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coupling-fx

VIEWS: 8 PAGES: 25

									                               Coupling Smiles
         Valdo Durrleman                Nicole El Karoui
                                             e               e
    Department of Mathematics Centre de Math´matiques Appliqu´es
        Stanford University           Ecole Polytechnique
     Stanford, CA 94305, USA        91128 Palaiseau, France
                                   March 15, 2007


                                        Abstract
          The present paper addresses the problem of computing implied volatilities
      of options written on a domestic asset based on implied volatilities of options
      on the same asset expressed in a foreign currency and the exchange rate. It
      proposes an original method together with explicit formulas to compute the
      at-the-money implied volatility, the smile’s skew, convexity, and term structure
      for short maturities. The method is completely free of any model specification
      or Markov assumption; it only assumes that jumps are not present. We also
      investigate how the method performs on the particular example of the currency
      triplet dollar, euro, yen. We find a very satisfactory agreement between our
      formulas and the market at one week and one month maturities.

Acknowledgements: The authors thank Andres Villaquiran for helpful discussions.


1     Introduction
Consider the problem of computing the implied volatility of an option on a domestic
asset based on implied volatilities of options on the same asset expressed in a foreign
currency and the exchange rate. It is well known that the information contained in the
implied volatility smile of options on a given underlying for a given maturity is exactly
the risk-neutral distribution of the random variable modelling the underlying asset’s
value at maturity (see [1]). In the present problem, we have information about the
risk-neutral distributions of the foreign asset and the exchange rate at different future
dates and we are interested in the law of the product of these two random variables.
This law is however far from being determined by the two marginal distributions.
    At this point, it is tempting to cast the problem into a copula problem, i.e.,
that of finding the risk-neutral copula of the two underlying assets. The space of
copulas is unfortunately very large and the option market gives comparatively little
information about the assets’ joint distribution. Statistical methods based on past
joint observations could also be used. These may be manageable with two assets but

                                             1
they quickly become untractable with a large number of variables. Once a copula
function has been chosen, it remains to compute the law of the product of the two
random variables. As attractive as it might look, the copula problem focuses only on
distributional properties and therefore completely forgets about the dynamics of the
underlying variables. This is rather unsatisfactory since modern finance is precisely
built on dynamic self-financing strategies.
    In the present paper, we propose a method to compute implied volatilities of op-
tions on a domestic asset based on implied volatilities of options on the same asset
expressed in a foreign currency and the exchange rate. The joint law is determined
by instantaneous correlations. These can be estimated from historical data and rein-
troduce a dynamical picture.
    The paper’s idea is based on results of [2] (see sections 2.1 and 2.2 of that paper)
that relate the spot volatility process to the shape of the implied volatility smile
around the money and for short maturities. These results provide a simple way to go
back and forth between the dynamics of the spot (spot volatility) and its distributional
properties (implied volatilities). The key observation is that the spot volatility of the
product of two underlying assets is simply the sum their volatilities. To go from
the individual implied volatilities to the implied volatility of the product, we use that
simple relation at the spot volatility level. Once granted the results of [2], the problem
is reduced to computing quadratic variations. The method is therefore very general
and flexible. We intend to present further applications in later papers.
    The paper is organized as follows. First, we present the relations between spot and
implied volatilities. We put emphasis on their financial interpretations leaving the
formal proofs to [2]. The second part provides the formulas to compute the implied
volatility smile of quanto options. Finally, we look at the cross-volatilities problem
and see how our solution performs on the currency triplet dollar, euro, yen.


2     Relating implied and spot volatilities
Let us imagine an investor who can continuously trade in a stock S and in European
options written on it. The implied volatility at time t of the European option with
strike K and maturity T is denoted by Σt (T, K). Because the investor can diversify
the risks in both the stock and in its possibly random volatility vector σ by trading
among these instruments, the investor is risk-neutral.
    The dynamics of the stock under the pricing measure is
                                     dSt
                                         = σ t dW t
                                      St
where W t is a multidimensional Brownian motion. The spot volatility is σt = |σ t |.
We have assumed, for simplicity, that interest rates are zero.




                                            2
2.1    The four fundamental contracts and their relation to the
       smile
P&L of a delta hedged option. Let us first recall a well-known result about the
profit or loss of a delta hedged option. Consider the portfolio consisting of being long
the option and short its Black-Scholes delta hedge computed with the implied volatil-
ity Σt (T, K) as input. Over a small time interval δt before maturity, the position’s
profit or loss is
                               1 2 2
                                 ΓS σ − Σ(T, K)2 δt.
                               2
Γ is the Black-Scholes gamma computed at Σ(T, K). (For a derivation of this result,
see [3].)

Straddle With that in mind, let us look at the particular case where the option
is an at-the-money straddle (i.e, long an at-the-money call and an at-the-money put
with same maturities.) The payoff and the Black-Scholes gamma are plotted against
S in the top left corner of Figure 1.




