Derivation of Local Volatility by Fabrice Douglas Rouah

Transcription

1 Derivation of Local Volatility by Fabrice Douglas Rouah The derivation of local volatility is outlined in many papers and textbooks (such as the one by Jim Gatheral []), but in the derivations many steps are left out. In this Note we provide two derivations of local volatility.. The derivation by Dupire [] that uses the Fokker-Planck equation.. The derivation by Derman et al. [3] of local volatility as a conditional expectation. We also present the derivation of local volatility from Black-Scholes implied volatility, outlined in []. We will derive the following three equations that involve local volatility = (S t ; t) or local variance v L = :. The Dupire equation in its most general form (appears in [] on page 9) T = C + (r T q T ) C r T C: (). The equation by Derman et al. [3] for local volatility as a conditional expected value (appears with q T = 0 in [3]) T = (r T q T ) q T C + E T js T = C : () 3. Local volatility as a function of Black-Scholes implied volatility, = (; T ) (appears in []) expressed here as the local variance v L v L = y w y + w y + 4 T 4 w + y w y : (3) where w = (; T ) T is the Black-Scholes total implied variance and y = R ln T F T where F T = exp 0 tdt is the forward price with t = r t q t (risk free rate minus dividend yield). Alternatively, local volatility can also be expressed in terms of as + T T + (r T q T ) h + y + T i: 4T + Solving for the local variance in Equation (), we obtain = (; T ) T (r T q T ) C = : (4) C

2 If we set the risk-free rate r T and the dividend yield q T each equal to zero, Equations () and () can each be solved to yield the same equation involving local volatility, namely = (; T ) = T : (5) C The local volatility is then v L = p (; T ): these equations are all explained in detail. In this Note the derivation of Local Volatility Model for the Underlying The underlying S t follows the process ds t = t S t dt + (S t ; t)s t dw t (6) = (r t q t ) S t dt + (S t ; t)s t dw t : We sometimes drop the subscript and write ds = Sdt + SdW where = (S t ; t). We need the following preliminaries: Discount factor P (t; T ) = exp r s ds : R T t Fokker-Planck equation. Denote by f(s t ; t) the probability density function of the underlying price S t at time t. Then f satis es the equation f t = S [Sf(S; t)] + S S f(s; t) : (7) Time-t price of European call with strike, denoted C = C(S t ; ) C = P (t; T )E (S T ) + (8) = P (t; T )E (S T ) (ST >) = P (t; T ) Z (S T )f(s; T )ds: where (ST >) is the Heaviside function and where E [] = E [jf t ]. In the all the integrals in this Note, since the expectations are taken for the underlying price at t = T it is understood that S = S T ; f(s; T ) = f(s T ; T ) and ds = ds T : We sometimes omit the subscript for notational convenience. Derivation of the General Dupire Equation (). Required Derivatives We need the following derivatives of the call C(S t ; t).

3 First derivative with respect to strike = P (t; T ) = P (t; T ) Z Z Second derivative with respect to strike (S T )f(s; T )ds (9) f(s; T )ds: C = P (t; T ) [f(s; T )] S= S= (0) = P (t; T )f(; T ): We have assumed that lim f(s; T ) = 0: S! First derivative with respect to maturity use the chain rule T = T P (t; T ) P (t; T ) Z Z (S T )f(s; T )ds + () (S T ) T [f(s; T )] ds: Note that P T = r T P (t; T ) so we can write (). Main Equation T = r T C + P (t; T ) Z (S T ) T In Equation () substitute the Fokker-Planck equation (7) for f t Z [f(s; T )] ds: () at t = T T + r T C = P (t; T ) (S T ) (3) S [ T Sf(S; T )] + S S f(s; T ) ds: This is the main equation we need because it is from this equation that the Dupire local volatility is derived. In Equation (3) have two integrals to evaluate Z I = T (S T ) [Sf(S; T )] ds; (4) S I = Z (S T ) S S f(s; T ) ds: Before evaluating these two integrals we need the following two identities. 3

