Extensions of Dupire Formula: Stochastic Interest Rates and Stochastic Local Volatility
aa r X i v : . [ q -f i n . M F ] M a y Extensions of Dupire Formula: Stochastic Interest Rates andStochastic Local Volatility
Orcan ¨Ogetbil ∗ Abstract
We derive generalizations of Dupire formula to the cases of general stochastic driftand/or stochastic local volatility. First, we handle a case in which the drift is given asdifference of two stochastic short rates. Such a setting is natural in foreign exchangecontext where the short rates correspond to the short rates of the two currencies, eq-uity single-currency context with stochastic dividend yield, or commodity context withstochastic convenience yield. We present the formula both in a call surface formulationas well as total implied variance formulation where the latter avoids calendar spreadarbitrage by construction. We provide derivations for the case where both short ratesare given as single factor processes and present the limits for a single stochastic rate orall deterministic short rates. The limits agree with published results. Then we derivea formulation that allows a more general stochastic drift and diffusion including oneor more stochastic local volatility terms. In the general setting, our derivation allowsthe computation and calibration of the leverage function for stochastic local volatilitymodels.
Risk neutral pricing frameworks aim to establish methodologies for producing prices consis-tent with market data available as of valuation time. As a standard approach, practitionersconsider parametric models to map market quotes to time and space dependent model pa-rameters. The single parameter Black-Scholes model, for instance, gives European vanillaoption prices as a function of implied volatility . In a sense, having an implied volatilitysurface spanning a range of strikes and maturities is equivalent to knowing the prices ofEuropean vanilla options, whose payoffs depend on the value of the underlier solely at matu-rity, for the same strikes and maturities. This in turn amounts to knowing the risk-neutralprobability density of the underlier at given future times conditioned at its present value. Inthis paradigm, the bulk of the work in developing a methodology to price European vanillaoption instruments written on the same underlier lies in the construction of the impliedvolatility surface.The benefits of formulating the risk-neutral probability density as a function of time andunderlying spot value, however, go beyond the ability of pricing European vanilla options.To price more complex options, whose payoffs depend not only on the terminal value of the ∗ Corporate Model Risk, Wells Fargo Bank, [email protected]
Dupire introduces a state dependent diffusion coefficient σ LV ( S t , t ) that uniquely describesthe distribution of the state variable S t for each time t , conditioned on the initial value S .Accordingly, there is a risk-neutral spot process that is compatible with observed marketskew and allows a complete model. This is commonly referred to as local volatility process [1], dS t = µ t S t dt + σ LV ( S t , t ) S t dW S (DRN) t . (2.1)Dupire’s formula gives the function σ LV ( · , · ) in terms of call and/or put option prices, orequivalently, implied volatility or total implied variance.The original work of Dupire [1] assumes zero interest rates ( µ t = 0), while the inde-pendent study of Derman and Kani [2] introduces deterministic interest rates ( µ t = µ ( t )).In the latter setup, the drift term µ t is assumed to be the instantaneous forward rate ofmaturity t implied from the yield curve, which is a deterministic function of time. In this2aper we relax this constraint and let this term be stochastic. In particular, we are inter-ested in a model with two stochastic terms that comprise a drift of the form µ t = µ t − µ t .One can consider the pair µ t , µ t as interest rate/dividend rate in equities setup, or domes-tic rate/foreign rate in foreign exchange setup. In this section, without loss of generality,we will use the conventions of the latter. In particular, µ t = r dt and µ t = r ft denote thedomestic and foreign short rates, respectively. These rates follow single factor processes ofthe generic form dr dt = α d ( S t , r dt , r ft , t ) dt + σ d ( S t , r dt , r ft , t ) dW d (DRN) t ,dr ft = α f ( S t , r dt , r ft , t ) dt + σ f ( S t , r dt , r ft , t ) dW f (DRN) t . (2.2)Our model admits three Brownian motions under the domestic risk neutral measure Q DRN and we set the three pairs of correlations as dW St dW dt = ρ Sd dt , dW St dW ft = ρ Sf dt , and dW dt dW ft = ρ df dt . Note that the short rate stochastic differential equations (SDEs) aretypically written in the risk neutral measure of their own currency. Here, drift adjustmentin the foreign short rate process due to the change from foreign risk neutral measure todomestic risk neutral measure is absorbed into the term α f ( S t , r dt , r ft , t ). As required byour computations below, we assume that the functions σ LV , α d , σ d , α f , and σ f are twicedifferentiable with respect to the arguments S t , r dt , and r ft over their entire ranges. The discounted price of an asset V T ≡ V ( S T , r dT , r fT , T ) is a martingale under the domesticrisk neutral measure Q DRN . In particular for any T , V B d = V = E Q DRN (cid:20) V T B dT (cid:21) , (2.3)where B dT = exp hR T r du du i is the domestic money market account, by means of which wecan define D T ≡ B dT as the corresponding discount factor. If instead of the money marketaccount we take the zero coupon bond P d (0 , T ) maturing at time T as the num´eraire, wecan write the corresponding expectation under the (domestic) T -forward measure Q T as V P d (0 , T ) = E Q T (cid:20) V T P d ( T, T ) (cid:21) = E Q T [ V T ] , (2.4)which can be written as V = P d (0 , T ) Z Z Z V ( S T , r dT , r fT , T )Φ T ( S T , r dT , r fT ) dS T dr dT dr fT , (2.5)where Φ T ( S T , r dT , r fT ) denotes the T -forward measure probability density. We assume theprobability density function to be sufficiently tractable; in particular, it is bounded; and In general, the correlations can be time-dependent; or they can even be generalized to stochastic processesas we shall see in Section 3. The nature of the correlations does not have any impact on our result, thus wekeep their notation simple.
