                   Volatility                        2
                                                            0   1        Remaining Periods

                                                    Jump Exposure

                              0

                        −0.1

                        −0.2

                        −0.3

                        −0.4


                                  0.75                                                       5
                                           0.5                                         4
                                                  0.25                         3
                                     Volatility                      2
                                                            0   1        Remaining Periods




       Figure 1: Factor Exposures as Share of Wealth under Monthly Rebalancing
The figures shows the optimal exposures to the B1, B2, and jump risk conditional on the number
of remaining time periods until the end of the investment horizon and the local volatility of stock
returns. The options used in the calculations are an ATM call and an ITM call with a strike of
89% of the stock price both with a time-to-maturity of one year (the overall best performing
choice). The length of the investment horizon is 6 months.



                                                                31
                                        Time to Maturity: 3 Months
                         1.8
                                                                                                            mean
                                                                                                            std. dev.
                                                                                                            skewness
                         1.6                                                                                kurtosis
                                                                                                            minimum
                                                                                                            R


                         1.4




                         1.2




                          1




                         0.8




                         0.6




                         0.4
                           0.9   0.92     0.94        0.96       0.98    1     1.02    1.04   1.06          1.08        1.1


                                        Time to Maturity: 6 Months
                         1.6
                                                                                mean
                                                                                std. dev.
                                                                                skewness
                                                                                kurtosis
                         1.4                                                    minimum
                                                                                R


                         1.2




                          1




                         0.8




                         0.6




                         0.4




                         0.2
                           0.9   0.92     0.94        0.96       0.98    1     1.02    1.04   1.06          1.08        1.1


                                         Time to Maturity: 1 Year
                         1.5
                                          mean
                                          std. dev.
                         1.4              skewness
                                          kurtosis
                                          minimum
                         1.3              R


                         1.2


                         1.1


                          1


                         0.9


                         0.8


                         0.7


                         0.6


                         0.5
                           0.9          0.95                 1          1.05          1.1            1.15




               Figure 2: Properties of the Distribution of Terminal Wealth
The figure shows standardized measures of distributional properties of terminal wealth for differ-
ent strikes in the one option case. Every variable has been divided by its mean in the respective
maturity category to be able to plot the variables conveniently in one graph. From the mean,
first the terminal wealth that would have been achieved by an investment in the risk-free asset
was deducted in order to make the maximum better visible.



                                                                        32
                                                        B1 Exposure


                       0.6


                       0.4


                       0.2


                         0

                                 0.75                                                         5
                                            0.5                                        4
                                                                               3
                                                     0.25            2
                                 Volatility
                                                            0   1        Remaining Periods

                                                        B2 Exposure

                             0

                       −0.1

                       −0.2

                       −0.3

                       −0.4

                       −0.5

                                  0.75                                                        5
                                              0.5                                      4
                                                                                3
                                                     0.25            2
                                        Volatility          0   1        Remaining Periods

                                                      Jump Exposure

                             0

                       −0.1

                       −0.2

                       −0.3

                       −0.4


                                  0.75                                                        5
                                              0.5                                      4
                                                                                3
                                     Volatility      0.25            2
                                                            0   1         Remaining Periods




Figure 3: Factor Exposures as Share of Wealth under Monthly Rebalancing for Variance
Contract and OTM Call
The figure shows the optimal exposures to the B1, B2, and jump risk conditional on the number
of remaining time periods until the end of the investment horizon and the local volatility of
stock returns. The derivatives used in the calculations are an OTM call with a strike of 116%
of the stock price and a variance contract both with a time-to-maturity of 1 year. The length of
the investment horizon is 6 months.

                                                                33
                                                                                           Price
                                                               0.4



                                                              0.35



                                                               0.3



                                                              0.25




                               Price
                                                               0.2



                                                              0.15



                                                               0.1



                                                              0.05




                                                                         0.1   0.2   0.3    0.4            0.5   0.6   0.7   0.8
                                                                                                  Volatility




                                                                                           Vega

                                                              0.5




                                                              0.4
                               Vega




                                                              0.3




                                                              0.2




                                                              0.1




                                                               0
                                                                        0.1    0.2   0.3    0.4            0.5   0.6   0.7   0.8
                                                                                                  Volatility




                            Price Change in Case of Jump
                                                               0.2


                                                              0.18


                                                              0.16


                                                              0.14
                               Price Change in Case of Jump




                                                              0.12


                                                               0.1


                                                              0.08


                                                              0.06


                                                              0.04


                                                              0.02


                                                                    0
                                                                         0.1   0.2   0.3    0.4            0.5   0.6   0.7   0.8
                                                                                                  Volatility




       Figure 4: Sensitivities of Variance Contracts for Different Volatility Levels
The figure shows price, vega and and the price change in case of a jump for variance contracts
with a time to maturity of 1 month (dotted line), 3 months (dashed line), 6 months (dashed-
dotted line) and 1 year (solid line) for different volatility levels.




                                                                                             34

				
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