4 .3 Two Useful Identities.3. First Identity From the call price Equation (8), we obtain C P (t; T ) = = Z Z (S T )f(s; T )ds (5) S T f(s; T )ds Z f(s; T )ds: From the expression for in Equation (9) we obtain Z f(s; T )ds = P (t; T ) : Substitute back into Equation (5) and re-arrange terms to obtain the rst identity Z S T f(s; T )ds = C P (t; T ) P (t; T ) : (6).3. Second Identity We use the expression for C.4 Evaluating the Integrals in Equation (0) to obtain the second identity C f(; T ) = P (t; T ) : (7) We can now evaluate the integrals I and I de ned in Equation (4)..4. First integral Use integration by parts with u = S T ; u 0 = ; v 0 = S [Sf(S; T )] ; v = Sf(S; T ) Z I = [ T (S T )S T f(s; T )] S= S= T Sf(S; T )ds Z = [0 0] T Sf(S; T )ds: We have assumed lim (S )Sf(S; T ) = 0. Substitute the rst identity (6) S! to obtain the rst integral I I = T C P (t; T ) + T P (t; T ) : (8) 4

5 .4. Second integral Use integration by parts with u = S T ; u 0 = ; v 0 = S S f(s; T ) ; v = S f(s; T ) S I = (S T ) S f(s; T ) S = [0 0] S f(s; T ) S= S= = f(; T ) S= S= Z where = (; T ). We have assumed that lim S! S S f(s; T ) ds S S f(s; T ) = 0. Substitute the second identity (7) for f(; T ) to obtain the second integral I.5 Obtaining the Dupire Equation I = C P (t; T ) : (9) We can now evaluate the main Equation (3) which we write as T + r T C = P (t; T ) I + I : Substitute for I from Equation (8) and for I from Equation (9) T + r T C = T C Substitute for T = r T Dupire equation () T = C + (r T q T ) C T + C q T (risk free rate minus dividend yield) to obtain the r T C: Solve for = (; T ) to obtain the Dupire local variance in its general form (; T ) = T + q T C + (r T C q ) Dupire [] assumes zero interest rates and zero dividend yield. Hence r T = q T = 0 so that the underlying process is ds t = (S t ; t)s t dw t : We obtain which is Equation (5). (; T ) = T : C 5

6 3 Derivation of Local Volatility as an Expected Value, Equation () We need the following preliminaries, all of which are easy to show S (S )+ = (S>) S (S>) = (S ) (S )+ = (S>) (S>) = (S ) = P (t; T )E (S>) C = P (t; T )E [(S )] In the table, () denotes the Dirac delta function. Now de ne the function f(s T ; T ) as f(s T ; T ) = P (t; T )(S T ) + : Recall the process for S t is given by Equation (6). By Itō s Lemma, f follows the process f df = T + f T S T + f f S T T S T ST dt + T S T dw T : (0) S T Now the partial derivatives are f T = r T P (t; T )(S T ) + ; f S T = P (t; T ) (ST >); f = P (t; T ) (S T ) : S T Substitute them into Equation (0) df = P (t; T ) () r T (S T ) + + T S T (ST >) + T ST (S T ) dt +P (t; T ) T S T (ST >) dwt Consider the rst two terms of (), which can be written as r T (S T ) + + T S T (ST >) = r T (S T ) (ST >) + T S T (ST >) = r T (ST >) q T S T (ST >): When we take the expected value of Equation (), the stochastic term drops out since E [dw T ] = 0. Hence we can write the expected value of () as dc = E [df] () = P (t; T )E r T (ST >) q T S T (ST >) + T ST (S T ) dt 6

7 so that dc dt = P (t; T )E r T (ST >) q T S T (ST >) + T ST (S T ). (3) Using the second line in Equation (8), we can write P (t; T )E S T (ST >) = C + P (t; T )E (ST >) so Equation (3) becomes dc dt = P (t; T )r T E[ (ST >)] q T C + P (t; T )E (ST >) + P (t; T )E T S T (S T ) = (r T q T ) q T C + P (t; T )E T S T (S T ) (4) where we have substituted for P (t; T )E[ (S T >)]. The last term in the last line of Equation (4) can be written P (t; T )E T S T (S T ) = P (t; T )E T S T js T = E[(S T )] = P (t; T )E T js T = E[(S T )] = E T js T = C where we have substituted C for P (t; T )E[(S T )]. We obtain the nal result, Equation () T = (r T q T ) q T C + E T js T = C : When r T = q T = 0 we can re-arrange the result to obtain E T js T = = T C which, again, is Equation (5). Hence when the dividend and interest rate are both zero, the derivation of local volatility using Dupire s approach and the derivation using conditional expectation produce the same result. 4 Derivation of Local Volatility From Implied Volatility, Equation (3) To express local volatility in terms of implied volatility, we need the three derivatives T, ; and C that appear in Equation (), but expressed in terms of 7