3t is differentiable with respect to time and twice differentiable with respect to its spatialarguments. Notationwise, here and in what is below, the integrals written without explicitlimits are meant to be taken over the entire domain, which is ( −∞ , ∞ ) in most cases.One can integrate the full T -forward probability density Φ T over the entire ranges of r dT and r fT to get the marginal T -forward probability density q T of S T over time, q T ( S T ) = Z Z Φ T ( S T , r dT , r fT ) dr dT dr fT . (2.6)The marginal T -forward distribution has the time derivative ∂q T ( S T ) ∂T = Z Z ∂ Φ T ( S T , r dT , r fT ) ∂T dr dT dr fT . (2.7)Next, we apply Itˆo’s lemma to the discounted asset price, d ( D T V T ) D T = (cid:20) ∂V T ∂T − r dT V T + 12 ( σ LV T ) S T ∂ V T ∂S T + ( r dT − r fT ) S T ∂V T ∂S T + 12 ( σ dT ) ∂ V T ∂y + α dT ∂V T ∂y + 12 ( σ fT ) ∂ V T ∂ ( r fT ) + α fT ∂V T ∂r fT + ρ Sd S T σ LV T σ dT ∂ V T ∂S T ∂r dT + ρ Sf S T σ LV T σ fT ∂ V T ∂S T ∂r fT + ρ df σ dT σ fT ∂ V T ∂r dT ∂r fT dT + σ LV T S T ∂V T ∂S T dW S (DRN) T + σ dT ∂V T ∂r dT dW d (DRN) T + σ fT ∂V T ∂r fT dW f (DRN) T . Here and below we use the convention σ LV T ≡ σ LV ( S T , T ), α dT ≡ α d ( S T , r dT , r fT , T ), σ dT ≡ σ d ( S T , r dT , r fT , T ), α fT ≡ α f ( S T , r dT , r fT , T ), and σ fT ≡ σ f ( S T , r dT , r fT , T ) for notational brevity.Since the discounted asset price is a martingale under Q DRN , the drift term of d ( D T V T )must vanish,0 = ∂V T ∂T − r dT V T + 12 ( σ LV T ) S T ∂ V T ∂S T + ( r dT − r fT ) S T ∂V T ∂S T + 12 ( σ dT ) ∂ V T ∂ ( r dT ) + α dT ∂V T ∂r dT + 12 ( σ fT ) ∂ V T ∂ ( r fT ) + α fT ∂V T ∂r fT + ρ Sd S T σ LV T σ dT ∂ V T ∂S T ∂r dT + ρ Sf S T σ LV T σ fT ∂ V T ∂S T ∂r fT + ρ df σ dT σ fT ∂ V T ∂r dT ∂r fT . (2.8)We differentiate (2.5) with respect to T to get0 = ∂P d (0 , T ) ∂T Z Z Z V T Φ T dS T dr dT dr fT + P d (0 , T ) Z Z Z (cid:20) ∂V T ∂T Φ T + V T ∂ Φ T ∂T (cid:21) dS T dr dT dr fT , Applying (2.8) into this gives0 = ∂P d (0 , T ) ∂T Z Z Z V T Φ T dS T dr dT dr fT + P d (0 , T ) Z Z Z V T ∂ Φ T ∂T dS T dr dT dr fT P d (0 , T ) Z Z Z Φ T (cid:20) r dT V T −
12 ( σ LV T ) S T ∂ V T ∂S T − ( r dT − r fT ) S T ∂V T ∂S T −
12 ( σ dT ) ∂ V T ∂ ( r dT ) − α dT ∂V T ∂r dT −
12 ( σ fT ) ∂ V T ∂ ( r fT ) − α fT ∂V T ∂r fT − ρ Sd S T σ LV T σ dT ∂ V T ∂S T ∂r dT − ρ Sf S T σ LV T σ fT ∂ V T ∂S T ∂r fT − ρ df σ dT σ fT ∂ V T ∂r dT ∂r fT dS T dr dT dr fT . Using the definition of the instantaneous forward rate f i (0 , T ) ≡ − ∂ log P i (0 , T ) ∂T = − P i (0 , T ) ∂P i (0 , T ) ∂T , with i = d, f , we can reformulate this as0 = Z Z Z " V T ∂ Φ T ∂T + Φ T ( (cid:16) r dT − f d (0 , T ) (cid:17) V T −
12 ( σ LV T ) S T ∂ V T ∂S T − ( r dT − r fT ) S T ∂V T ∂S T −
12 ( σ dT ) ∂ V T ∂ ( r dT ) − α dT ∂V T ∂r dT −
12 ( σ fT ) ∂ V T ∂ ( r fT ) − α fT ∂V T ∂r fT − ρ Sd S T σ LV T σ dT ∂ V T ∂S T ∂r dT − ρ Sf S T σ LV T σ fT ∂ V T ∂S T ∂r fT − ρ df σ dT σ fT ∂ V T ∂r dT ∂r fT ) dS T dr dT dr fT . (2.9)The collection of zero coupon bond prices P i with maturities sequenced over a time gridis called a discount curve . The instantaneous forward rates f i can be evaluated along givendiscount curves which are used as standard input data in various pricing and other financialmodels.We integrate by parts the terms that have the partial derivatives of V T appearing in(2.9). Noting that the boundary terms vanish as we assume Φ T and its derivatives tendto zero fast enough as its arguments approach