Local Volatility Models. Copyright Changwei Xiong June last update: October 28, 2017

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1 Local Volatility Models Copyright Changwei Xiong 016 June 016 last update: October 8, 017 TABLE OF CONTENTS Table of Contents ologorov Forward and Backward Equations ologorov Forward Equation ologorov Backward Equation...5. Local Volatility Local Volatility by Vanilla Call Local Volatility by Forward Call Local Variance as a Conditional Expectation of Instantaneous Variance Forula in Log-oneyness Local Volatility by Iplied Volatility Forula in Log-strike Forula in Log-oneyness Equivalency in Forulas Local Volatility: PDE by Finite Difference Method Date Conventions of Equity and Equity Option Deterinistic Dividends Forward PDE Treatent of Deterinistic Dividends Backward PDE PDE in Centered Log-spot Treatent of Deterinistic Dividends Vanilla Call PDE in Log-spot Treatent of Deterinistic Dividends Local Volatility Surface Barrier Option Pricing...5 References...7

2 Changwei Xiong, October 017 The note is prepared for the purpose of suarizing local volatility odels frequently encountered in derivative pricing. It at first derives the ologorov forward and backward equations, which fundaentally govern the transition probability density of the diffusion process in derivative price dynaics. Subsequently, it introduces the local volatility odel in the context of Dupire forula and then presents a PDE based local volatility odel, in which the local volatility function is paraetrized to be piecewise linear in log-oneyness and piecewise constant in tie. 1. OLMOGOROV FORWARD AND BACWARD EQUATIONS The tie evolution of the transition probability density function is governed by ologorov forward and backward equations, which will be introduced as follows, without loss of generality, in ulti-diension ologorov Forward Equation Let s consider the following -diensional stochastic spot process S t R driven by an ndiensional Brownian otion W t whose correlation atrix ρ is given by ρdt = dw t dw t ds t 1 = A(t, S t ) dt + B(t, S t) dw t n n 1 (1) We derive the dynaics of h, where h: R R in this case is a scalar-valued Borel-easurable function only on variable S t dh(s t ) 1 1 = J h ds t + 1 ds t 1 1 H h ds t 1 1 = J h Adt + J h BdW t + 1 dw t B H h BdW t () where J h is the 1 Jacobian (i.e. the sae as gradient if h is a scalar-valued function) and H h the Hessian (with subscripts of S now denoting the indices of vector coponents) [J h ] i = h S i and [H h ] ij = h S i S j (3) Expanding the expression in (), we have

3 Changwei Xiong, October 017 dh = h A S i dt + h B i S ik dw k + 1 i h B S i S ik ρ ij B jk dt j n k=1 n k=1 = ( h A S i + 1 i h Σ S i S ij ) dt + h B j S ik dw k i n k=1 (4) where Σ = BρB is the instantaneous variance-covariance atrix of ds. Integrating on both sides of (4) fro t to T, we have T h(s T ) h(s t ) = ( h A S i + 1 i h Σ S i S ij ) du + h B j S ik dw k i t t T n k=1 (5) Taking expectation on both sides of (5), we get (using notation E t [ ] = E[ F t ]) LHS = E t [h(s T )] h(s t ) = h y p T,y t,x dy h x T RHS = E t [ ( h A S i + 1 i h Σ S i S ij ) du] + E t [ h B t j S ik dw k ] t i T n k=1 =0 (6) T = E t [ h A S i ] du + 1 i E t [ h Σ S i S ij ] du j t t T where p T,y t,x is the transition probability density having S T = y at T given S t = x at t (i.e. if we solve the equation (1) with the initial condition S t = x R, then the rando variable S T = y has a density p T,y t,x in the y variable at tie T). Differentiating (6) with respect to T on both sides, we have p T,y t,x h y dy = E t [ h A S i ] + 1 i E t [ h Σ S i S ij ] j = h y A y i p T,y t,x dy + 1 i h y y Σ i y ij p T,y t,x dy j (7) 3

4 Changwei Xiong, October 017 If we assue R and also assue the probability density p and its first derivatives p/ y i vanish at a higher order of rate than h and h/ y i as y i ± i = 1,,, then we can integrate by parts for the right hand side of (7), once for the first integral and twice for the second h y y i A i pdy = h y A i p + yi = i =0 dy i (A i p) h y dy y i and h y y Σ i y ij pdy = h y Σ j y ij p i j =0 + y i = + dy i h y (Σ ij p) dy y j y i (Σ ij p) (Σ ij p) = h y dy j + h y y dy j i y y j = i y j =0 (8) where ( )dy i i = ( )dy 1 dy i 1 R R R R dy i+1 dy Plugging the results of (8) into (7), we have h y p dy (A i p) = h y dy + 1 y i h (Σ ij p) y dy y i y j h y ( p + (A ip) 1 y i (Σ ij p) ) dy = 0 y i y j (9) By the arbitrariness of h, we conclude that for any y p + (A ip) 1 y i (Σ ij p) = 0, Σ = BρB (10) y i y j This is the Multi-diensional Fokker-Planck Equation (a.k.a. ologorov Forward Equation) [1]. In this equation, the t and x are held constant, while the T and y are variables (called forward variables ). In the one-diensional case, it reduces to p + (Ap) y 1 (B p) y = 0 (11) where A = A(T, y) and B = B(T, y) are then scalar functions. 4