8 implied volatility. Following Gatheral [] we de ne the log-moneyness y = ln F T R T where F T = S 0 exp 0 tdt is the forward price ( t = r t q t, risk free rate minus dividend yield) and is the strike price, and the "total" Black-Scholes implied variance w = (; T ) T where (; T ) is the implied volatility. be written as where The Black-Scholes call price can then C BS (S 0 ; ; (; T ) ; T ) = C BS (S 0 ; F T e y ; w; T ) (5) d = and d = d p w = yw w. = F T fn(d ) e y N(d )g ln S0 + R T 0 (r t q t ) dt + w p w = yw + w (6) 4. The Reparameterized Local Volatility Function To express the local volatility Equation () in terms of y, we note that the market call price is C(S 0 ; ; T ) = C(S 0 ; F T e y ; T ) and we take derivatives. The second derivative we need is since by the chain rule A third derivative we need is The rst derivative we need is, by the chain rule y = y = : (7) C y = y y = A T + y = C + y ; y, so that y = T + T = T + T = T + y T 8 (8) = C y = C. The (9)

9 R T since = S 0 exp 0 tdt + y that so that T = T. Equation (8) implies C = C y y : Now we substitute into Equation (), reproduced here for convenience = T C + T C T y T = C y + y T C y which simpli es to T = v L C y where v L = (; T ) is the local variance. []. 4. Three Useful Identities + y T C (30) This is Equation (.8) of Gatheral Before expression the local volatility Equation () in terms of implied volatility, we rst derive three identities used by Gatheral [] that help in this regard. We use the fact that the derivatives of the standard normal cdf and pdf are, using the chain rule, N 0 (x) = n(x)x 0 and n 0 (x) = xn(x)x 0. We also use the relation n(d ) = = p e (d + p w) p e (d +d p w+w) = n(d )e d p w w = n(d )e y : From Equation (5) the rst derivative with respect to w is BS = F T [n(d )d w e y n(d )d w ] = F T n(d )e d y w + w = F T e y h n(d )w i e y n(d )d w 9

10 where d w is the rst derivative of d with respect to w and similarly for d. The second derivative is C BS = F T e y n(d )d d w w n(d )w 3 (3) = F T e y n(d )w d d w w = BS yw + w yw 3 4 w w = BS 8 w + y w : This is the rst identity we need. C BS y = F T w The second identity we need is y [ey n(d )] (3) = F T w [e y n(d ) e y n(d )d d y ] = BS [ d d y ] = BS y w where d y = w is the rst derivative of d with respect to y. To obtain the third identity, consider the derivative BS y = F T [n(d )d y e y N(d ) e y n(d )d y ] = F T e y [n(d )d y N(d ) n(d )d y ] = F T e y N(d ): The third identity we need is C BS y = F T [e y N(d ) + e y n(d )d y ] (33) = F T e y N(d ) + F T e y n(d )w = BS y + BS : We are now ready for the main derivation of this section. 4.3 Local Volatility in Terms of Implied Volatility We note that when the market price C(S 0 ; ; T ) is equal to the Black-Scholes price with the implied volatility (; T ) as the input to volatility C(S 0 ; ; T ) = C BS (S 0 ; ; (; T ); T ): (34) 0