                    Figure 1: Payoffs and Black-Scholes gammas.


    It is easily seen from the Black-Scholes formula that the price of the at-the-money
straddle is approximately linear in the implied volatility

                                  Straddle ∝ Σ(T, S).

On the other hand, in view of the large positive gamma around the money, a delta
hedged straddle will lead to a profit or loss that is proportional to the difference of the


                                           3
realized variance σ 2 and the square of the implied Black-Scholes volatility Σ(T, K)2 .
(“The investor is long volatility.”) We therefore expect Σ(T, St ) and σ to be closely
related for short maturities. The correct relationship actually is

(1a)                            ΣAT M := lim Σt (T, St ) = σt .
                                 t
                                           T ↓t


ΣAT M is the limit of at-the-money implied volatilities when the time to maturity
  t
shrinks to zero.

Risk-reversal Next consider the case where the option is a risk-reversal. A risk-
reversal is a combination of a short position in an out-of-the-money put with strike
K1 and a long position in an out-of-the-money call with strike K2 > K1 . See the top
right corner of Figure 1 for the payoff and gamma of a risk-reversal.
    When strikes are not too far from the money, the price of a risk-reversal is ap-
proximately proportional to the difference between the two implied volatilities

                         Risk-Reversal ∝ Σ(T, K2 ) − Σ(T, K1 ),

or, equivalently, to the skew.
    On the other hand, in view of the positive gamma at large strikes and negative
gamma at low strikes, its value is in direct relation with the correlation between moves
of S and σ. Indeed, in the region of positive gamma, the holder of a delta-hedged
risk-reversal expects the volatility σ to be as high as possible. In other words, he
expects a positive correlation between moves in S and σ. Likewise, in the region of
negative convexity, a positive correlation will be in the favor of the investor, since a
low stock price will come with a low volatility. We therefore expect a relationship
between skewness and spot/volatility correlation. The correct relationship actually is

                                     ∂Σt            1 1            dS
(1b)                  St := lim St       (T, St ) = 2        dσ;            .
                             T ↓t    ∂K            2σt dt          S    t

St is the skew for short maturities, i.e., the short maturity limit of the derivative with
respect to moneyness K/St .
    To continue the discussion, we fix some notation for the stochastic evolution of σ
under the pricing measure:
                                  2                 2
                                dσt = 2σt δt dt + 2σt ν t dW t .

We refer to δt as the drift and to νt = |ν t | as the volatility of volatility (V-vol). ν t is
the coefficient in front of dW t in dσt /σt , it therefore has the dimension of a volatility.
With this notation, (1b) rewrites
                                                  ν tσt
                                          St =          .
                                                   2σt




                                                  4
Butterfly spread Consider now a butterfly spread, where the investor is long an
out-of-the-money call, an out-of-the-money put, and short an at-the-money straddle.
See the bottom left corner of Figure 1 for the payoff and gamma of a butterfly spread.
   When strikes are not too far from the money, the price of a risk-reversal is ap-
proximately proportional to the convexity of the smile

                 Butterfly Spread ∝ Σ(T, K1 ) − 2Σ(T, S) + Σ(T, K2 )

    As before, we relate the price to the profit or loss of the trade. First, if S moves up
(say, to 110, in case of the figure) a long position in the butterfly spread will mostly
look like a long position in a risk-reversal (similar gamma profiles). Also, if S moves
down, a long position in the butterfly spread will mostly look like a short position in
a risk-reversal. The price of the butterfly spread should therefore be proportional to
the correlation between changes in S and changes in the price of a risk-reversal, or,
as we just saw, changes in skewness S.
    Second, assume that the spot/volatility correlation is zero. Suppose that the V-
vol ν is large. This implies that σ varies a lot, either up or down. If σ turns out to
be large as well, then the spot experiences a large volatility and moves up and down,
but in each case the spot will be in a region of positive gamma. Since σ is large, the
trade will result in a profit. Similarly, in case σ turns out to be small, S will not
be volatile and will stay around its current value, which is in the region of negative
gamma. The trade again results in a profit because volatility is small. The price of
the butterfly spread should therefore be proportional to ν.
    Third, let us look at what might happen if the spot/volatility correlation is not
zero. Whatever the sign, this will lower the profit of the investor in a butterfly spread.
Indeed, it may lead to low volatility in regions of positive gamma and high volatility
in the region of negative gamma. The price of the butterfly spread should therefore
be proportional to − |S| or −S 2 . The correct relationship actually is
                           ∂ 2 Σt            2 1           dS           νt2  2S 2
(1c)       Ct := lim St2          (T, St ) = 2       dS;            +       − t − St .
                 T ↓t      ∂K 2             3σt dt          S   t       3σt   σt
Ct is the smile’s convexity for short maturities, i.e., the limit of the second derivative
with respect to moneyness as the time to maturity shrinks to 0. The reason for the
last term on the right-hand side (−St ) comes from the fact that we differentiate with
respect to moneyness and not log-moneyness.