5 Changwei Xiong, October ologorov Backward Equation Let s express conditional expectation of h(s t ) by g(t, S t ) = E t [h(s T )]. Because for o t T we have g(o, S o ) = E o [h(s T )] = E o [E t [h(s T )]] = E o [g(t, S t )] (1) the g(t, S t ) is a artingale by the tower rule (i.e. If H holds less inforation than G, then E[E[X G] H] = E[X H]). The dynaics of the g(t, S t ) is given by dg = g t dt + J g ds t + 1 ds t 1 1 H g ds t 1 1 = g t dt + J gadt + J g BdW t + 1 dw t B H g BdW t (13) where J g is the Jacobian (i.e. the sae as gradient if g is a scalar-valued function) and H g the Hessian of g with respect to S (with subscripts denoting the indices of vector coponents) J g = ( g S 1 g S ), H g = g g S 1 S 1 S g ( S S 1 g S ) (14) Expanding (13), we have dg = ( g t + g A S i i + 1 g Σ S i S ij ) dt + g B j S ik dw k i n k=1 (15) Since g(t, S t ) is a artingale, the dt-ter ust vanish, which gives g t + g A S i i + 1 g Σ S i S ij j = 0 (16) This is the ulti-diensional Feynan-ac forula 1. Using the transition probability density p T,y t,x, we can write the expectation as 1 5

6 Changwei Xiong, October 017 g t,x = E t [h(s T )] = h y p T,y t,x dy (17) The forula (16) defines that t h ypdy + A i h x y pdy + 1 i Σ ij x i x j h y pdy h y ( p t + A p i + 1 x i Σ p ij ) dy = 0 x i x j = 0 (18) By the arbitrariness of h, we have p t + A p i + 1 x i Σ p ij x i x j = 0, Σ = BρB (19) This is the ulti-diensional ologorov Backward Equation. In this equation, the T and y are held constant, while the t and x are variables (called backward variables ). In the 1-D case, it reduces to p p + A t x + 1 p B x = 0 (0) where A = A(t, x) and B = B(t, x) are then scalar functions.. LOCAL VOLATILITY In local volatility odels, the volatility process is assued to be a function of both the spot level and the tie. It is one step generalization of the well-known Black-Scholes odel. Under risk neutral easure, the spot process (e.g. an equity or an FX rate) is assued to follow a geoetric Brownian otion ds t S t = μ t dt + σ(t, S t )dw t, μ t = r t q t (1) with cash rate r t and dividend rate q t (or foreign cash rate for FX)..1. Local Volatility by Vanilla Call Under the assuption of deterinistic r t, the European (vanilla) call option price can be expressed as a function of aturity tie T and strike 6

7 Changwei Xiong, October 017 C T, t,x = E t[d t,t (S T ) + ] = D t,t (y )p T,y t,x dy () T t where D t,t = exp ( r u du) is the deterinistic discount factor and p T,y t,x is the transition probability density having spot S T = y at T given initial condition S t = x at t. Differentiating () with respect to, we have the first order and second order partial derivative C T, t,x = D t,t p T,y t,x dy, C T, t,x = D t,t p T, t,x (3) which gives the transition probability density function by p T, t,x = 1 D t,t C T, t,x (4) The (4) is also known as Breeden-Litzenberger forula. Taking the first derivative of C T, t,x in () with respect to T, we find C T, t,x = r T C T, t,x + D t,t (y ) p T,y t,x dy = r T C T, t,x + D t,t (y ) ( 1 (σ T,y y p T,y t,x ) y (μ Typ T,y t,x ) ) dy y (5) using the ologorov Forward Equation (11) p T,y t,x = 1 (σ T,y y p T,y t,x ) y (μ Typ T,y t,x ) y (6) Applying integration by parts to the integrals on the right hand side of (5) yields (y ) (μ Typ T,y t,x ) dy y and = (y )μ T yp T,y t,x y= =0 = μ T ( (y )p T,y t,x dy μ T yp T,y t,x dy + p T,y t,x dy) = μ T C T, t,x D t,t μ TC T, t,x D t,t (7) 7