the integration limits, we can derive thefollowing identities by integrating by parts Z Φ T f ( · ) ∂ V T ∂u du = Z ∂ (Φ T f ( · )) ∂u V T du, Z Φ T f ( · ) ∂V T ∂u du = − Z ∂ (Φ T f ( · )) ∂u V T du, Z Φ T f ( · ) ∂ V T ∂u∂v dudv = Z ∂ (Φ T f ( · )) ∂u∂v V T dudv, f of spatialcoordinates u and v , e.g. representing S T , r dT , r fT in our setup. Thus (2.9) can be writtenas0 = Z Z Z V T (cid:26) ∂ Φ T ∂T + Φ T (cid:16) r dT − f d (0 , T ) (cid:17) − ∂ (Φ T ( σ LV T ) S T ) ∂S T + ( r dT − r fT ) ∂ (Φ T S T ) ∂S T − ∂ (Φ( σ dT ) ) ∂ ( r dT ) + ∂ (Φ T α dT ) ∂r dT − ∂ (Φ T ( σ fT ) ) ∂ ( r fT ) + ∂ (Φ T α fT ) ∂r fT − ∂ (Φ T ρ Sd S T σ LV T σ dT ) ∂S T ∂r dT − ∂ (Φ T ρ Sf S T σ LV T σ fT ) ∂S T ∂r fT − ∂ (Φ T ρ df σ dT σ fT ) ∂r dT ∂r fT (cid:27) dS T dr dT dr fT . Since the above equation holds for any asset V T , the term inside the braces must vanish.This leads us to the Fokker-Planck (forward Kolmogorov) equation [9], which describes theevolution of the probability density function Φ T ( S T , r dT , r fT ) of the underlying factors overtime, 0 = ∂ Φ T ∂T + Φ T (cid:16) r dT − f d (0 , T ) (cid:17) − ∂ (Φ T ( σ LV T ) S T ) ∂S T + ( r dT − r fT ) ∂ (Φ T S T ) ∂S T − ∂ (Φ( σ dT ) ) ∂ ( r dT ) + ∂ (Φ T α dT ) ∂r dT − ∂ (Φ T ( σ fT ) ) ∂ ( r fT ) + ∂ (Φ T α fT ) ∂r fT − ∂ (Φ T ρ Sd S T σ LV T σ dT ) ∂S T ∂r dT − ∂ (Φ T ρ Sf S T σ LV T σ fT ) ∂S T ∂r fT − ∂ (Φ T ρ df σ dT σ fT ) ∂r dT ∂r fT . (2.10) We integrate (2.10) over the entire ranges of r dT and r fT . As before, as its argumentsapproach their limits the probability distribution function Φ T and its derivatives go to zerofast enough to make the boundary terms vanish,0 = ∂q T ∂T + Z Z Φ T (cid:16) r dT − f d (0 , T ) (cid:17) dr dT dr fT − ∂ ( q T ( σ LV T ) S T ) ∂S T + ∂∂S T (cid:18)Z Z ( r dT − r fT )Φ T S T dr dT dr fT (cid:19) . (2.11)So far we did not make any assumptions about the payoff function of the asset V T .We now concentrate on a European vanilla call option C with strike K , which pays offmax( S T − K,
0) at time T . Following (2.5), the time zero value of this option is given by C = P d (0 , T ) Z Z Z ∞ K ( S T − K )Φ T dS T dr dT dr fT = P d (0 , T ) E Q T [( S T − K ) S T >K ] . (2.12)6e compute the first two derivatives of the call price with respect to strike K , ∂C∂K = P d (0 , T ) Z Z " − ( S T − K )Φ T (cid:12)(cid:12)(cid:12)(cid:12) ∞ S T = K − Z ∞ K Φ T dS T dr dT dr fT = − P d (0 , T ) Z Z Z ∞ K Φ T dS T dr dT dr fT = − P d (0 , T ) E Q T [ S T >K ] , (2.13) ∂ C∂K = P d (0 , T ) Z Z Φ T ( K, r dT , r fT ) dr dT dr fT = P d (0 , T ) q T ( K ) . (2.14)Next, we differentiate the call price with respect to time. Here we make use of (2.11) andintegration by parts, ∂C∂T = ∂P d (0 , T ) ∂T Z Z Z ∞ K ( S T − K )Φ T dS T dr dT dr fT + P d (0 , T ) Z Z Z ∞ K ( S T − K ) ∂ Φ T ∂T dS T dr dT dr fT = − f d (0 , T ) C + P d (0 , T ) Z ∞ K ( S T − K ) ∂q T ∂T dS T = − f d (0 , T ) C + P d (0 , T ) Z ∞ K ( S T − K ) (cid:26) − Z Z Φ T (cid:16) r dT − f d (0 , T ) (cid:17) dr dT dr fT + 12 ∂ ( q T ( σ LV T ) S T ) ∂S T − ∂∂S T (cid:18)Z Z ( r dT − r fT )Φ T S T dr dT dr fT (cid:19) (cid:27) dS T = P d (0 , T ) Z Z Z ∞ K Φ T ( Kr dT − S T r fT ) dS T dr dT dr fT − P d (0 , T ) q T S T ( σ LV T ) (cid:12)(cid:12)(cid:12)(cid:12) ∞ S T = K . Plugging (2.14) into this expression yields ∂C∂T = P d (0 , T ) E Q T h ( Kr dT − S T r fT ) S T >K i + 12 K ∂ C∂K ( σ LV T ) . (2.15)Thus we arrive at the extended Dupire formula under stochastic rates σ LV ( K, T ) = ∂C∂T − P d (0 , T ) E Q T h ( Kr dT − S T r fT ) S T >K i K ∂ C∂K . (2.16)There is no known method to compute the expectation above analytically, yet it can beevaluated by numerical methods such as Monte Carlo or finite differences. Note also thatthe term with the expectation corresponds to the price of an option with maturity T andpayoff ( Kr dT − S T r fT ) S T >K . 