11 We can also reparameterize the Black-Scholes price in terms of the total implied volatility w = (; T ) T and = F T e y. Since w depends on and depends on y, we have that w = w(y) and we can write C(S 0 ; ; T ) = C BS (S 0 ; F T e y ; w(y); T ): (35) We need derivatives of the market call price C(S 0 ; ; T ) in terms of the Black- Scholes call price C BS (S 0 ; F T e y ; w(y); T ). From Equation (35), the rst derivative we need is y = BS + BS y y = a(w; y) + b(w; y)c(y): (36) It is easier to visualize the second derivative we need, C y, when we express the partials in y as a; b; and c: C y = a y + a c b + b(w; y) y y + y + b c(y) (37) y = C BS y = C BS y + C BS y + C BS y The third derivative we need is T y + BS w y + y + BS w y + C BS = BS T + BS T T : = T C + BS C BS y + C BS : y y y (38) Gatheral explains that the second equality follows because the only explicit dependence of C BS on T is through the forward price F T, even though C BS depends implicitly on T through y and w. The reparameterized Dupire equation (30) is reproduced here for convenience T = v L C y + y T C: We substitute for T ; C y ; and y from Equations (38), (37), and (36) respectively and cancel T C from both sides to obtain " BS = v L C BS T y + C BS y y + BS w y + C BS y BS y + BS y : (39)

12 Now substitute for C BS ; C BS y ; and C BS y from the identities in Equations (3), (3), and (33) respectively, the idea being to end up with terms involving BS BS on the right hand side of Equation (39) that can be factored out. T = v L BS + w y Remove the factor BS " T = v L " + : y y w y + 8 from both sides and simplify to obtain y w y + w y w + y w y # w + y : w y Solve for v L to obtain the nal expression for the local volatility expressed in terms of implied volatility w = (; T ) T and the log-moneyness y = ln F T v L = y w y + w y + 4 T 4 w + y w y : 4.4 Alternate Derivation In this derivation we express the derivatives ; C ; and T in the Dupire equation () in terms of y and w = w(y), but we substitute these derivatives directly in Equation () rather than in (30). This means that we take derivatives with respect to and T, rather than with y and T: Recall that from Equation (35), the market call price is equal to the Black-Scholes call price with implied volatility as input C(S 0 ; ; T ) = C BS (S 0 ; F T e y ; w(y); T ): Recall also that from Equation (5) the Black-Scholes call price reparameterized in terms of y and w is C BS (S 0 ; F T e y ; w(y); T ) = F T fn(d ) e y N(d )g where d is given in Equation (6), and where d = d p w. The rst derivative we need is = BS y y + BS = BS + BS y : (40)

13 The second derivative is Let A = y Similarly C = BS y + BS BS + y + BS w for notational convenience. Then BS y BS y : (4) = A and = A = A y y + A = C BS y + C BS y : = C BS y (4) + C BS : (43) Substituting Equations (4) and (43) into Equation (4) produces C = BS + C BS y y + C BS y C BS + y + C BS = C BS BS y + C BS y y + C BS + BS + BS w w : (44) The third derivative we need is T = BS T + BS y y T + BS T = T C BS + BS y T + BS T ; (45) again using the fact that BS T depends explicitly on T only through F T : Now substitute for ; C ; and T from Equations (40), (44), and (45) respectively into Equation (4) for Dupire local variance, reproduced here for convenience. = T T CBS C : 3

14 We obtain, after applying the three useful identities in Section 4., T C BS + BS y T + BS T T hc BS BS y = h C BS BS y y + C BS y + C BS + BS + BS i i: w Applying the three useful identities in Section 4. allows the term BS to be factored out of the numerator and denominator. The last equation becomes T = + T h + y w + 8 w + y i: (46) w + w Equation (46) can be simpli ed by considering deriving the partial derivatives of the Black-Scholes total implied variance, h w = (; T ) T. We have T = T T +, = T, and w i = T + : Substitute into Equation (46). The numerator in Equation (46) becomes + T T + T (47) and the denominator becomes y + T w + T " # + T + : 8 w + y w Replacing w with T everywhere in the denominator produces y + T T + T 8 T + " # + T + = + T = y + T " + y y T 4 y + + y y 4 T # : (48) Substituting the numerator in (47) and the denominator in (48) back to Equation (46), we obtain + T T + T + y + T h 4 T i + 4

15 See also the dissertation by van der amp [4] for additional details of this alternate derivation. References [] Gatheral, J. (006). The Volatility Surface: A Practitioner s Guide. New York, NY: John Wiley & Sons. [] Dupire, B. (994). "Pricing With a Smile." Risk 7, pp [3] Derman, E., ani, I., and M. amal (996). "Trading and Hedging Local Volatility." Goldman Sachs Quantitative Strategies Research Notes. [4] van der amp, Roel (009). "Local Volatility Modelling." M.Sc. dissertation, University of Twente, The Netherlands. 5