8 Changwei Xiong, October 017 (y ) (σ T,y y p T,y t,x ) y dy = (y ) (σ T,y y p T,y t,x ) y =0 y= (σ T,y y p T,y t,x ) dy y = σ T,y y p T,y t,x = σt, p y= T, t,x = σ T, C T, t,x D t,t where we have p T, t,x = 0 and p T, t,x / y = 0 assuing the probability density p T,y t,x and its first derivative vanish at a higher order of rate as y. Plugging (7) into (5), we find C T, t,x = r T C T, t,x + σ T, C T, t,x μ T C T, t,x + μ TC T, t,x = σ T, C T, t,x μ T C T, t,x q TC T, t,x (8) and eventually the Dupire forula for the local volatility σ T, expressed in ters of vanilla call price C T, t,x σ T, = + μ T C T, t,x + q TC T, t,x C T, t,x (9).. Local Volatility by Forward Call Soeties, it is ore convenient to express the Dupire forula in ters of a forward (i.e. undiscounted) call, which is defined as C T, t,x = E t[(s T ) + ] = (y )p T,y t,x dy = C T, t,x D t,t (30) with C T, t,x given in () (note that (30) is true only if the interest rate is deterinistic). Following a siilar derivation, we find that C T, t,x C T, t,x = p T,y t,xdy, = (y ) p T,y t,x dy C T, t,x = p T, t,x and = 1 σ T, C T, t,x + μ T (C T, t,x C T, t,x ) (31) 8

9 Changwei Xiong, October 017 Therefore the Dupire forula for σ T, expressed in ters of forward call price reads C T, t,x σ T, = + μ T C T, t,x μ TC T, t,x C T, t,x (3)..1. Local Variance as a Conditional Expectation of Instantaneous Variance The forward call (30) can be expressed as C T, t,x = E t[(s T ) + ] = E t[h(s T )(S T )] = E t[h(s T )S T ] E t[h(s T )] (33) where h is the Heaviside step function 1. Differentiating once with respect to, we get C T, t,x = p T,y t,xdy = E t[h(s T )] E t[h(s T )S T ] = C T, t,x C T, t,x (34) Differentiating again with respect to, we have C T, t,x = p T, t,x = E t[δ(s T )] (35) where δ is the Dirac delta function. Applying Ito s Lea to the terinal payoff of the option gives the identity d(s T ) + = h(s T )ds T + 1 δ(s T )σ T S T dt (36) = h(s T )μ T S T dt + 1 δ(s T )σ T S T dt + h(s T )σ T S T dw T Taking conditional expectations on both sides gives dc T, t,x = de t[(s T ) + ] = μ T E t[h(s T )S T ]dt + 1 E t[δ(s T )σ T S T ]dt C T, t,x = μ T E t[h(s T )S T ] + 1 E t[δ(s T )σ T S T ] (37) Notice that 1 0, x < 0 Heaviside step function: h(x) = { 1, x 0 Dirac delta function can be viewed as the derivative of the Heaviside step function: δ(x) = dh(x) which is also constrained to satisfy the identity: δ(x)dx R = 1. dx, x = 0 = { 0, x 0, 9

10 Changwei Xiong, October 017 E t[δ(s T )σ T S T ] = E t[δ(s T )σ T ] (38) and fro Bayes rule E t[δ(s T )σ T ] = E t[σ T S T = ]E t[δ(s T )] (39) we can derive fro (37) C T, t,x = μ T C T, t,x μ T C T, t,x + 1 C T, t,x E t[σ T S T = ] E t[σ T S T = ] = C T, t,x + μ T C T, t,x C T, t,x μ TC T, t,x (40) This is identical to (3). It eans that the conditional expectation of the stochastic variance ust equal the Dupire local variance []. That is, local variance is the risk-neutral expectation of the instantaneous variance conditional on the final stock price S T being equal to the strike price [3].... Forula in Log-oneyness In real applications, nuerical ethods are often in favor of log-oneyness to be the spatial variable, which can be regarded as a centered log-strike k = ln F t,t where = 1, = μ T (41) We want to express the Dupire forula in the (T, k)-plane using call option price C T,k (short for C T,k t,z for z = ln x F t,t ) equivalent to the forward call C T, (short for C T, t,x ). Note that although the C T,k and C T, are equivalent, they are two different functions. The conversion fro (T, )-plane to (T, k)-plane is achieved by using the following partial derivatives derived by chain rule C T, C T, = C T,k = C T,k + C T,k = C T,k μ C T,k T + C T,k = 1 C T,k (4) C T, = ( C T,k ) + ( C T,k ) = 1 (1 C T,k ) = 1 C T,k ( C T,k ) 10