7 ingle stochastic rate limit In the limit where the foreign rates r fT are deterministic, this equation becomes (see [5]for an alternative derivation) σ LV ( K, T ) = ∂C∂T − P d (0 , T ) K E Q T (cid:2) r dT S T >K (cid:3) + P d (0 , T ) r fT E Q T [ S T S T >K ] K ∂ C∂K . (2.17)The second expectation in the numerator can be evaluated using (2.12) and (2.13) P d (0 , T ) r fT E Q T [ S T S T >K ] = r fT h C + P d (0 , T ) K E Q T [ S T >K ] i = r fT (cid:20) C − K ∂C∂K (cid:21) , which reduces (2.17) to σ LV ( K, T ) = ∂C∂T − P d (0 , T ) K E Q T (cid:2) r dT S T >K (cid:3) + r fT (cid:2) C − K ∂C∂K (cid:3) K ∂ C∂K . (2.18) Deterministic rates limit
In the limit where both the domestic rates r dT and the foreign rates r fT are deterministic,one can evaluate the expectation in the above numerator using (2.13) − P d (0 , T ) E Q T [ S T >K ] = ∂C∂K . This allows us to reproduce the standard
Dupire formula , σ LV ( K, T ) = ∂C∂T + ( r dT − r fT ) K ∂C∂K + r fT C K ∂ C∂K . (2.19) Quotes for various European call options with a range of strikes and maturities are requiredfor the evaluation of the Dupire formula (2.19) or the extended Dupire formula (2.16) tocreate a local volatility surface . In practice, one can create a call price surface interpo-lator to evaluate the call price and its derivatives in these equations along the grid thelocal volatility surface is being constructed. However the method for interpolation whileevaluating the Dupire formula is a concern as the interpolated values might introduce arbi-trage to the model. One way to address this problem is to construct a Black-Scholes totalimplied variance surface and interpolate that instead. As a matter of fact, practitioners typ-ically work with market data that is in the form of parametrized or dense implied volatilitysurfaces that are calibrated with such penalty functions that aim to avoid or at least tominimize arbitrage. The absence of calendar spread arbitrage implies that the total impliedvariance surface is a monotonically increasing function of time [10, 11]. By construction,interpolating the total implied variance surface and using these values in the Dupire for-mula avoids calendar spread arbitrage. In this section we derive the total implied varianceparametrization of the extended Dupire formula.8he Black-Scholes European call option price function C BS can be parametrized in termsof log-moneyness y ( K, T ) = log KF T , where F T ≡ S P f (0 ,T ) P d (0 ,T ) is the forward price at time T , and the total implied variance w ( y ( K, T ) , T ) = Σ( K, T ) T as [12] C BS ( T, y, w ) = P d (0 , T ) F T ( N ( d ) − e y N ( d )) (2.20)with d = − yw − + 12 w ,d = d − w . Here Σ(