Calendar spread Let us finally study the term structure of the implied volatility
surface. To this end, we consider a calendar spread, where the investor is short
an at-the-money straddle with maturity T1 and long another at-the-money straddle
with longer maturity T2 > T1 . The price of the calendar spread is approximately
proportional to the difference between the two implied volatilities

                           Calendar Spread ∝ Σ(T2 , S) − Σ(T1 , S)

   Because of the positive gamma between T1 and T2 , the profit or loss of the calendar
spread is proportional to a relative increase of volatility σ. This longer term effect is

                                                5
related to the drift and we expect the price of the calendar spread to be proportional
to δ. On the other hand, this long term effect will be less pronounced in case the
V-vol is high because the random fluctuations will be more important. We therefore
expect the price of the calendar spread to be proportional to −νt . Also, a non zero
spot/volatility correlation will dampen the long term effect by driving the spot far
away from its current value to regions of positive but small gamma as volatility
changes. The correct relationship actually is
                                     ∂Σt            1
(1d)                  Mt := lim          (T, St ) =         2
                                                      δt − σt Ct − 3σt St2 .
                              T ↓t   ∂T             2
Mt is the smile’s term structure for short maturities, i.e., the limit of the derivative
with respect to maturity T as the time to maturity shrinks to 0.


    Equations (1a)–(1d) are consequence of no arbitrage restrictions among the spot
and European call or put options. They are rigorously derived in [2]. The statements
are Theorems 1, 2, and 3 in [2]. They are presented at the beginning of that paper in
sections 2.1 and 2.2. To prove these formulas, we worked under a set of assumptions,
which essentially insure that the implied volatility surface is smooth and behaves
nicely for short maturities. In particular, they insure that the limits of derivatives
showing in the above equations do exist in some precise sense. These conditions do
not depend on any particular model but hold, for instance, in the case of stochastic
volatility models with analytic coefficients. Precise statements and assumptions can
be found in [2] Theorem 7. Let us just stress that the market information is modelled
by the filtration generated by a finite dimensional Wiener process and therefore no
jumps are allowed in that framework.

2.2     Implied volatilities approximation
We can use formulas (1a)–(1d) to compute an approximation of the implied volatility
smile in a given model. By model, we mean a particular stochastic evolution for σt .
   Indeed, first parameterize the stochastic differential equation by δt and ν t ,
                                   2                 2
                                 dσt = 2σt δt dt + 2σt ν t dW t .

Then, compute ΣAT M , St , Ct , and Mt using (1a)–(1d). We get the following approx-
                  t
imation for the implied volatility smile, valid for t < T :
                                                                             2
                                           K − St               K − St           Ct
(2)    Σt (T, K) =   ΣAT M
                      t      + (T − t)Mt +        St +                              + error terms.
                                             St                   St             2
                                                                        2
                                                                 K−St
This approximation will be best if both |T − t| and               St
                                                                            are small compared to
                                                                2
                                                         K−St
the typical scales of the problem. That means             St
                                                                    1. The time scale is given
                                                         2
by the inverse of the volatility so |T − t| small means σt |T − t|                 1.

                                                 6
An example: Heston model Heston’s model reads
                              2          2
                            dσt = κ µ − σt dt + εσt dWt

where W is a Brownian motion with correlation ρ with that driving the stock. This
model is easily rewritten using a two dimensional Wiener process W t and (1a)–(1d)
lead to the following:
                           ερ
                   St =
                           4σt
                            ε2              ερ
                    Ct   =     3
                                  2 − 5ρ2 −
                           24σt             4σt
                            κ         2    ε2               ερσt
                  Mt     =       µ − σt −     3
                                                2 − ρ2 /2 +      .
                           4σt            48σt               8
These formulas give a lot of insight about the roles of the model parameters.

2.3    From implied to spot volatilities
The nice feature of (1a)–(1d) is that they can be inverted to give information about the
underlying process when we observe the smile dynamics. We saw that the dynamics
of the smile is well summarized by the dynamics of the short-dated and at-the-money
implied volatility ΣAT M , the skewness S, the convexity C, and the term structure M.
Given that we have an idea about their dynamics, here is what we can say about the
underlying process S and its volatility σ.
    The instantaneous volatility is precisely the implied volatility of the at-the-money
and short-dated option

(3a)                                  σt = ΣAT M .
                                            t

    The stochastic differential equation driving σt can be expressed using these deriv-
atives:

(3b)                2              2                     2
                  dσt = 2σt 2Mt + σt Ct + 3σt St2 dt + 2σt ν t dW t ,

where ν t is the V-vol and, as usual, νt = |ν t |.
     The inner product between σ t and ν t is given, up to a proportionality factor, by
St :

(3c)                                 σ t ν t = 2σt St .

