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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 Bakshi, G., C. Cao, and Z. Chen, 1997, Empirical Performance of Alternative Option Pricing Models, Journal of Finance 52, 2003–2049. Bondarenko, O., 2004, Market Price of Variance Risk and Performance of Hedge Funds, Working Paper. Brandt, M., and P. Santa-Clara, 2006, Dynamic Portfolio Selection by Augmenting the Asset Space, Journal of Finance 59, 2187–2217. Branger, N., B. Breuer, and C. Schlag, 2006, Discrete-Time Implementation of Continuous-Time Portfolio Strategies, Working Paper. Branger, N., C. Schlag, and E. Schneider, 2006, Optimal Portfolios When Volatility can Jump, Working Paper. 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, Working Paper. 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 and Optimal Variance Swap Investments, Working Paper. Goetzmann, W., J. Ingersoll, M. Spiegel, and I. Welch, 2002, Sharpening Sharpe Ratios, Working Paper. Interactivebrokers.com, 2006, Description of Margin Requirements for Interactivebro- kers.com, http://www.interactivebrokers.com/en/trading/marginRequirements/fundamentals.php. 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, Journal of Finance 58, 231–259. Liu, J., and J. Pan, 2003, Dynamic Derivative Strategies, Journal of Financial Economics 69, 401–430. 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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