11 Changwei Xiong, October 017 Plugging these partial derivatives into (3), we have the Dupire forula expressed in k C T,k σ T,k = C T,k μ TC T,k C T,k (43) where σ T,k is the local volatility in (T, k)-plane equivalent to σ T,..3. Local Volatility by Iplied Volatility It is a arket standard to quote option prices as Black-Scholes iplied volatilities. Hence, it is ore straightforward to express the local volatility in ters of the iplied volatilities rather than option prices. Taking t as of today, we can define the forward price as T F t,t = S t exp ( μ u du) (44) t The forward call price in Black-Scholes odel is then given by X T,,ξ = F t,t Φ(d + ) Φ(d ) with d ± = ln F t,t ξ T, τ ± ξ T, τ, τ = T t (45) where ξ T, is the Black-Scholes iplied volatility derived fro arket quotes of vanilla options and Φ the standard noral cuulative density function. Its partial derivatives can be derived as X T,,ξ = μ T F t,t Φ(d + ) + F t,t φ(d + ) d + φ(d ) d = μ TF t,t Φ(d + ) + φ(d )ξ τ X T,,ξ = F t,tφ(d + ) d + φ(d ) d Φ(d ) = Φ(d ), X T,,ξ = F t,t φ(d + ) d + φ(d ) d = φ(d ) τ, X T,,ξ X T,,ξ = φ(d ) ξ τ = d +d φ(d ) τ ξ (46) X T,,ξ = φ(d )d + ξ where we have used d ± = μ T ξ τ d τ, d ± = 1 ξ τ, d ± = d ξ, d ± = 1 ξ τ (47) 11

12 Changwei Xiong, October 017 and the identity F t,t φ(d + ) = φ(d ) (48) Further using the partial derivatives C T, C T, = X T,,ξ = X T,,ξ + X T,,ξ, + X T,,ξ C T, = X T,,ξ + X T,,ξ + X T,,ξ ξ + X T,,ξ ( ) (49) the local volatility given by iplied volatility can be derived fro (3) as σ T, = C T, + μ T C T, μ TC T, C T, = X + X ( T, + μ T T, ) + μ T X μ TX X ( + X T, + X ξ T, + X ( ) ) μ T F t,t Φ(d + ) + φ(d )ξ τ = φ(d ) ξ τ + φ(d )d + ξ + = 1 X ξτ + d + ξ τ + X ( + μ T ) μ TΦ(d ) μ T X X X ( ξ τ + + μ T ) X + X ξ + ξ + d + d ξ d + d φ(d ) τ ( ξ ) X ( ) (50) = ξ + ξτ ( + μ T ) 1 + τd + + d +d τ ( ) + ξτ ξ Nuerical ethods, e.g. PDE or Monte Carlo siulation, often deand a local volatility function constructed on a D grid to perfor pricing. In these ethods, it is often ore nuerically stable and convenient to work with a spatial diension in log-strike or in log-oneyness Forula in Log-strike The local volatility forula in log strike x = ln can be derived fro (50) 1

13 Changwei Xiong, October 017 σ T,x = ξ + ξτ ( + μ T x ) 1 + τd + x + d +d τ ( x ) + ξτ ( ξ x x ) = ξ + ξτ ( + μ T x ) 1 + (ξτ k ξ ) x + (k ξ ξ τ 4 ) ( x ) + ξτ ( ξ x x ) (51) ξ + ξτ ( = + μ T (1 k ξ x ) ( ξτ x ) providing that we have the following identities = x x = 1 x, d ± = k ξ τ ± ξ τ, k = ln F t,t.3.. Forula in Log-oneyness x ) + ξτ ξ x ξ = x (1 x ) x = 1 ξ ( x x ), x = 1 (5) We ay also want to change the spatial variable to log-oneyness k. Defining a new quantity, iplied total variance v T,k, which is equivalent to ξ T, τ, the Black-Scholes call price that is equivalent to (45) then transfors into X T,k,v = F t,t (Φ(d + ) exp(k) Φ(d )) with d ± = k ± v T,k v T,k (53) The partial derivatives of X T,k,v can be derived as X T,k,v = F t,t ( Φ(d +) d + d + exp(k) Φ(d ) d d ) = F t,tφ(d + ) ( d + d ) = F t,tφ(d + ) v X T,k,v = X T,k,v ( 1 v d d + + ) = X T,k,v ( 1 v ( k v + v ) ( k v v )) (54) = X T,k,v ( k v 1 v 1 8 ) 13

14 Changwei Xiong, October 017 X T,k,v = F t,t (φ(d + ) d + exp(k) Φ(d ) exp(k) φ(d ) d ) = F t,t exp(k) Φ(d ) X T,k,v exp(k) φ(d ) = F t,t exp(k) Φ(d ) + F t,t = X T,k,v + X T,k,v v X T,k,v = (F t,tφ(d + ) ) = X T,k,v v d + d + = X T,k,v (1 k v ) X T,k,v = (Φ(d +) exp(k) Φ(d )) F t,t = μ TX T,k,v We ay connect the local volatility σ T,k to the iplied total variance v T,k via two steps. Firstly we bridge the σ T,k to X T,k,v by (43) using the chain rule C T,k = X T,k,v + X T,k,v, C T,k = X T,k,v + X T,k,v C T,k = ( X T,k,v = X T,k,v = X T,k,v + X T,k,v + X T,k,v ) + ( X T,k,v + X T,k,v ) + X T,k,v + X T,k,v v + X T,k,v + X T,k,v ( ) + X T,k,v v + X T,k,v ( ) (55) This gives the local volatility expressed in ters of derivatives of X T,k,v σ T,k = X T,k,v + X T,k,v ( X T,k,v + X T,k,v μ TX T,k,v ) + X T,k,v v + X T,k,v ( ) X T,k,v X T,k,v (56) Secondly we substitute the partial derivatives in (54) into (56) and reach the final equation (μ T X + X σ T,k = μ TX) X + X + X (1 k v ) + X v + X ( k v 1 v 1 8 ) ( ) σ T,k = 1 k v (k v 1 v 1 4 ) ( ) + 1 v X X (57).3.3. Equivalency in Forulas 14