   Finally, νt is related to the joint quadratic variation between the smile’s slope and
the return on S:
                                                    2 1           dS
(3d)                νt2 = 3σt (Ct + St ) + 6St2 −           dS;            .
                                                    σt dt          S   t

  (3a), (3c), and (3d) are enough to characterize ν t and σ t up to rotations. Since
Wiener measure is invariant under them, the process (S, σ) is completely specified.

                                             7
Example To illustrate how this methodology could be put to use, we look at data on
options on the euro-yen exchange rate. The data we used is explained in section 4.1.
Here, we simply compute S EU RJP Y and C EU RJP Y and plot them against ΣAT M,EU RJP Y
in Figure 2. It shows that, while S EU RJP Y should be modelled as a stochastic process
on its own, C EU RJP Y can reasonably be modelled as a simple power function of the
exchange rate’s volatility.




Figure 2: S EU RJP Y and C EU RJP Y vs. ΣAT M,EU RJP Y from 06-Oct-2003 to 30-Sep-2005




2.4    Martingale representation and hedging
The previous asymptotic result on implied volatilities is tightly linked to an asymp-
totic result on the martingale representation of option prices. We recall it here. It
can be skipped in a first reading since we will only use it in section 3.4.
    Let us consider a call or a put option with strike K and maturity T . Its time-t


                                          8
price, Ot (T, K), is
                               Ot (T, K) = E (ST − K)± Ft
(Recall that interest rates are zero.) Under the pricing measure Ot (T, K) is a mar-
tingale and we would like to get an expression for its martingale representation. Let

                            dΣt (T, K) = (· · · )dt + αt (T, K)dW t .

                    o
Then, by applying Itˆ’s formula, we get

               dOt (T, K) = [∆t (T, K)St σ t + Vegat (T, K)αt (T, K)] dW t

where ∆t and Vegat are respectively the option’s Black delta and vega evaluated at
implied volatility Σt (T, K). In view of equation (10) in [2], α = Σξ. We decompose
it into two parts, a first part along dSt and a part that is orthogonal to it:
                                                       αt σ t
                                        αt = β t +             σt.
                                                       |σ t |2

By construction,
                                         β t (T, K)σ t = 0.
Now,
                                              1                       dS
                            αt (T, K)σ t =             dΣ(T, K);               .
                                              dt                      S    t
Therefore, we can rewrite the martingale representation as follows:

                                  Vegat (T, K) 1                        dS
  dOt (T, K) =     ∆t (T, K) +         2
                                                          dΣ(T, K);             dSt
                                     σt St    dt                         S t
                                                                      + Vegat (T, K)β t (T, K)dW t .

Moreover,
                                                                                        2
                        2   d              1                1                  dS
             |β t (T, K)| =    Σ(T, K) t − 2                         dΣ(T, K);              .
                            dt            σt                dt                  S   t

We would like to use this martingale representation to hedge the option with a position
in S and other options. The adjusted delta hedge is well identified in the first term
in the martingale representation. The orthogonal part is a pure volatility risk that
can be hedged with a position in other options. In the case of at-the-money and
short-dated options we further know from Theorem 2 in [2] that

(4)                             lim β t (T, St ) = σt ν t − 2St σ t .
                                 T ↓t




                                                   9
3     Quanto options
This section is devoted to quanto options, i.e., equity linked forex options. These are
discussed in details in [5] in the constant volatility case.
    We are going to apply the results of the previous section to the coupling of the
smile of a stock in the foreign market and the smile of the exchange rate to get the
smile of options on the stock in the domestic market.
    We now assume deterministic (but non necessarily zero) interest rates. This is a
minor assumption; in the short maturity regime, the effect of interest rates can be
altogether neglected. To use the results presented in section 2, we simply have to
replace S by the forward price.

3.1    Notations
Consider a foreign economy where the forward contract for delivery at time T of some
underlying asset trades at Ftf (T ) at time t. Its volatility is denoted by σ f :
                                                                              t


                                  dFtf (T )
                                              = σ f dW f
                                                  t    t
                                   Ftf (T )

where W f is Brownian motion under the foreign pricing measure.
         t
   The type of quanto options we consider here are options on the foreign forward
contract with strike in the domestic currency. As usual, we introduce the forward
exchange rate from the foreign to the domestic currency for delivery at time T . We
denoted it by Xt (T ) and its volatility by σ X :
                                              t

                                  dXt (T )
                                           = σ X dW d
                                               t    t
                                  Xt (T )

where W d is Brownian motion under the domestic pricing measure. Standard ar-
          t
bitrage arguments show that Xt (T )Ftf (T ) is nothing else than the arbitrage price
of the forward contract on the foreign underlying asset for delivery at time T and
denominated in domestic currency, Ftd (T ), i.e.,

                                Ftd (T ) = Xt (T )Ftf (T ).