K, T ) is the market implied volatility at strike K and maturity T , and N ( · ) is thestandard Gaussian cumulative distribution function. Noting that both C BS and w dependon the strike K indirectly through y ( K, T ), that is C BS = C BS ( T, y ( K, T ) , w ( y ( K, T ) , T )),the first two derivatives of the call price with respect to the strike can be computed as ∂C BS ∂K = (cid:18) ∂C BS ∂y + ∂C BS ∂w ∂w∂y (cid:19) ∂y∂K ,∂ C BS ∂K = (cid:20) ∂ C BS ∂y + (cid:18) ∂ C BS ∂w∂y + ∂ C BS ∂w ∂w∂y (cid:19) ∂w∂y + ∂C BS ∂w ∂ w∂y (cid:21) (cid:18) ∂y∂K (cid:19) + (cid:18) ∂C BS ∂y + ∂C BS ∂w ∂w∂y (cid:19) ∂ y∂K . Since ∂y∂K = K and ∂ y∂K = − K the second expression can be written as K ∂ C BS ∂K = ∂ C BS ∂y + (cid:18) ∂ C BS ∂w∂y + ∂ C BS ∂w ∂w∂y − ∂C BS ∂w (cid:19) ∂w∂y + ∂C BS ∂w ∂ w∂y − ∂C BS ∂y . (2.21)The right hand side of this equation demands evaluation of the derivatives of the callprice with respect to the log-moneyness and the total implied variance. Using the identity N ′ ( d ) = e y N ′ ( d ) we compute the first w -derivative as ∂C BS ∂w = P d (0 , T ) F T (cid:20) N ′ ( d ) ∂d ∂w − e y N ′ ( d ) ∂d ∂w (cid:21) = 12 P d (0 , T ) F T e y N ′ ( d ) w − ; (2.22)and, since N ′′ ( x ) = − xN ′ ( x ), the second w -derivative evaluates as ∂ C BS ∂w = 12 P d (0 , T ) F T e y (cid:20) − N ′ ( d ) d ∂d ∂w w − − N ′ ( d ) w − (cid:21) = 12 ∂C BS ∂w (cid:20) − − w + y w (cid:21) . (2.23)9urthermore, the remaining derivatives are ∂ C BS ∂w∂y = 12 P d (0 , T ) F T e y N ′ ( d ) w − (cid:20) − d ∂d ∂y + 1 (cid:21) = ∂C BS ∂w (cid:20) − yw + 12 (cid:21) , (2.24) ∂C BS ∂y = P d (0 , T ) F T (cid:20) N ′ ( d ) ∂d ∂y − e y N ( d ) − e y N ′ ( d ) ∂d ∂y (cid:21) = − P d (0 , T ) F T e y N ( d ) , (2.25) ∂ C BS ∂y = − P d (0 , T ) F T e y (cid:20) N ( d ) + N ′ ( d ) ∂d ∂y (cid:21) = ∂C BS ∂y + 2 ∂C BS ∂w . (2.26)Plugging in equations (2.22), (2.23), (2.24), (2.25), and (2.26) into (2.21) we arrive at12 K ∂ C BS ∂K = ∂C BS ∂w " − yw ∂w∂y + 12 ∂ w∂y + 14 (cid:18) ∂w∂y (cid:19) (cid:18) − − w + y w (cid:19) . (2.27)Finally, we use the identities ∂y∂T = − S F T ∂ P f (0 ,T ) P d (0 ,T ) ∂T = f f (0 , T ) − f d (0 , T ) ,∂ ( P d (0 , T ) F T ) ∂T = S ∂ ( P f (0 , T )) ∂T = − f f (0 , T ) P d (0 , T ) F T , to formulate the time derivative of the call price as ∂C BS ∂T = − f f (0 , T ) C BS + ∂C BS ∂w ∂w∂T + (cid:18) ∂C BS ∂y + ∂C BS ∂w ∂w∂y (cid:19) ( f f (0 , T ) − f d (0 , T )) . (2.28)Plugging in equations (2.27) and (2.28) into (2.16) gives us the extended Dupire formula inthe log-moneyness/total implied variance parametrization. σ LV ( K, T ) = ∂C BS ∂T − P d (0 , T ) E Q T h ( Kr dT − S T r fT ) S T >K i ∂C BS ∂w (cid:20) − yw ∂w∂y + ∂ w∂y + (cid:16) ∂w∂y (cid:17) (cid:16) − − w + y w (cid:17)(cid:21) , (2.29)where the explicit forms of C BS , ∂C BS ∂w , ∂C BS ∂y , and ∂C BS ∂T are given by (2.20), (2.22), (2.25),and (2.28) respectively. Single stochastic rate limit
In the limit where the foreign rates r fT are deterministic, this equation becomes σ LV ( K, T ) = ∂C BS ∂w ∂w∂T − f d (0 , T ) (cid:16) ∂C BS ∂y + ∂C BS ∂w ∂w∂y (cid:17) − P d (0 , T ) K E Q T (cid:2) r dT S T >K (cid:3) ∂C BS ∂w (cid:20) − yw ∂w∂y + ∂ w∂y + (cid:16) ∂w∂y (cid:17) (cid:16) − − w + y w (cid:17)(cid:21) . (2.30)10 eterministic rates limit In the limit where both the domestic rates r dT and the foreign rates r fT are deterministic,the equation further simplifies to the form given in [12] σ LV ( K, T ) = ∂w∂T − yw ∂w∂y + ∂ w∂y + (cid:16) ∂w∂y (cid:17) (cid:16) − − w + y w (cid:17) . (2.31) In (2.1) we considered a standard local volatility process of a particular form. Namely it isgeometric and the drift term is a linear combination of two stochastic rates, each modeledby a single factor process. Here we relax these constraints and study the following generalmodel with drift and diffusion functions that allow arbitrary number