SSkewing Quanto with Simplicity
George Hong
Credit Suisse [email protected]
First Version: 8-Jul-2018This Version: 25-Mar-2019
Abstract
We present a simple and highly efficient analytical method for solving the QuantoSkew problem in Equities under a framework that accommodates both Equity and FXvolatility skew consistently. Ease of implementation and extremely fast performance ofthis new approach should benefit a wide spectrum of market participants.
Quanto derivatives have existed for almost as long as derivatives quants. They are particularlypopular in the Equities OTC and structured product markets where a significant portion ofthe trades globally are quantos. It has become such a common and fundamental productfeature nowadays that people often take the market participants’ ability to properly priceand risk manage quantos for granted. In the Black-Scholes model, an elegant covariance driftadjustment enables one to accurately value and compute risks of a quanto derivative withoutincreasing the dimensionality of the problem. This insight is applicable to valuing generalpayoffs in the absence of volatility skew and have therefore become not just an integral partof many production pricing libraries but also the basis for model intuition for quants andtraders alike: the hedging cost for the cross gamma due to the quanto feature is captured bythe covariance between the underlying and the FX assets in the drift.It is in fact common for practitioners to retain the simplified approach, even when volatilityskew is assumed elsewhere for equity, rates or other risk factors in a model (with the exceptionof the FX asset class where the vol skews for all the triangular pairs are too obvious to beignored). By keeping the drift term of the asset dynamics constant and devising some ad-hocrules on picking some volatility level, one could potentially model the volatility skew in the diffusion part only, while staying well-marked on the tradable quanto forward prices using animplied correlation as an fudge factor. Although simple and fast, this approach has significantdrawbacks, as explained in [9] and [13].The key reasons for the popularity of the ”ad-hoc” adjustment approach lie in its sim-plicity, the (perceived) universality, low implementation effort and zero incremental computecosts. Hence it would be strongly desirable to address its deficiencies while maintaining itsadvantages where we can. With those goals in mind, we apply the Stochastic Collocationmethod recently introduced to finance from the field of Uncertainty Quantification by [5, 6],1 a r X i v : . [ q -f i n . M F ] S e p here the authors developed novel techniques to speed up Monte Carlo sampling and removevol arbitrages.Our contribution lies in the formulation of a simple and highly efficient recipe for comput-ing quantities related to quanto valuations using the stochastic collocation techniques. Themethod can be used to quickly compute quanto option prices, calibrate to observed quantoforward for an implied correlation and determine the local drift function in the sense ofMarkovian projection as defined in [12]. The first two results allow a wide audience of marketparticipants to price European quanto options with minimal implementation effort and noreliance on time-consuming numerical methods. The last result, on the other hand, enablesmodellers to incrementally improve their quanto methodology without incurring additionalcosts to their risk management systems and quoting platforms.Recently several other authors have also proposed novel approaches to overcome the extracompute and speed requirement. For example, [13] and [8] have applied perturbation methodsto obtain so-called Proxy expansion formulae for European quanto option prices. This paperjoins the quest for an efficient methodology that bypass PDE or Monte Carlo methods. How-ever, our approach differs in several aspects with the key difference being that the perturbationapproach is ”local” in nature as the proxy expansion is centered around one point only (oftenthe ATM spot or the option strike). This means one could have two input volatility surfacesthat differ materially on the wings but agree well locally (around the current spot, say) interms of their low-order derivatives and the Taylor-like expansion formulae would producevery similar quanto option prices. On the other hand, the
Polynomial expansion approach weproposed below is ”global” in nature as it makes use of volatilities across a wide strike rangevia Lagrange interpolation. Finally, in contrast to other research employing methods such asFast Fourier Transform within the Affine diffusion framework, no specialised assumption ismade here on the underlying processes.