Its volatility is denoted by σ d and
                               t

                                  dFtd (T )
                                            = σ d dW d .
                                                t    t
                                  Ftd (T )

The Brownian motions W d and W f are related by
                       t       t

                                                       t
(5)                             Wf = Wd −
                                 t    t                    σ X ds.
                                                             s
                                                   0




                                              10
and the Radon-Nikodym derivative is given on FT by
                                     T                          T
                                                       1                 2
                         exp             σ X dW d −
                                           s    t                    σ X ds .
                                                                       s
                                 0                     2    0

Then,
                                          σd = σf + σX ,
                                           t    t    t

and if let γt = σ X σ f be the instantaneous covariance between returns on X and F f ,
                  t   t
                                        d
we get the following expression for (σt )2 :
                                          f
(6)                              d                       X
                               (σt )2 = (σt )2 + 2γt + (σt )2 .

The quanto option can be thought of as an option in the domestic market on the
                                                      d
domestic forward contract F d . If we assume that σt is a deterministic function of
time, then, a call or a put quanto option is priced and hedged using Black’s formula.

3.2     Bringing smiles into the picture
We now wish to incorporate the smile effects on both the foreign market and the
foreign exchange market into the pricing and hedging of quanto options. In general,
  d
σt is not deterministic and we would like to compute the volatility parameter to be
plugged into Black’s formula.
                                                                                  f
    As in section 2.3, we introduce the stochastic differential equations driving σt and
  X
σt :
                f       f            f          f
            d(σt )2 = 2σt 2Mf + (σt )2 Ctf + 3σt (Stf )2 dt + 2(σt )2 ν f dW f
                               t
                                                                  f
                                                                        t    t

and
        d(σt )2 = 2σt 2MX + (σt )2 CtX + 3σt (StX )2 dt + 2(σt )2 ν X dW d .
           X        X
                        t
                              X            X                 X
                                                                    t    t
                      f      X
We sometimes write δt and δt for the terms in parenthesis in the drifts.
    We shall also need to specify the dynamics of the instantaneous correlation ρt =
           f X
σ f σ X /(σt σt ):
  t t
                                dρt = µρ dt + ω ρ dW d .
                                       t        t    t

It will be useful to compute the dynamics of the instantaneous covariance γt = σ f σ X :
                                                                                 t t

               f        X
              δt       δt  1                 2
  dγt = γt         +      − νf − νX              − ν f σX              f X
                                                                    + σt σt µρ + ω ρ (ν f + ν X )   dt
               f
              σt        X
                       σt  2 t    t                  t t                     t     t    t     t


                                                                + γt (ν f + ν X ) + σt σt ω ρ dW d .
                                                                        t     t
                                                                                     f X
                                                                                            t    t


The drift of γt will be denoted by µt and its diffusion term by ω t .




                                                  11
3.3       Formulas for Std , Ctd , and Md
                                        t
We now compute Std , Ctd , and Md in terms of the corresponding quantities for F f
                                    t
and X and various instantaneous covariances.
    Using Itˆ’s calculus, we get a stochastic differential equation for (σt )2 from (6)
            o                                                            d

(7)
                    f   f    f
d(σt )2 = 2 µt + σt δt − σt ν f σ X + σt δt dt+2 (σt )2 ν f + (σt )2 ν X + ω t dW d
    d
                                t t
                                          X X             f
                                                              t
                                                                    X
                                                                          t            t


Therefore, F d ’s V-vol is
                                  f                            f X
                                (σt )2 + γt f (σt )2 + γt X σt σt ρ
                                                X
(8)                      νd
                          t   =       d
                                           νt +    d
                                                         ν t + d 2 ωt .
                                  (σt )2         (σt )2       (σt )
From this, we can compute the three derivatives of the implied volatility surface.

Proposition 3.1. Std , Ctd , and Md are given by the following
                                  t

            f
          (σt )2 + γt                    (σ X )2 + γt                    σf σX
 Std =                  f
                      2σt Stf + ν f σ X + t d 3
                                  t t                 2σt StX + ν X σ f + t dt 3 ω ρ σ d
                                                        X
                                                                      t
                d
            2(σt )3                        2(σt )                 t
                                                                         2(σt ) t t