15 Changwei Xiong, October 017 The σ T, in (50) is in fact equivalent to the σ T,k in (57). This can be shown as follows σ T,k = 1 k v + ( k 4v 1 4v 1 16 ) ( ) + 1 v ξ + ξτ ( + μ = ) 1 ξτ k v + (k v 1 v 1 4 ) (ξτ ) + τ (( ) + ξ + ξ ξ ) = ξ + ξτ ( + μ ) 1 + (1 k ) ξτ v + (k v v 4 ) τ ( ) + ξτ ξ (58) = ξ + ξτ ( + μ ) 1 + τd + + d +d τ ( = σ T, ) + ξτ ξ where by definition we have k = ln, v = ξ τ, d F ± = ln F t,t ± ξ τ t,t ξ τ = k v ± v, d +d = k v v 4 (59) and also have the identities = (ξ τ) = ξ + ξτ ( + ) = ξ + ξτ ( + μ ) = (ξ τ) v (ξτ = = (ξ τ) + (ξ τ) = ξτ = ξτ ) (ξτ + ) = τ (ξ + + ξ ξ ) (60) = τ (( ) + ξ + ξ ξ ) considering the fact that in (T, k)-plane the T and are no longer independent = (F t,t exp(k)) = μ T, = (F t,t exp(k)) = (61) 15

16 Changwei Xiong, October LOCAL VOLATILITY: PDE BY FINITE DIFFERENCE METHOD In this chapter, we will present a PDE based local volatility odel, in which the local volatility surface is constructed as a -D function that is piecewise constant in aturity and piecewise linear in log-oneyness (for equity) or delta (for FX). Due to great siilarity between FX and equity processes, our interest lies priarily in the context of equity derivatives, the conclusions and forulas drawn fro our discussion here are in general applicable to FX products with inor changes. In contrast to the traditional way to construct the local volatility by estiating highly sensitive and nuerically unstable partial derivatives in Dupire forulas, this ethod relies heavily on solving forward PDE s to calibrate a paraetrized local volatility surface to vanilla option prices in a bootstrapping anner. Once the local volatility surface is calibrated, the backward PDE can then be used to price exotic options (e.g. barrier options) that are in consistent with the arket observed iplied volatility surface. Before proceeding to the PDE s, it is iportant to have an overview of the date conventions for equity and equity options. The date conventions for FX products are defined in a siilar anner Date Conventions of Equity and Equity Option The diagra illustrates the date definitions for an equity and its associated option. The quantities appeared in the diagra are listed in Table 1. τ o = t o, t 0 t o, T o,p Δ o,s t o,s Δ o,p t Δ e,s t e,s t i,e t i,p Δ e,p t 0 τ e = t e, t 0 t e, T e,p 16

17 Changwei Xiong, October 017 Table 1. Dates of Equities and Options attribute sybol description reark/exaple trade date t 0 on which the equity/option is traded today equity spot lag Δ e,s equity preiu settleent lag 3D equity spot date t e,s on which the equity preiu is settled t e,s = t 0 Δ e,s equity aturity date t e, equity aturity date t e, = t 0 1Y equity pay lag 1 Δ e,p lag between t e, and t e,p e.g. sae as Δ e,s equity pay date t e,p on which the equity payoff is settled t e,p = t e, Δ e,p i-th dividend θ i dividend payent aount i-th ex- div. date t i,e ex-dividend date i-th div. pay date t i,p dividend pay date option spot lag Δ o,s option preiu settleent lag D option spot date t o,s on which the option is settled t o,s = t 0 Δ o,s option aturity date t o, option aturity date t o, = t 0 1Y option pay lag Δ o,p lag between t o, and t o,p e.g. sae as Δ o,s option pay date t o,p on which the equity payoff is settled t o,p = t o, Δ o,p day rolling rolling with convention following Following calendar defining business days and holidays US / U / H As ost of the quantities are self-explanatory, our discussion focuses ore on the treatent of equity dividends. 3.. Deterinistic Dividends In our exaple, we can assue both the short rate and the dividend rate are deterinistic and continuous, e.g. tie-dependent r u and q u as in (1). the equity forward in this case can be calculated by F(t 0, t e, ) = S(t 0 ) P q(t e,s, t e,p ) P r (t e,s, t e,p ) T P q (t, T) = exp ( q u du t where T ), P r (t, T) = exp ( r u du t ) (6) In a ore realistic ipleentation, we ay assue the underlying equity issues a series of discrete dividends with fixed aounts in a foreseeable future. It is obvious that the equity spot still follows the SDE (1) with q u = 0 in between two adjacent ex-dividend dates (There is discontinuity in 1 Equity settleent delay 17