We follow the general convention and call the equity asset foreign and its base currency the
Foreign currency. The payoff currency, which the structure is ”quantoed into”, is called the
Domestic currency. The FX rate, X t , is the number of unit of Foreign currency per 1 unit ofDomestic currency. Consider the risk-neutral dynamics in the foreign measure, F : dSS = ( r F − δ ) dt + σ S ( S t , t ) dW F S dXX = ( r F − r D ) dt + σ X ( X t , t ) dW F X . where r, δ, σ ( · , · ) denote the deterministic risk-free rates, equity dividend yield and local volatil-ity functions, and E F [ dW F S dW F X ] = ρ dt . Note that the market reality remains that the onlyliquid instrument with which one can calibrate to and hedge the Equity-FX covariance iscurrently still limited to Quanto forwards or futures. This implies that the modelling of cor-relation skew is perhaps less pressing than say, in the FX world where the volatility surface ofthe ”Cross” also needs to be repriced as in [1]. Hence a model with constant correlation focus-ing on the volatility skews serves as a good step forward that provides a consistent frameworkfor valuing equity quantos. 2nder the Domestic measure, D , we have: dSS = ( r F − δ + ρ σ X ( X t , t ) σ S ( S t , t )) dt + σ S ( S t , t ) dW D S dXX = ( r F − r D + σ X ( X t , t ) ) dt + σ X ( X t , t ) dW D X . Consider valuing Quanto, vanilla equity and FX calls under Domestic and Foreign mea-sures: C Q ( K, T ) = B TD · E D (cid:104)(cid:0) S T − K (cid:1) + (cid:105) = B TF · E F (cid:104) X T X · (cid:0) S T − K (cid:1) + (cid:105) ,C S ( K, T ) = B TD · E D (cid:104) X X T ( S T − K ) + (cid:105) = B TF · E F (cid:104)(cid:0) S T − K (cid:1) + (cid:105) ,C X ( K, T ) = B TD · E D (cid:104) X X T ( X T − K ) + (cid:105) = B TF · E F (cid:104)(cid:0) X T − K (cid:1) + (cid:105) , with Domestic/Foreign discount factors as B TD := e ( − r D T ) ; B TF := e ( − r F T ) . For any general European payoff paying V T ( S T , X T ) at time T , its price in domesticcurrency is: V = B TD · E D (cid:104) V T (cid:105) = B TF · E F (cid:104) X T X · V T (cid:105) , d D d F = X T X B TF B TD = X T E F (cid:2) X T (cid:3) . We briefly recall the well-known technique of distribution mapping widely used on the street(a.k.a. quantile transform). Given two random variables, Y and Z , whose CDF, F Y ( y ) := P ( Y < y ) and F Z ( z ) := P ( Z < z ) are monotonic and right-continuous, there exist an inversefunction, called quantile function , in the following sense: F − Y ( q ) = inf { y | F Y ( y ) ≥ q, < q < } . If we define the random variable Y (cid:48) := g ( Z ) with the Quantile transform function: g ( z ) := F − Y ( F Z ( z )), then the fact that the CDF of a random variable is uniformly distributed impliesthat Y (cid:48) has the same distribution as Y .Now let F X T and F S T be the CDF’s of the FX and Equity underlyings at maturity T in theforeign measure, F . They are determined by the full Vanilla European call prices, C X ( K, T )and C S ( K, T ): F X T ( K ) = 1 + e ( r F T ) · ∂C X ( K,T ) ∂K F S T ( K ) = 1 + e ( r F T ) · ∂C S ( K,T ) ∂K (1)which can be equivalently re-expressed in terms of the calibrated implied volatility surface,ˆ σ X ( K, T ) and ˆ σ S ( K, T ). Care needs to be taken in the tail extrapolation of the implied volto prevent arbitrages and ensure accuracy and stability (see [3] for detailed discussions). Thisleads to the following distribution maps for the underlyings: g ( z ) = F − X T ( N ( z )) , g ( z ) = F − S T ( N ( z )) (2)where we map the F -distributions of X T and S T onto the standard Normal distribution.So working with a standard Normal Z will give us g ( Z ) that has F X T as the distributionfunction. 