                                      f
          6(Std )2         (νtd )2 (σt )2 + γt
  Ctd
    =−       d
                        d
                    − St +      d
                                  +       d 5
                                                  f        f
                                                (σt )2 3σt (Ctf + Stf ) + 18(Stf )2 − (νtf )2
            σt              3σt      3(σt )
    f 1            dX        f 1    ν f σ X dF d
+2σt       dS f ;         + σt     d X ; d           + 2ν f σ X 5σt Stf + σt StX + ν f σ X
                                                          t t
                                                                   f        X
                                                                                       t t
      dt            X t        dt     σ       F    t
   (σ X )2 + γt                                                        X 1          dF f
 + t d 5              X      X
                   (σt )2 3σt (CtX + StX ) + 18(StX )2 − (νtX )2 + 2σt       dS X ; f
      3(σt )                                                             dt          F       t
          1        ν X σ f dF d                                                     (ν X σ f )(ν f σ X )
   X
 +σt           d          ; d         + 2ν X σ f 5σt StX + σt Stf + ν X σ f
                                                            f
                                           t   t
                                                   X
                                                                      t   t     +      t   t
                                                                                              d
                                                                                                 t t
          dt         σf    F      t                                                      3(σt )3
            γt                                          2   σf σX 1               dF d
      −      d
                    f
                  2σt Stf + ν f σ X − 2σt StX − ν X σ f + t dt 5
                              t t
                                        X
                                                  t   t               d(ω ρ σ d ); d
          3(σt )5                                           3(σt ) dt             F     t
                                         2γt   ρ d      f f      f X                   f
                                     +                                  X
                                              ω σ 2σt St + ν t σ t + 2σt StX + ν X σ t
                                       3(σt )5 t t
                                           d                                        t




                 d          d
               (σt )2 Ctd 3σt (Std )2   1
  Md = −
   t                     −                  f
                                      + d 2σt Mf + (σt )3 Ctf + 3(σt Stf )2 + 2σt MX
                                               t
                                                     f             f            X
                                                                                   t
                   2          2        2σt
                                                                                      f
                                                   +(σt )3 CtX + 3(σt StX )2 + µt − (σt )2 ν f σ X
                                                      X             X
                                                                                             t t


   These formulas are remarkable in the sense that they do not depend on a stochastic
volatility model.
   The equation giving Std has the following interpretation. The product of the
volatility and the smile’s slope in the domestic market is an average of the same
quantities in the foreign and foreign exchange markets plus correction terms involving

                                                    12
instantaneous covariances. These correction terms have an interesting role. Assume
that the X-smile and F f -smile have a U-shape with minimum at the money. This
means that StX = Stf = 0. This also means, in view of (3c), that each forward is
uncorrelated with its volatility. On the other hand, the same might not be true in
the domestic market if the forwards are correlated with each other’s volatility, or if
the instantaneous covariance between the forwards is correlated with the domestic
forward.
Proof. To compute Std , use (1b) to get

                                                ν dσd
                                                  t t
                                        Std =       d
                                                      ,
                                                 2σt

Then replace σ d = σ f + σ X and ν d from (8). Finally, use (1b) for Stf and StX . To
                 t     t    t        t
compute Ctd , start by using (1b) and (1c) to get

                           6St2        ν2  2 1                        dF
                  Ct = −        − St + t + 5              d(σ 3 S);
                            σt        3σt 3σt dt                      F    t

and then compute the quadratic covariation from the formula just found for Std .
                                            d
Finally, to compute Md , find the drift of (σt )2 from (7) and use (1d).
                     t


3.4    Hedging
In this section, we study how an option on F d can be hedged. As explained in section
2.4 we compute the adjusted delta, which gives the number of domestic contracts to
be held. We denote by Πd (T, K) the portfolio consisting in being short one option
and long the adjusted delta number of domestic contracts. The risk of this portfolio
is a pure volatility risk:

                               Πd (T, K)
                           d    t
                                   d
                                         = Vegad β d (T, K)dW d
                                               t              t
                                Bt (T )

with β d (T, K)σ d = 0. Bt (T ) denotes the price of the domestic zero coupon bond
                 t
                         d

with maturity T . Using (4), this remaining risk is approximated (for at-the-money
and short-dated option) by

                         Πd (T, K)
                          t
                     d       d
                                   ≈ Vegad σt ν d − 2Std σ d dW d .
                                         t
                                            d
                                                t          t    t
                          Bt (T )

Thanks to (8), it rewrites
                            f                           f X
   Πd (T, K)                              X
                          (σt )2 + γt f (σt )2 + γt X σt σt ρ
  d t d      ≈ Vegad
                   t            d
                                     νt +     d
                                                   νt +  d
                                                            ω t − 2Std σ d dW d .
                                                                         t    t
    Bt (T )                   σt            σt          σt




                                             13
Part of this risk can be approximately replicated with positions in delta hedged op-
tions in the foreign market Πf (T f , K f ), delta hedged options in the foreign exchange
market ΠX (T X , K X ), and in Ftf , Xt and Ftd :