18 Changwei Xiong, October 017 spot process on ex-dividend dates that deands special treatent. This will be discussed in detail in due course). With fixed dividends, the equity forward becoes F(t 0, t e, ) = S(t 0) i θ i P r (t e,s, t i,p ) P r (t e,s, t e,p ) for t 0 < t i,e t e, (63) where θ i is the fixed aount of the i-th dividend issued on ex-dividend date t i,e. Discrete dividend can also be odeled as proportional dividend. It assues that at each exdividend date, the dividend payent will result in a price drop in equity spot proportional to the spot level. For exaple, the equity spot before and after the dividend fall has the relationship S(t i,e + Δ) = S(t i,e Δ)(1 η i ) (64) where Δ denotes an infinitesial aount of tie and η i the proportional dividend rate at ex-dividend date t i,e. By this relationship, we can write the equity forward as F(t 0, t e, ) = S(t 0 ) i (1 η i) P r (t e,s, t e,p ) for t 0 < t i,e t e, (65) Soeties it is often ore convenient to approxiate the fixed dividends by proportional dividends. The conversion can be achieved by equating the equity forward in (63) and in (65), such that (1 η i ) i = 1 1 S(t 0 ) θ ip r (t e,s, t i,p ) i for t 0 < t i,e t e, (66) The proportional dividend η i can then be bootstrapped fro a series of fixed dividends θ i starting fro the first ex-dividend date Forward PDE In the following, our derivation is based on the spot process S t defined in (1) and its variants. For exaple, depending on the application we ay write the SDE (1) in ters of log-spot z u = ln S u or in ters of centered log-spot z u = ln S u F t,u dz u = (μ u 1 σ(u, z) ) du + σ(u, z)dw u and dz u = 1 σ(u, z) du + σ(u, z)dw u (67) 18

19 Changwei Xiong, October 017 where σ(u, z) and σ(u, z) are the local volatility function of z and z, respectively. Let s denote the forward teporal variable by u for t < u < T, the spatial variable by logoneyness k = ln F t,u noralized forward call can be defined as (as in (41)) and the spot by z = ln S u F t,u. Given that z t = 0, the value of a V u,k t,z = C u,k t,z F t,u = E [(S u ) + t, S t ] F t,u (68) Let σ u,k be the local volatility function of variable k equivalent to σ T,. We can derive the forward PDE for V u,k t,z fro (43) σ u,k = F V u,k t,z t,u + μ u F t,u V u,k t,z μ u C u,k t,z F t,u V u,k t,z F t,u V u,k t,z = V u,k t,z V u,k t,z V u,k t,z (69) V u,k t,z = σ u,k V u,k t,z ( V u,k t,z ) with initial condition V t,k t,z = C E + [(S t,k t,z t F = t,tek ) t, St ] = (1 e k ) + (70) F t,t F t,t using the partial derivatives V u,k t,z = 1 C u,k t,z μ F t,u u V u,k t,z, V u,k t,z = 1 C u,k t,z, F t,u V u,k t,z = 1 F t,u C u,k t,z (71) The PDE (69) appears drift-less and provides ore robust calibration stability at low volatility and/or high drift due to the transparency of drift in the PDE Treatent of Deterinistic Dividends A (discrete) dividend pay-out will typically result in a drop in equity price on the ex-dividend date. Suppose that tie u is the ex-dividend date, the no-arbitrage condition states that at u + the tie right after the ex-dividend date (e.g. the difference between u and u + can be infinitesial), we ust have 19

20 Changwei Xiong, October 017 S u+ = S u θ u (7) where θ u is the value of dividend issued at u (note that in a rigorous setup the value ust take into account the discounting effect due to dividend payent delay). Since a forward is expectation of spot under risk neutral easure 1, we ay write F t,u+ = E t[s u+ ] = E t[s u θ u ] = F t,u E t[θ u ] (73) Under the assuption that θ u is a fixed aount, it reads F t,u+ = F t,u θ u (74) In our finite difference ethod, the spatial grid for log-oneyness k is assued unifor such that k i k i 1 is constant for all i. Dividend payent causes discontinuity in the underlying spot. Evolving the forward PDE (69) fro initial tie t produces a state vector V u,k t,z at tie u. Iediately after the issuance of dividend at tie u +, the spot and forward drop the sae θ u aount and hence the state vector V u+,k t,z ust be realigned to reflect the dividend fall. This can be done using the option noarbitrage condition, such that C u+,k t,z = E t [(S u+ ) + ] = E t [(S u θ u F t,u+ e k ) + ] = E t [(S u F t,u e k ) + ] = C u,k t,z where k = ln F t,u + e k + θ u F t,u (75) Subsequently we can use k to interpolate fro the V u,k t,z state vector and transfor the interpolated value to for V u+,k t,z vector by V u+,k t,z = C u +,k t,z F t,u+ = C u,k t,z F t,u F t,u F t,u+ = F t,u F t,u+ V u,k t,z (76) If the dividend is proportional, we ust have spot price S u+ = S u (1 η u ) for a rate η u and hence forward price F t,u+ = F t,u (1 η u ) before and after the dividend fall. Because we can show that 1 Strictly speaking, a forward on tie T spot is an expectation of the spot under T-forward easure, i.e. F t,t = E tt [S T ]. However since the interest rate is assued deterinistic, the T-forward easure coincides with the risk neutral easure. 0