3 .3 Stochastic Collocation We briefly introduce the concept and refer the readers to [5] and the references therein fora general introduction to the field of Polynomial Chaos. Consider the Lagrange polynomialapproximation for a general function u ( · ): u ( x ) ≈ ˜ u ( x ) := N (cid:88) i =0 u ( x i ) L i ( x ) , L i ( x ) = N (cid:89) j =1 ,j (cid:54) = i x − x j x i − x j where { x i } are the nodes and L i ( x ) are the Lagrange interpolation polynomials satisfying theorthogonal property L i ( x j ) = δ ij . We re-expressed this as:˜ u ( x ) = a + a x + · · · + a N − x N − = a (cid:62) poly ( N − ( x ) , where we denote poly ( N − ( x ) = (1 , x, x , · · · , x N − ) (cid:62) . The coefficients a = ( a , a , a , · · · , a N − ) (cid:62) solves the linear system with the Vandermonde matrix, V, below: V a = u ( x ) (3) x x .. x N − x x .. x N − ... ... ... .. ...1 x N x N .. x N − N a a ... a N − = u ( x ) u ( x )... u ( x N ) Following [5] we let u ( · ) be the distribution mapping function (2) and the nodes { x i } bethe zeroes of the Gauss-Hermit polynomials (a natural choice given the mapping to Normaldistributions). Solving the two set of coefficients a and a gives us polynomial approximationsto the distribution mapping (e.g. to solve a small Vandermonde system see [11] for simpleroutines or simply do matrix inversion with double precision). By choosing the BivariateNormal as the driving random variables, Z , Z , correlated by ρ and the marginal distributionmappings, we now have a pair of transformed random variables with joint distribution in theGaussian Copula setting (where we may choose the polynomials to have different orders): X T d = g ( Z ) ≈ ˜ g ( Z ) := (cid:80) N − n =0 a ,n · Z n S T d = g ( Z ) ≈ ˜ g ( Z ) := (cid:80) N − n =0 a ,n · Z n As articulated in [2] on the resemblance between multi-underlying constant correlation LocalVolatility models and a model with Gaussian copula linking the marginal distributions, thisin turns leads us to the approximation of the T -distribution for the local vol processes of theEquity and FX assets by the method that follows.4 Quanto Skew Quantified
First note the following result regarding the conditional moments of a Gaussian on anothercorrelated Gaussian: E [ Z n | Z = z ] = E [ ¯ Z ( z ) n ] = (cid:98) n (cid:99) (cid:88) j =0 (cid:18) n j (cid:19) (2 j − (cid:112) − ρ ) (2 j ) ( ρz ) ( n − j ) =: n (cid:88) i =0 q i ( n ; ρ ) · z i = q ( n ; ρ ) (cid:62) poly n ( z ) (4)which comes from the conditioned variable again being Normally distributed: ¯ Z ∼ N ( ρz, (cid:112) − ρ )(and n !! is the double factorial function which multiplies all integers up to n with the sameparity). The conditional moments are polynomials with coefficients q i ( n ; ρ ) dependent on thecorrelation.Next we compute the conditional expectation of FX price given the Equity Gaussiandriver , Z : E F [ X T | Z ] ≈ E F (cid:104) ˜ g ( Z ) (cid:12)(cid:12)(cid:12) Z (cid:105) = N − (cid:88) n =0 a ,n · E F (cid:104) Z n (cid:12)(cid:12)(cid:12) Z (cid:105) = N − (cid:88) n =0 a ,n (cid:16) n (cid:88) j =0 q j Z j (cid:17) = N − (cid:88) n =0 b n Z n = b (cid:62) poly ( N − ( Z ) b n := N − (cid:88) j = n a ,j q n ( j ; ρ ) (5)The key is to approximate the conditional expectation of X as a polynomial where the newcoefficients b are scaled from the original coefficients a by a multiplier calculated from theconditional moments of the Bivariate Normal distribution, allowing an expression as a functionof the Equity price: E F [ X T | S T = S ] = E F [ X T | Z = g − ( S )] ≈ N − (cid:88) n =0 b n · (cid:0) g − ( S ) (cid:1) n When both X T and S T are Log-Normal, this conditional expectation is proportional to asimple power function of S where the power is the correlation times the ratio of the volatilities.Introducing Equity and FX skews complicates the function