For a Heston model, take mean reversion of 15%. Calibrate , , (t) to the prices

For a Heston model, take mean reversion of 15%. Calibrate η, ρ, θ (t) to the prices of strangles and risk reversals over 5 years (using the characteristic function method for pricing European options in Section 6.3.3.). What does the implied vol smile look like for 5 years versus 1 month? Repeat this for mean reversion of 200 %.

Section 6.3.3.

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Just as the SABR method has acquired popularity to the existence of an asymptotic expansion to allow efficient pricing of European options, a major reason for the popularity of the Heston model is Density 30 20 Volatility (%) 10 Vt = 0 the existence of a Fourier transform method to price European call and put options efficiently. A(u) = = B(u) Let us rewrite the equations in a way more amenable to inter- est rates, i.e. with a displaced diffusion process for the forward (as lognormal is not ideal for interest rates) and a stochastic volatility multiplier so that we can renormalise the parameters of the SDE: 12.3696 24.7392 37.1088 where t = bFt+ (1 -b) Fo, = 6 6 %t, dFt = zto(bFt + (1 - b) Fo)dWt, dzt = k(1zt)dt +nzdUt, and dWtdUt = pdt. Specifically, in the Heston model, the price of a European call option with strike K and expiry T is given by K' D(0, T)ET [(FT K)+] = D(0, T) 0, T) { $0 - 2T = - 49.4784 61.848 K' =bK + (1 - b) Fo, (u) = A(u)(u+1/4)B(u)vo, u R, +(u)T + 2log (n') K= K - -1.4 1 a = b0, n = |b|on, and p' = |b|p. Normalised Log Spot X 1 -s(u)T v_(u) + +(u)e-S(u)T p'n', = (u) = F(iup'n' + k) + ((u), 5(u) = /u(n)(1 (P)) + 2iun/p/k+k + roo e(-iu+1/2) log(o/K') u + 1/4 fo - (u) du v_(u) + +(u)e-s(u)T 25(u) (n') This formula is attributed to Lipton [Lip02]. To avoid disrupting the flow of the text, we shall hive off the derivation into a separate appendix at the end of this chapter. (Note that unlike the asymp- totic expansion for the implied vol of a European option under the SABR model, the Fourier transform method gives an exact price for a European option under the Heston model.) At this stage, however, it is worth remarking that the transfor- mations Et bFt + (1 - b) Fo and vt bo2t (from the original F and zt) in the formula give = det = ttdWt, dvt= k(a' - vt)dt +n'd, and }, dWtdU!= p'dt using the definitions of a', n', and p' above. Notice that this transformed set of SDEs gives the more typical formulation of the Heston model in an FX or equity setting, so our earlier definition is a generalisation. It is worth pointing out that we can extend the Heston model to include time-dependent parameters (t), n(t), and p(t) to get the system of SDES dFt = V+FtdWt, dVt = k(t)(0(t) Vt)dt + n(t)tdUt, and dWtdUt p(t)dt. Elices [Eli07] has shown that we can also price European options efficiently in this setup via Fourier transform methods. Arguably the time-dependent parameters give the necessary degrees of freedom to fit the prices of vanilla options across all expiries and for multiple strikes. However, it is not clear that bootstrapping these parameters from the prices of vanilla options will be stable, and in the interests of brevity, we shall not discuss this approach here.