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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