shape, adding more convexitiesand turns. This is illustrated here by a graph of the conditional expectation function using2017 market skew for Nikkei and USDJPY alongside the ”No Skew” case with Log-Normaldistribution given by ATM vols. We see that our successive polynomial approximation withincreasing order quickly converges to the exact function solved by a 2-d integration. We now value the Quantos. Given the potential oscillatory nature of polynomial approxi-mation impacting the accuracy of the analytical method, we use the non-quanto European5 .400.550.700.85
N=3N=4N=5
N=6N=8
N=10No Skew
Figure 1: Approximation of conditional expectation E F [ X T X | S T ] as a function of S T S call (of the same strike) as a control variate for ”noise cancelation”. To simplify notation,we work with undiscounted option values scaled by initial FX, ˆ C Q T,K := C Q T,K /B TF · X andˆ C S T,K := C S T,K /B TF · X and consider the equivalent strike in the Gaussian space by mapping K via the inverse of the distribution mapping g ( · ): κ := g − ( K ) . The idea is to re-express in terms of the driving Gaussians and the conditional moments(5): ˆ C Q T,K = E F (cid:104) X T (cid:0) S T − K (cid:1) + (cid:105) ≈ E F (cid:104) (cid:101) g ( Z ) · (cid:0) (cid:101) g ( Z ) − K (cid:1) { g ( Z ) >K } (cid:105) = E F (cid:104) E F [ (cid:101) g ( Z ) | Z ] · (cid:0) (cid:101) g ( Z ) − K (cid:1) · { g ( Z ) >K } (cid:105) = E F (cid:20) N − (cid:88) i =0 b i Z i · (cid:18) N − (cid:88) j =0 a ,j Z j − K (cid:19) · { Z >κ } (cid:21) = E F (cid:20)(cid:18) N + N − (cid:88) n =0 c n Z n − K · N − (cid:88) n =0 b n Z n (cid:19) · { Z >κ } (cid:21) (6) c n := n (cid:88) k =0 a ,k b n − k , ∀ n = 0 , · · · , N + N − c is the convolution of a and b which makes intuitive sense as it defines thepolynomial coefficients for X T × S T via conditioning. Let’s denote N − N + N − N .Now following [5] we make use the recursive relation for the moment formulae for truncatednormal distribution: m i ( κ ) := E [ Z i | Z > κ ] , Z ∼ N (0 , , = ( i − · m i − ( κ ) + κ i − φ ( κ )1 − Φ( κ ) (8) m ( κ ) = 1 , m − ( κ ) = 0where the Normal CDF Φ( · ) and PDF φ ( · ) are evaluated once only regardless of N .6his allows us to compute each term in the formula above: E (cid:104) Z i · { Z >κ } (cid:105) = E (cid:2) Z i (cid:12)(cid:12) Z > κ (cid:3) · P (cid:2) Z > κ (cid:3) = m i ( κ ) · (cid:2) − Φ( κ ) (cid:3) (9)For the (Foreign) Equity Vanilla Call, we have a similar formula in the coefficient { a ,n } N − :ˆ C S T,K ≈ X · E F (cid:104)(cid:0) (cid:101) g ( Z ) − K (cid:1) + (cid:105) = X · E F (cid:20)(cid:18) N − (cid:88) n =0 a ,n Z n − K (cid:19) · { Z >g − ( K ) } (cid:21) (10)Combining (6) and (10), we get the Quanto-Vanilla Spread with strike K as a polynomialexpansion: ˆ C QS T,K := ˆ C Q T,K − ˆ C S T,K = E F (cid:104) X T (cid:0) S T − K (cid:1) + (cid:105) − X E F (cid:104)(cid:0) S T − K (cid:1) + (cid:105) = E F (cid:20)(cid:18) N − (cid:88) n =0 c n Z n − K · N − (cid:88) n =0 b n Z n − X · N − (cid:88) n =0 a ,n Z n + X · K (cid:19) · { Z >κ } (cid:21) (11)= E F (cid:20)(cid:18) N − (cid:88) n =0 e n ( K ) Z n (cid:19) · { Z >κ } (cid:21) = N − (cid:88) n =0 e n ( K ) · E F (cid:20) Z n · { Z >κ } (cid:21) = (cid:18) N − (cid:88) n =0 e n ( K ) · m n ( κ ) (cid:19) · (cid:104) − F S T ( K ) (cid:105) (12)where we have collected the coefficients c , b , a in (11) into e : e ... e ( N − .e ( N − ... e ( N − := c ... c ( N − .c ( N − ... c ( N − − K · ( b − X )... b ( N − . − X · a , ...... .a , ( N − ...0 e n := c n − K · b n n Quanto Local Drift function, ρ σ XS ( S, t ) σ S ( S, t ).Noting the expression (14) is under Domestic measure but all the implied volatility infor-mation for { S t } is under Foreign measure, we move to F : σ XS ( S, t ) = E F (cid:104) X t E F [ X t ] · σ X ( X t , t ) (cid:12)(cid:12)(cid:12) S t = S (cid:105) (15)The key idea is to approximate the well-behaved forex local volatility σ X ( X t , t ) again withthe Lagrange polynomial. Mapping each t slice of the distribution of X t to normals Z with8 ( · ) (dropping t assumed implicitly in the notation), we have: ν t ( Z ) := (cid:16) g ( Z ) X · σ X ( g ( Z ) , t ) · B TF B TD (cid:17) d = X t E F [ X t ] · σ X ( X t , t ) ≈ ˜ ν t ( Z ) := N t − (cid:88) n =0 a t,n · Z n The coefficients a t = ( a , a , a , · · · , a N t − ) (cid:62) again solves the linear equation V a t = ν t ( x )where x are the Gaussian quadrature points which the re-mapped local vol functions ν ( z ) areapplied. In practice we use the same N t for all t .Note that we do not need to construct the full FX local vol grid here. For each t we onlyneed to evaluate the ν ( z ) function N t times where N t is typically small ( ≤ σ XS ( S, t ) ≈ E F (cid:20) N t − (cid:88) n =0 a t,n · Z n (cid:12)(cid:12)(cid:12)(cid:12) Z = g − ( S ) (cid:21) = N t − (cid:88) n =0 a t,n · E F (cid:2) Z n (cid:12)(cid:12) Z = z S (cid:3) = N t − (cid:88) n =0 a t,n · (cid:16) n (cid:88) i =0 q i ( n ; ρ ) · ( z S ) i (cid:17) = N t − (cid:88) n =0 b t,n · ( z S ) n b t,n := N t − (cid:88) j = n a t,j q n ( j ; ρ )where the re-mapped variable z S := g − ( S ) is cheap to evaluate.With the Quanto Local Drift function at hand, we can now replace the ad-hoc constantquanto drift adjustments. Note that for strongly path-dependent payoffs, the local drift ap-proach will not yield identical results as the full-blown 2-factor local volatility model in Equityand FX because Markovian projection guarantees the invariance in the terminal distributiononly. On the other hand, for products such as basket, dispersion or rainbow options, the ap-proach performs very well and allows easy integration into existing Monte Carlo/PDE enginesin a pricing library. Similar to the local volatility in the diffusion term, one can simply use anew spot-dependent drift to advance the Monte Carlo path or propagate the PDE. It retainsthe computational cost of the original numerical method but allows one to capture FX andEquity skew so that European Quanto options can be consistently valued. We illustrate the accuracy of the new approach with test results where several alternativebenchmark methods were used to ensure the testing is comprehensive: • • Monte Carlo with 2 ( ∼ • g i and the CDF functions were obtained from parametrisedimplied volatility surface fitted to the market with arbitrage removal techniques applied toensure the tails are well behaved.Starting with the USD quanto 2 year option on Nikkei using market data as of January2017 and comparing the alternative approaches in Figure 2, we see that the ad-hoc adjustmentis over pricing the vols across all strikes versus the other methods, even if the quanto forwardsmatch between all of them. The effect comes from the positive correlation between Nikkeiand USDJPY, which remains strongly positive since the financial crisis (around 70% in theexample above) as well as the volatility skew. It is clear that our method shows excellentagreement with the other three standard (slower) benchmark methods. Similar pattern isobserved for an equally popular pair, Nikkei quantoed into Australian dollar, where Figure 3showing our method remains accurate for extended maturity and strikes. Ad-hocCopulaMCPDENew Figure 2: Results: .N225 USD 2Y Quanto, Q1 2017 (in implied vol) Ad-hocCopulaMCPDENew Figure 3: Results: .N225 AUD 5Y Quanto, Q1 2017 (in implied vol)The Quanto skew effect is directly proportional to the Equity-FX correlation and some ofthe major index-currency pairs can easily go through regimes of positive, negative or