        Πd (T, K)                                              f
                   σ f + ρt σ X Vegad Πf (T f , K f ) σt + ρt σt Vegad ΠX (T X , K X )
                                                       X
    d    t
            d
                  ≈ t d t           t
                                    f
                                      d t f          +    d          X
                                                                      t
                                                                        d t d X
         Bt (T )        σt      Vegat    Bt (T  f)       σt      Vegat     Bt (T )
                    f
                   σt + ρt σt f dFtf
                            X         X       f
                                     σt + ρt σt X dXt    d dFt
                                                              d  f X
                                                                σt σt ρ
    +    2Vegad               St f +            St    − St d +        ω dW d .
              t         d
                       σt       Ft       d
                                        σt         Xt      Ft    2σt t
                                                                    d      t



From the last formula, it becomes apparent that this scheme is actually a trade on
the volatility of the correlation between X and F f . Indeed, if this correlation is
deterministic, ω ρ = 0. If correlation is stochastic and ω ρ cannot be written as a
                 t                                             t
linear combination of σ f , σ X , ν f , and ν X , part of the correlation’s volatility risk
                         t    t     t         t
cannot be hedged.


4         Application to cross-volatilities
As an illustration of Proposition 3.1, we look at the problem of cross-volatilities. Take
the three major currencies EUR, USD, and JPY, and their corresponding exchange
rates EURJPY, EURUSD, and USDJPY. They satisfy the no arbitrage relation

                             EURJPY = EURUSD × USDJPY.

We are going to reconstruct the implied volatility smile on exchange rate EURJPY
from the smiles on EURUSD and USDJPY. In this case, the smile on EURJPY is
also observable so that we can see how Proposition 3.1 performs in practice.

4.1        The data
The OTC market is particularly well suited for our purposes. Contracts have ma-
turities 1 week, 1, 2, 3, and 6 months, and 1 and 2 years. We will mostly focus on
contracts having one week or one month maturity since our results are valid for short
maturities. The available contracts are straddles, risk reversals, and butterfly spreads.
As it customary in these markets, the strikes are quoted in terms of the correspond-
ing delta. Straddles are at-the-money forward. Risk reversals and butterfly spreads
have strikes at ±25 or ±10 delta. (For contracts with a maturity of one week, only
±25 delta risk reversals and butterfly spreads are available.) The rule to compute
the implied volatilities given mid-market quotes for the at-the-money straddle (a),
25-delta risk reversal (r), and 25-delta butterfly spread (b) is reported in Figure 3.
    There are 520 data points from 06-Oct-2003 to 30-Sep-2005. Figure 4 displays
the observed volatility smiles against moneyness for the three exchange rates at a
particular date.




                                             14
Figure 3: Rule to compute implied volatilities given mid-market quotes for the at-
the-money straddle (a), 25-delta risk reversal (r), and 25-delta butterfly spread (b).


4.2    Results
We first compute the observed St and Ct by fitting a parabola through the mid-market
quotes, and Mt by linear interpolation between at-the-money implied volatilities.
We then use the formulas of Proposition 3.1 to reconstruct these quantities for the
EURJPY based on EURUSD and USDJPY. The various instantaneous correlations
are computed with a 30 trading day window. We did not estimate the correlation ρ
between returns on EURUSD and USDJPY, but instead used the implied correlation
based on (6) with the at-the-money implied volatilities. The drift of ρ is set to 0. It
appears in µ and only plays a role in Mt and we do not have a reasonable way of
estimating it without knowing the risk premia.
    We then use the same method to reconstruct EURUSD implied volatilities from the
EURJPY and USDJPY smiles, and USDJPY implied volatilities from the EURJPY
and EURUSD smiles. To apply Proposition 3.1 in these cases, we need to compute
the smile of 1/F based on that of F . We make use of the following formulas that are
consequence of the domestic call/foreign put symmetry:
                                        1/F
                                       St      = −StF
                               1/F      1/F
                              Ct     + St      = CtF + StF
                                        1/F
                                      Mt       = MF .
                                                  t

                                                                  1/F
They can also be recovered from (1a)–(1d) using the fact that σ t = −σ F .  t
   Figures 5, 6, 7, 8, 9, and 10 report results for the period from 01-Mar-2004 to
06-Oct-2005. In each graph, we plot two estimated quantities against the actually
observed values. We first compute St , Ct , and Mt based on the formulas of Proposition