21 Changwei Xiong, October 017 V u+,k t,z = E t [(S u+ ) + ] (1 η u )E t [(S u F t,u e k ) + ] = = E t [(S u F t,u e k ) + ] = V F t,u+ F t,u (1 η u ) F u,k t,z (77) t,u the state vector reains unchanged before and after the issuance of dividend. With continuous dividend q u, the realignent of state vector is unnecessary because there is no discontinuity in equity spot Backward PDE Again we assue the spot follows the SDE (1). Without loss of generality, let s denote G(S T ) an arbitrary payoff function with paraeter, whose value is contingent on S T at tie T. One exaple of such function would be the payoff function of a call option: G(S T ) = (S T ) +. Let U u,x T, be the expectation of the function G(S T ) at tie u with spatial variable x = S u, which can be written as U u,x T, = E [G(S T ) u, x] = G(y )p T,y u,x dy R (78) where the transition probability p T,y u,x follows the ologorov backward equation (0) p T,y u,x = μ u x p T,y u,x x + σ u,x x p T,y u,x (79) x In turn, we can derive the backward PDE for the U u,x T, such that U u,x T, = G(y ) p T,y u,x dy = G(y ) (μ u x p T,y u,x x R = μ u x U u,x T, x R σ u,x x U u,x T, x + σ u,x x p T,y u,x x ) dy (80) with terinal condition U T,x T, = G(x ) (81) PDE in Centered Log-spot Assuing the spatial variable is z u = ln x at tie u, we ay write U F u,z T,k in the (u, z)-plane t,u equivalent to U u,x T,. The backward PDE (80) can then be transfored into 1

22 Changwei Xiong, October 017 U u,z T,k = σ u,z U u,z T,k ( U u,z T,k ) (8) with terinal condition U T,z T,k = G(F t,t e z F t,t e k ) (83) by using the following partial derivatives derived fro the chain rule x = 1 x, U u,x T, x = μ u, = U u,z T,k U u,x T, x = 1 U u,z T,k, x = U u,z T,k U u,x T, x + U u,z T,k = U u,z T,k U u,z T,k μ u = 1 x ( U u,z T,k U u,z T,k ) (84) Treatent of Deterinistic Dividends With fixed dividend θ u, we have S u+ = S u θ u and F t,u+ = F t,u θ u (85) The no arbitrage condition shows that for the spatial grid z U u,z T,k = E [G(S T F t,t e k ) u, F t,u e z ] = E [G(S T F t,t e k ) u +, F t,u e z θ u ] = E [G(S T F t,t e k ) u +, F t,u+ e z ] = U u+,z T,k where z = ln F t,ue z θ u F t,u+ (86) It is likely that if z is sufficiently sall (e.g. at lower boundary of spatial grid) we ay end up with F t,u e z θ u < 0, which akes the z not well defined. A solution is to floor it to a sall positive nuber, e.g. taking ax(10 10, F t,u e z θ u ). This is valid because equity spot ust be positive and the U u+,z T,k flattens as z goes to negative infinity. After the special treatent, we can use the z to interpolate fro the U u,z T,k state vector and convert the interpolated value into vector U u+,z T,k. With proportional dividend, the conclusion drawn for forward PDE still applies here and the state vector reains unchanged before and after the dividend fall. With continuous dividend, the realignent of state vector is unnecessary because there is no discontinuity in equity spot.

23 Changwei Xiong, October Vanilla Call Due to the duality between the forward and backward PDE, it is evident that vanilla calls (or puts) ust adit the identity: U t,z T,k = V T,k t,z F t,t, where U t,z T,k is the forward call solved fro backward PDE (8) and V T,k t,z the noralized forward call solved fro forward PDE (69). This relationship can be used to check the correctness of ipleentation of the nuerical engines of forward and backward PDE PDE in Log-spot For pricing soe exotic options, e.g. barrier options, it is ore convenient to use log-spot z = ln x as the spatial variable. Siilarly we can define k = ln. Let s denote the (discounted) price of a derivative product by X u,z T,k = D u,t U u,z T,k = E [D u,t G(S T e k ) u, e z ] (87) By taking into account the discount factor, it ust follow the following backward PDE X u,z T,k = r u X u,z T,k + D u,t U u,z T,k = r u X u,z T,k + D u,t ( μ u x 1 U u,z T,k x σu,z 1 U u,z T,k x ( U u,z T,k )) (88) = σ u,z X u,z T,k where the partial derivatives below have been used + ( σ u,z μ u) X u,z T,k + r u X u,z T,k x = 1 x, = 0, U u,x T, = U u,z T,k + U u,z T,k = U u,z T,k U u,x T, x U u,x T, x = U u,z T,k = x (1 x x + U u,z T,k x = 1 U u,z T,k x U u,z T,k ) = 1 U u,z T,k x + 1 U u,z T,k x x + 1 U u,z T,k x x (89) = 1 x ( U u,z T,k U u,z T,k ) 3