near-zero10ealized correlation as influenced by supply-demand or political events. We therefore examinethe effect of changing signs of correlation by testing S&P500 index quantoed into EUR andstressing the correlation to be highly negative (-80%) in Figure 4. As we flip the sign, thead-hoc bias changes direction as expected while our method retains its close match with othermethods, confirming its robustness. Ad-hocCopulaMCPDENew Figure 4: Results: .SPX EUR 2Y Quanto, Q1 2017 ( ρ = − t = 6 months and 2 years. Similarly, we demon-strate close match between our very fast approach with the other two benchmark methods:PDE and Copula integration. In contrast to the no-skew case assuming a constant drift de-rived from the ad-hoc approach, the non-trivial shape of the function is accurately capturedby our polynomial. NewCopulaPDE (a) Quanto Local Drift at t =6 Month NewCopulaPDE (b) Quanto Local Drift at t =2 Year Figure 5: Local Drift Adjustment: Conditional expectation of Quanto Covariance E D (cid:2) ρ · σ S ( S, t ) · σ X ( X t , t ) (cid:12)(cid:12) S t = S (cid:3) as a function of S Finally we report the speed of the proposed method for the Quanto Vanilla option valua-tion. Implementing the proposed pricing recipe in Section 3.2 in C++ and running the testswith CPU Intel Core i7-4770 3.4 GHz and 32 GB RAM, the average compute time is shown11n Table 1. We report in the last three rows the results of combining hundreds of options inone calculation. no. of no. of no. of Total time Per-optionoptions maturities strikes (seconds) (seconds)1 1 1 0.000536119 0.000536119100 1 100 0.000919493 0.000009194100 10 10 0.005316215 0.0000531621000 10 100 0.008435730 0.000008436Table 1: Compute time for Quanto Vanilla option spread using proposed pricing recipePricing a single option takes only about half a millisecond. This extremely fast perfor-mance gets even better if one computes options with the same maturity (and different strikes)simultaneously by only updating the coefficients needed: less than 1 millisecond to compute100 options (in row 2) and hence a hundredth of that per option. This clearly beats all theother benchmark numerical methods mentioned above by orders of magnitude.In closing we note that while the method delivers high accuracy and speed in the localvolatility/Gaussian copula setting, less precision is expected in stochastic volatility models asobserved in [3]. This is because the joint distribution of the two drivers post the marginalquantile transforms in stochastic volatility models deviates from the Gaussian assumption.Potential extension of our approach to address this issue is left as future research. In this article, a new approach of valuing quanto derivatives analytically has been developed,with the aims of high performance and low implementation effort in mind, by applying thestochastic collocation methods introduced by [5, 6]. Our hope is that the approach is simpleenough for participants across different segments of the market and asset classes to startincorporating quanto skew into their pricing and risk management decisions, beyond theuse of simplistic constant drift adjustments. This includes trading desks, for which fastcalculation is critical, as well as hedge funds and banks, which lean towards simple solutionsfree from the heavy machinery of sophisticated numerical methods and highly specialisedmodel assumptions. Acknowledgement The author would like to thank his team members and fellow quants for stimulating discussionson the subject and is particularly grateful to Yonglan Zhu, David Wilkinson and GhislainVong.The views expressed in this article are those of the author alone and do not necessarilyrepresent those of Credit Suisse Group. 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