                                              15
Figure 4: Implied volatilities for each exchange rate on September 23, 2005. A cross
‘+’ denotes a mid-market quote.


3.1, it is called the full estimator. The partial estimator is obtained by setting to 0
all instantaneous covariances.
    The overall agreement very good. It is better with one week maturity, as it was
expected, since we are using asymptotic formulas valid in the short maturity regime.
Using estimates of the instantaneous covariances does not seem to really improve the
quality of the estimators. Surprisingly, our formulas predict a somewhat more convex
EURUSD smile. A first explanation could be that option traders in the EURUSD
market may not trade options involving the Japanese Yen. In other words, markets
may be segmented.
    To complement these results, we focus on the period from 31-Dec-2003 to 24-
Mar-2004. The reconstruction formula does not work very well during this period as
seen in Figure 11 for the EURJPY at one week maturity. For instance, it predicts
negative values for the smile’s convexity. The problem probably comes from the
markets’ extreme volatility during the period as seen in Figure 12: we plotted the daily
returns and the one week at-the-money implied volatilities. Due to the suddenly low
at-the-money volatilities, the historical instantaneous covariance between σ U SDJP Y
and ρU SDJP Y /EU RU SD exhibits a very usual behavior. The bottom part of Figure 12
displays this instantaneous covariance against another one that is more stable.
    Let us conclude by a few words on jumps. The absence of jumps is the essential
assumption behind the formulas (1a)–(1d). It is somewhat surprising that we get such
a good fit with empirical data ignoring jumps. Indeed, practioners and academics alike
mostly agree on the presence of jumps in foreign exchange markets, although they
might not agree on their frequency.


                                          16
    What is quite surprising is that the data tell us that, most of the time, the market
is actually not pricing the jump risk in the underlying. To see that, we look at the
term structure effect in the ATM implied volatilities. Using a result in [4] p. 435, the
presence of jumps implies the following asymptotic behavior when T ↓ t:
                                      π            √
                 Σt (T, St ) ∼ σt +     λt E{|Jt |} T − t + Mt (T − t).
                                      2
λt is the intensity of jumps and Jt is the size of a jump in the asset’s return. What
is important here is that for short maturities, the dominant term when jumps are
                         √
present is the term in T − t and it is increasing. What we see by looking at the
maturity effect on Figures 5, 7, and 9 at one week maturity is that, about 50% of
the time, the ATM implied volatilities are decreasing at short maturities. When they
are decreasing, we are sure that the market is not pricing jumps. When they are
increasing, we cannot conclude because the the jump term might well be zero and
Mt be positive.


5    Conclusion
The present paper addresses the problem of computing implied volatilities of options
written on a domestic asset based on implied volatilities of options on the same asset
expressed in a foreign currency and the exchange rate.
    It proposes an original method together with explicit formulas to compute the
ATM implied volatility, the smile’s skew, convexity, term structure for short maturi-
ties. The method is completely free of any model specification or Markov assumption.
It only assumes that jumps are not present.
    We also investigate how the method performs on the particular example of the
currency triplet dollar, euro, yen. We find a very satisfactory agreement between our
formulas and the market at one week and one month maturities.


References
[1] D. Breeden and R. Litzenberger. Prices of state-contingent claims implicit in
    option prices. J. Bus., 51(4):621–651, 1978.
[2] V. Durrleman. From implied to spot volatilities. Technical report, 2005. Available
    at http://math.stanford.edu/∼valdo/research.html.
                                     e
[3] N. El Karoui, M. Jeanblanc-Picqu´, and S. Shreve. Robustness of the Black and
    Scholes formula. Math. Finance, 8(2):93–126, 1998.
[4] A. Medvedev and O. Scaillet. Approximation and calibration of short-term im-
    plied volatilities under jump-diffusion stochastic volatility. Rev. Financ. Stud.,
    20(2):427–459, 2007.
[5] E. Reiner. Quanto mechanics. RISK, 5(10):59–63, 1992.


                                           17
Figure 5: S EU RJP Y , C EU RJP Y , and MEU RJP Y at one week maturity reconstructed
with full formula (dashed line) and with partial formula (dotted line) against the
observed values (solid line).


                                        18
Figure 6: S EU RJP Y , C EU RJP Y , and MEU RJP Y at one month maturity reconstructed
with full formula (dashed line) and with partial formula (dotted line) against the
observed values (solid line).


                                         19
Figure 7: S U SDJP Y , C U SDJP Y , and MU SDJP Y at one week maturity reconstructed
with full formula (dashed line) and with partial formula (dotted line) against the
observed values (solid line).


                                        20
Figure 8: S U SDJP Y , C U SDJP Y , and MU SDJP Y at one month maturity reconstructed
with full formula (dashed line) and with partial formula (dotted line) against the
observed values (solid line).


                                         21
Figure 9: S EU RU SD , C EU RU SD , and MEU RU SD at one week maturity reconstructed
with full formula (dashed line) and with partial formula (dotted line) against the
observed values (solid line).


                                        22
Figure 10: S EU RU SD , C EU RU SD , and MEU RU SD at one month maturity reconstructed
with full formula (dashed line) and with partial formula (dotted line) against the
observed values (solid line).


                                         23
Figure 11: S EU RJP Y , C EU RJP Y , and MEU RJP Y at one week maturity reconstructed
with full formula (dashed line) and with partial formula (dotted line) against the
observed values (solid line).


                                         24
Figure 12: Daily returns (top), one week at-the-money implied volatilities (middle),
and instantaneous covariances between σ U SDJP Y and ρU SDJP Y /EU RU SD , and between
σ EU RU SD and ρU SDJP Y /EU RU SD (bottom).


                                         25

								
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