of stochastic factors.It is constructive to write down this SDE system in terms of N independent Brownianmotions, dS t = µ ( S t , Y t , t ) dt + L ( S t , t ) N X k =1 ˆ σ Sk ( S t , Y t , t ) d ˆ W kt ,dy jt = µ j ( S t , Y t , t ) dt + N X k =1 ˆ σ jk ( S t , Y t , t ) d ˆ W kt ,d ˆ W kt d ˆ W lt = δ kl dt.Y t ≡ ( y t , . . . , y Mt ) is the set of additional Itˆo processes in the SDE system for which we donot assume any special form other than the above. L ( S t , t ) is the local volatility or leveragefunction we want to compute. In this setup, we observe that the correlation structureof the underlying assets is absorbed into the functions ˆ σ Sk ( S t , Y t , t ) and ˆ σ jk ( S t , Y t , t ). Thecorrelations themselves can be Itˆo processes, in which case they are assigned to particular y jt s. The (domestic) risk free rate r dt is an adapted function of Y t ; yet in general we do notassume a particular mapping .The SDE system can also be written in terms of correlated Brownian motions split intothose driving the processes of S t and Y t separately, with N = N S + N Y , as dS t = µ ( S t , Y t , t ) dt + L ( S t , t ) N S X k =1 σ Sk ( S t , Y t , t ) dW Skt ,dy jt = µ j ( S t , Y t , t ) dt + N Y X k =1 σ jk ( S t , Y t , t ) dW Y kt . (3.1) In the foreign exchange setting of Section 2, r dt and r ft are each direct components of Y t . We will returnto this particular case in Section 3.4. As another example, in case r dt follows a multi-factor short rate model,it can be written as a function of the factors that are a subset of Y t . .2 Fokker-Planck Equation Following the same methodology from Section 2.2, omitting repetitive parts of the compu-tation, we derive the corresponding Fokker-Planck equation. Since the discounted price ofan asset V T = V ( S T , Y T , T ) is a martingale, the drift term of its Itˆo differential must vanish,leading to0 = ∂V T ∂T − r dT V T + 12 L T ¯ σ T ∂ V T ∂S T + µ T ∂V T ∂S T + terms involving Y T derivatives of V T , (3.2)where we defined¯ σ T ≡ N S X l,m =1 (cid:0) σ Sl ( S T , Y T , T ) ρ Slm σ Sm ( S T , Y T , T ) (cid:1) = N X k =1 (cid:0) ˆ σ Sk ( S T , Y T , T ) (cid:1) , and to keep the notation compact we denoted µ T = µ ( S T , Y T , T ) and L T = L ( S T , T ). Here ρ Slm denotes the correlation function between the Brownian motions W Slt , i.e. dW Slt dW Smt = ρ Slm dt . Applying this to the expression for the time derivative of the discounted value of theasset price (2.2) yields0 = ∂P d (0 , T ) ∂T Z Z V T Φ T dS T dY T + P d (0 , T ) Z Z V T ∂ Φ T ∂T dS T dY T + P d (0 , T ) Z Z Φ T h r dT V T − L T ¯ σ T ∂ V T ∂S T − µ T ∂V T ∂S T + terms involving Y T derivatives of V T i dS T dY T , (3.3)where Φ T ( S T , Y T ) is the T -forward measure probability density, with the correspondingmarginal density q T ( S T ) = R Φ T ( S T , Y T ) dY T , which can be used to formulate the derivativesof the call option price with respect to strike, analogous to (2.13) and (2.14) as ∂C∂K = − P d (0 , T ) E Q T [ S T >K ] ,∂ C∂K = P d (0 , T ) q T ( K ) . Moreover, to elucidate the notation, we make a note that the integrals along the stochasticfactors Y T Z f ( · ) dY T ≡ Z . . . Z f ( · ) dy T dy T . . . are taken over their entire domains.Since ∂P d (0 ,T ) ∂T = − f d (0 , T ) P d (0 , T ), (3.3) becomes0 = Z Z h V T ∂ Φ T ∂T + Φ T n (cid:16) r dT − f d (0 , T ) (cid:17) V T − L T ¯ σ T ∂ V T ∂S T − µ T ∂V T ∂S T + terms involving Y T derivatives of V T oi dS T dY T .
