Matytsin s Weak Skew Expansion Jim Gatheral, Merrill Lynch July, Linking Characteristic Functionals to Implied Volatility In this section, we follow the derivation of Matytsin ) albeit providing more detail here than is provided in the original. First, define the characteristic functional φ T u) = The risk-neutral pdf is given by iu ln K/F ) dkpk) e pk) = C Define the implied standard deviation K) = σ BS K, T ) T and the moneyness variable z through z d = lnf/k) K) K) We can think of z as expressing how many standard deviations out-of-themoney a given strike is. To get the risk-neutral probability of exceeding the strike, we compute Now C = Nz) + N z)k K = z K z ) )
and z = K K + lnk) Substituting into equation ) gives Then Noting that we see that K = K ) + z) K = + + z) = + + z Nz) = e z π N z) = zn z) Substituting back into equation ) gives Now pk)dk = C dk C = Nz) N z) + + z = N z)dz N z)dz = N z)dz + + + z N z) z z + + z z + + z + + z dz Integration by parts gives φ T u) = dzn z) e iu +z) + iu ) 3) Here we have a formula which relates the characteristic functional to the implied volatility as a function of z. Why is this useful? Suppose the volatility skew is small. Then, z is approximately proportional to the logstrike k and z) is to first order just σ BS k).
. Change of Variable Consider the change of variable z w = z + iuz). We have z + iuz)) = z + iuz)z u z) so Then N z + iuz)) = N z)e iuz)z e u z) φ T u) = dw N w) e uu+i) w) 4) where we define so w satisfies w) = z) w = z + iuz) = lnf/k) z) = lnf/k) w) z) + iuz) + iu ) w) As Matytsin points out, we see from equation 4) that the characteristic function φ T u) is related to the analytic continuation of the implied volatility smile.. Matytsin s Weak Skew Expansion We start with equation 3) and suppose that z) = +ψz) with ) constant and ψ small. Then φ T u) dzn z) e iu e iu z iu ψ iuzψ + iu ψ ) 5) Integrating by parts, we have that dzn z) e iuz zψ = dz N z) e iuz ψ = dz N z) e iu z iu ψ + ψ } Substituting into equation 5) gives φ T u) dzn z) e iu e iu z iu ψ + iu) ψ ) 3
Integrating the first term on the right hand side exactly and rearranging gives dzn z) e iu z ψz) Let k = LogK/F ). Then e u φ T u)e iu } uu + i) z = k + Oψ) Rewriting equation 6) in terms of k gives dkn k ) e iuk e iu ψk) e u φ T u)e iu } uu + i) Taking the inverse Fourier Transform and rearranging, we obtain ψk) π e k + ) 6) du uu + i) e iuk e uu+i) φ T u) } 7) It is easy to show that this formula is equivalent to the one given in Matytsin ). So, in the weak skew limit, implied volatility as a function of log-strike k is given by a simple integral of the characteristic function φ T u). Applications.. Efficient computation of implied volatility We may apply equation 7) to the efficient computation of implied volatility as follows. We start off with an initial guess and compute the implied volatility as + ψk). Explicitly, for any choice of sufficiently close to ), we have k) + π e k + ) du uu + i) e iuk e uu+i) φ T u) } 8) It turns out that equation 8) provides a very efficient way of computing implied volatility for strikes not too far from at-the-money for any given characteristic function. 4
Note that when = k), we must have du uu + i) e iuk e uu+i) = du uu + i) e iuk φ T u) although this relationship is strictly formal because both left and right-hand sides diverge... Computing the at-the-money volatility skew Differentiating equation 8) with respect to k and evaluating at k = gives k k) k= e ) du iu) e uu+i) φ T u) } π uu + i) = [) ] i e ) du e uu+i) φ T u) } π u + i 9) Examples..3 The Heston model First, let s compute the at-the-money volatility and skew in the limit t. Denoting the Heston characteristic function by φ H u), and letting = vt as a first approximation we have that φ H u) e uu+i) + 4 uu + i) λv v) ρηviu) t } Then, performing the integration in equation 8) gives ) vt + π e ) du = vt t 4v uu + i) e λv v) )} ρηv uu+i) 4 uu + i) λv v) ρηviu) t } We recognize this as the small t expansion of the expression that we derived earlier: σbst) v v ) t e λ + v λ t 5
with λ = λ ρη and v = v λ λ. As for the at-the-money volatility skew, applying equation 9) with the choice = vt gives ) e ) du π uu + i) = ρηt 4 vt iu) e uu+i) 4 uu + i) λv v) ρηviu) t } We recognize this as another version of the by-now familiar property of the Heston model: k σ BSk) ρη k= as t Now in the limit t, with = vt we have λ φ H u) exp vtuu + i) λ iuρη + λ iuρη) + uu + i)η For small η, we can expand this further to first order in η to get φ H u) exp uu + i) vt } uu + i) vtiuρη λ Substituting into equations 8) and 9) gives and ) vt + π e ) du = vt ρη } 4λ uu + i) e uu+i) } uu + i) vtiuρη λ ) = e ) du iu) π vt ρη λ uu + i) e uu+i) } uu + i) vtiuρη λ 6
Variance Swap Pricing We can apply equation 4) to the valuation of a variance swap as follows. First, recall that [ ] E [lns T /F )] = E σt dt We then note that E [lns T /F )] = i u E [ e ] iu lns T /F ) = i u= u φ T u) From equation 3), it follows that [ ] E T σ t dt = T dwn w) w) Noting that w u= = z, we see that [ ] T E σt dt = dz N z) z) ) In principle, equation ) allows us to go directly from a functional form for the implied volatility smile to the value of a variance swap. Typically however, we parameterize the volatility smile in terms of the log-strike not z = d. Suppose that we have a parameterization of the smile in terms of the logstrike k = ln K/F ). To make the computation practical, change variables in equation ) from z to k. Then [ ] T ) E σt dt = dz N z) z) = dk N zk z k ) wk) k with z k = wk) k wk) ; wk) = σ k)t This technique is computationally very efficient! References Matytsin, Andrew,, Perturbative analysis of volatility smiles, Columbia Practitioners Conference on the Mathematics of Finance. u= u= 7