24 Changwei Xiong, October Treatent of Deterinistic Dividends With fixed dividend θ u, the no arbitrage condition states that X u,z T,k = E [D u,t G(S T e k ) u, e z ] = E [D u+,tg(s T e k ) u +, e z θ u ] = E [D u+,tg(s T e k ) u +, e z ] = X u+,z T,k where z = ln(e z θ u ) (90) Again, extreely sall z ay result in z that is not well defined, we ay floor the difference e z θ u to a sall positive nuber, e.g. taking ax(10 10, e z θ u ). The vector X u,z T,k can then be interpolated fro the known X u+,z T,k using the z. With proportional dividend η u, again the no arbitrage condition shows X u,z T,k = E [D u,t G(S T e k ) u, e z ] = E [D u+,tg(s T e k ) u +, e z (1 η u )] = E [D u+,tg(s T e k ) u +, e z ] = X u+,z T,k where z = z + ln(1 η u ) (91) The vector X u,z T,k can be interpolated fro the X u+,z T,k using the z. With continuous dividend, the realignent of state vector is unnecessary because there is no discontinuity in equity spot Local Volatility Surface This section is devoted to discussing the construction of local volatility surface σ(u, k). There are various ways to define the local volatility surface. The one that we would like to discuss is a -D function that is piecewise constant in aturity u and piecewise linear in log-oneyness k = ln F t,u (or in delta for FX). The volatility surface coprises a series of volatility siles σ j (k) for aturity t < u 1 < < u j < < u = T. At each aturity u j, volatility sile σ j (k) is constructed by linear interpolation between log-oneyness pillars k i = ln i F t,u for strikes 1 < < i < < n and flat extrapolation where the volatility values at k 1 and k n are used for all k < k 1 and k > k n, respectively. The sile σ j (k) constructed at u j is assued to reain constant over tie for any u between the two adjacent aturities u j 1 < u u j. 4

25 Changwei Xiong, October 017 Calibration of the local volatility surface is conducted in a bootstrapping anner starting fro the shortest aturity u 1. It is done by solving the forward PDE such that the local volatility surface is able to reproduce the vanilla call prices at the prescribed log-oneyness pillars k i for each of the aturities u j. The PDE can be solved using finite difference ethod 1 on a unifor grid defined on logoneyness k that extends to ±5 standard deviations of the underlying spot. The choice of boundary condition has little ipact to the solutions of vanilla option prices because at ±5 standard deviations the transition probability becoes negligibly sall. Our application uses linearity boundary condition for its siplicity. To allow a higher tolerance to arket data input and soother calibration process, the objective function ay include a penalty ter to suppress unfavorable concavity of a local volatility sile. Again, there can be any ways to define the objective function as well as the penalty function. In this essay, we will only focus on the siplest objective (e.g. at aturity u j ): the least square iniization of vanilla call prices argin σ j (k i ) n BS (U t,z uj,k i PDE U t,z uj,k i ) (9) where U t,z T,k is the noralized forward call price defined in (68), the superscript BS denotes the theoretical price by Black-Scholes odel and the PDE denotes the nuerical value by forward PDE. Note that without a penalty ter, the iniization can lead to an exact solution given a proper iplied volatility surface Barrier Option Pricing In contrast to the calibration, the pricing of a barrier option relies on the backward PDE (88) in line with proper terinal condition (i.e. payoff function) and boundary conditions defined by the characteristics of the barrier option. Barrier options often deand a spatial grid defined on log-spot z = ln S u, which allows an easier fit of tie-invariant barrier (e.g. with European or Aerican type of 1 A brief introduction to finite difference ethod can be found in y notes Introduction to Interest Rate Models, which can be downloaded fro. A proper iplied volatility surface should well behave and adit no arbitrage. 5

26 Changwei Xiong, October 017 observation window) into the doain. For exaple, an up-and-out barrier option would be priced on a doain with upper bound at the barrier level b where Dirichlet boundary condition is applied (the lower bound and its boundary condition reain the sae as for vanilla options). 6

27 Changwei Xiong, October 017 REFERENCES 1. Clark, I., Foreign Exchange Option Pricing - A Practitioner s Guide, Wiley-Finance, 011, pp.8. Online resource: 3. Gatheral, J., The Volatility Surface: A Practitioner s Guide, Wiley-Finance, 006, pp