12s before, we integrate by parts the above integrals to factor out V T . This leads to thefollowing Fokker-Planck equation,0 = ∂ Φ T ∂T + Φ T (cid:16) r dT − f d (0 , T ) (cid:17) − ∂ (Φ T L T ¯ σ T ) ∂S T + ∂ (Φ T µ T ) ∂S T + terms involving Y T derivatives of Φ T . (3.4) As in Section 2.3.1, we integrate the Fokker-Planck equation (3.4) over the entire rangesof Y T . The probability distribution function Φ T goes to zero fast enough as its argumentsapproach their limits, making the boundary terms that involve the Y T derivatives vanish,0 = ∂q T ∂T + Z Φ T (cid:16) r dT − f d (0 , T ) (cid:17) dY T − ∂ ∂S T (cid:18) L T Z Φ T ¯ σ T dY T (cid:19) + ∂∂S T (cid:18)Z Φ T µ T dY T (cid:19) . (3.5)At this point we note that the terms involving the correlation coefficients are all integratedout, therefore we conclude that the nature of the correlations will not have any impact onour result. Next we compute the time derivative of the price of a European vanilla calloption C with strike K . Here we make use of the definition of conditional expectation,Φ T ( Y | S T = X ) ≡ Φ T ( X,Y ) q T ( X ) , as well as (3.5) and integration by parts, ∂C∂T = ∂P d (0 , T ) ∂T Z Z ∞ K Φ T dS T dY T + P d (0 , T ) Z Z ∞ K ( S T − K ) ∂ Φ T ∂T dS T dY T = − f d (0 , T ) C + P d (0 , T ) Z ∞ K ( S T − K ) ∂q T ∂T dS T = − f d (0 , T ) C + P d (0 , T ) Z ∞ K ( S T − K ) (cid:26) − Z Φ T (cid:16) r dT − f d (0 , T ) (cid:17) dY T + 12 ∂ ∂S T (cid:18) L T Z Φ T ¯ σ T dY T (cid:19) − ∂∂S T (cid:18)Z Φ T µ T dY T (cid:19) (cid:27) dS T = P d (0 , T ) Z Z ∞ K Φ T h µ T − ( S T − K ) r dT i dS T dY T + 12 P d (0 , T ) q T ( K ) L ( K, T ) Z Z Φ T ( Y T | S T = K )¯ σ T dY T = P d (0 , T ) E Q T hn µ T − ( S T − K ) r dT o S T >K i + 12 L ( K, T ) ∂ C∂K E Q T (cid:2) ¯ σ T | S T = K (cid:3) . This gives us the generalized form of the Dupire formula, L ( K, T ) = ∂C∂T − P d (0 , T ) E Q T (cid:2)(cid:8) µ T − ( S T − K ) r dT (cid:9) S T >K (cid:3) ∂ C∂K E Q T (cid:2) ¯ σ T | S T = K (cid:3) . (3.6)13 .4 Examples For simplicity, we consider the underlier S t to be driven by a single Brownian motion( N S = 1) in this section, dS t = µ ( S t , Y t , t ) dt + L s ( S t , t )¯ σ ( S t , Y t , t )) dW St , where L s denotes the leverage function of the simplified model, to distinguish it from thegeneralized model.The special case of this model with ¯ σ t = S t is of special interest where the SDE becomesa simple local volatility model. In this case (3.6) becomes σ LV ( K, T ) = ∂C∂T − P d (0 , T ) E Q T (cid:2)(cid:8) µ T − ( S T − K ) r dT (cid:9) S T >K (cid:3) K ∂ C∂K . (3.7)Comparison of (3.6) with (3.7) gives us the following relationship between the generalizedmodel and its corresponding simple local volatility simplification, σ LV ( K, T ) K = L ( K, T ) E Q T (cid:2) ¯ σ T | S T = K (cid:3) . (3.8)To recover the simpler FX local volatility model with two stochastic rates ( Y t = r dt , r ft )from Section 2.1, one can set µ t = ( r dt − r ft ) S t and ¯ σ t = S t . In this case it is straightforwardto show that (3.6) reduces to (2.16).An extension of this model is the stochastic local volatility (SLV) model with ¯ σ t = S t √ U t where U t is the variance process, dU t = µ U ( U t , t ) dt + σ U ( U t , t ) dW Ut , which is typically calibrated to near-the-money options, and the leverage function L ( S t , T )serves as a correction for the behavior in the wings. A common choice is to use a Cox-Ingersoll-Ross (CIR) process [13] in which case in the context of FX derivatives the SDEsystem becomes [8, 14], dS t =( r dt − r ft ) S t dt + L s ( S t , t ) S t p U t dW S (DRN) t dU t = κ ( θ − U t ) dt + ξ p U t dW U (DRN) t , (3.9)where mean reversion speed κ , long term mean θ , and vol-of-vol ξ are (possibly time-dependent) CIR parameters.For this model, the generalized Dupire formula (3.6) simplifies to L s ( K, T ) = ∂C∂T − P d (0 , T ) E Q T h ( Kr dT − S T r fT ) S T >K i K ∂ C∂K E Q T [ U T | S T = K ] . (3.10)Here we emphasize that the above equation assumes only the particular form of the SDEfor the underlier S t , and is not restricted to the case where the SDE for the variance U t isof type CIR. 14omparing (2.16) to (3.10) allows us to write the relationship between the local volatilityfunction of the pure local volatility model (2.1) and the leverage function of the stochasticlocal volatility model (3.9) as σ LV ( K, T ) = L s ( K, T ) E Q T [ U T | S T = K ] . (3.11)This result is reached independently from but is consistent with Gy¨ongy’s finding [15]that links the stochastic process with Itˆo differential dX t = α ( t, ω ) dt + β ( t, ω ) dW Gt , where α and β are bounded functions of a general set of stochastic factors ω , to anotherstochastic process with deterministic coefficients a and b , dZ t = a ( t, Z t ) dt + b ( t, Z t ) dW Gt , in that the two processes have the same marginal probability distribution for every t if a ( t, z ) = E [ α ( t, ω ) | X t = z ] ,b ( t, z ) = E [ β ( t, ω ) | X t = z ] . Applying this to our example, the marginal distribution of the stochastic local volatilitymodel (3.9) must be the same as the distribution of the pure local volatility model (2.1)if (3.11) holds. This implies that having computed the function σ LV ( K, T ) for the purelocal volatility model using (2.16), one can obtain the leverage function L s ( K, T ) of thestochastic local volatility model by evaluating the conditional expectation E Q T [ U T | S T = K ].One utilizes a numerical method such as multi-dimensional finite difference or Monte Carlosimulation to estimate this conditional expectation as there is no straightforward way toevaluate it analytically.Equation (3.11) allows us to write the extended Dupire formula for the two stochasticrates and stochastic local volatility model (3.9) in the total implied variance surface formu-lation as well. Since the deterministic local volatility limiting case was already computedin this formulation as in (2.29), we can write the leverage function for the stochastic localvolatility generalization as L s ( K, T ) = ∂C BS ∂T − P d (0 , T ) E Q T h ( Kr dT − S T r fT ) S T >K i ∂C BS ∂w (cid:20) − yw ∂w∂y + ∂ w∂y + (cid:16) ∂w∂y (cid:17) (cid:16) − − w + y w (cid:17)(cid:21) E Q T [ U T | S T = K ] , where, as before, the explicit forms of C BS , ∂C BS ∂w , ∂C BS ∂y , and ∂C BS ∂T are given by (2.20), (2.22),(2.25), and (2.28), respectively. Acknowledgments
The author is grateful to Bernhard Hientzsch and Dooheon Lee fornumerous enlightening discussions and guidance about the construction and the flow of thispaper. The author would also like to thank Agus Sudjianto for supporting this research,and Vijayan Nair for suggestions, feedback, and discussion regarding this work. The authoris appreciative for Paul Feehan’s original introduction to the subject and his endorsement.Any opinions, findings and conclusions or recommendations expressed in this material arethose of the author and do not necessarily reflect the views of Wells Fargo Bank, N.A., itsparent company, affiliates and subsidiaries. 15 eferences [1] Bruno Dupire. Pricing with a smile.
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