Calibration Lecture 4: LSV and Model Uncertainty

Transcription

1 Calibration Lecture 4: LSV and Model Uncertainty March 2017

2 Recap: Heston model Recall the Heston stochastic volatility model ds t = rs t dt + Y t S t dw 1 t, dy t = κ(θ Y t ) dt + ξ Y t dw 2 t, where Y t is the variance process of the asset. ξ > 0 governs the volatility of variance, and κ, θ > 0 the mean reversion. The Wiener processes Wt 1, Wt 2 have correlation ρ. A (slightly idealised) hedging argument gives the pricing PDE [ ] V t +1 S 2 Y 2 V 2 S 2 + 2ργYS 2 V S Y + ξ2 Y 2 V Y 2 +rs V V +κ(θ Y ) S Y rv = 0 The model does not fit vanillas for all strikes and maturities, but captures leverage (negative correlation between spot and vol) and has sensible dynamic properties (reasonably stable under re-calibration and for hedging).

3 Recap: Local volatility Also recall the local volatility model ds S = µ dt + σ(s, t) dw. The function (S, t) σ(s, t) is called local volatility. The risk neutral transition density satisfies p T = S 2 (σ(s, T ) 2 S 2 p ) and the option values the Dupire PDE V T ( rs S p ) + rk V K = 1 2 K 2 σ 2 (K, T ) 2 V K 2. The model can exactly fit vanillas with different strikes and maturities, but is unstable in the calibration and dynamically inconsistent (mismatch of re-calibrated surfaces).

4 Stochastic local volatility Combining the above, we get a model of the form ds t S t = r dt + σ(s t, t) Y t dw 1 t, dy t = κ(θ Y t ) dt + ξ Y t dw 2 t, with a stochastic volatility component Y t and leverage function σ. Flexibility of local volatility for pricing, but better dynamics. A fairly flat σ suffices to correct the Heston model. Can use this for a two step calibration procedure: first fix the Heston parameters, then fine-tune by boot-strapping the function σ; see later (as per Ren, Madan and Qian (2007)).

5 The Gyöngy (1986) result Consider a general Itô process (under some technical conditions) ds t = µ t dt + σ t dw t. Then the Markov process dx t = a(x t, t) dt + b(x t, t) dwt X, b 2 (x, t) = E[σt 2 S t = x], a(x, t) = E[µ t S t = x], has the same marginal distribution, S t law = Xt, for all t. For the SLV model to be consistent with European call/put option prices, it has to hold that σ 2 LV (K, T ) = σ(k, T )2 E [ Y T S T = K ], where σ LV is the Dupire local volatility function calibrated to the calls and puts.

6 The joint density and forward equation To find ψ(k, T ) = E [ Y T S T = K ], need the two-dimensional KFE (Kolmogorov forward equation) p t xp y)p +r +κ (θ x y Can then compute = (σ 2 yp) x (σξyρp) (ξ 2 yp) x y 2 y 2 ψ(x, t) = 0 yp dy 0 p dy. Know that ψ 2 (K, 0) = 1. A complex highly non-linear calibration problem. In practice, iterative solution or stepwise decoupling (see Ren et al. paper).

7 Finite difference approximation Apply finite differences δ x f = f m i+1,j f m i 1,j 2 x δ y f = f i,j+1 m f i,j 1 m 2 y, δ 2 xf = f m i+1,j f m i,j + f m i 1,j x 2, etc, to obtain, e.g. implicit (forward equation!) P m i,j Pm 1 i,j t + r(δ x XP m ) i,j + κ(δ y (θ Y )P m ) i,j = 1 2 (δ2 xσ 2 X 2 YP m ) i,j + (δ x δ y σξy ρp m ) i,j (δ2 y ξ 2 YP m ) i,j, where X = (0, x,..., x max ), Y = (0, y,..., y max ), and, e.g., (δ y (θ Y )P m ) i,j = (θ y j+1)pi,j+1 m (θ y j 1)Pi,j 1 m. 2 y

8 Finite difference approximation The genuinely new term is the cross term. Discretisation matrices have 9 non-zero elements per line, not tridiagonal. Direct solution of discretised systems costly ( fill-in, i.e. destruction of sparsity see next slide for a possible solution). Need to solve problem for Dirac data, which causes stability problems for Crank-Nicolson (recall the Rannacher start-up). Depending on the parameters, the y-density can be singular near zero (if the Feller condition is not satisfied often the case in practice).

9 ADI splitting Split every timestep in three steps and approximate by: an explicit step involving the cross-term: P m,0 i,j P m 1 i,j t an implicit split-step in x-direction, = (δ x δ y σξy ρp m 1 ) i,j, P m,1 i,j P m,0 i,j + r(δ x XP m,1 ) i,j = 1 t 2 (δ2 xσ 2 X 2 YP m,1 ) i,j, an implicit split-step in y-direction, P m i,j Pm,1 i,j t + κ(δ y (θ Y )P m ) i,j = 1 2 (δ2 y ξ 2 YP m ) i,j. Extensions for better stability and accuracy to Douglas scheme, Craig-Sneyd scheme, etc.

10 Stepwise decoupling Write this as P m = L(σ(, t m 1 ))P m 1 with a linear operator L depending on σ at time t m 1. From this, approximate J ψ(s i, t m ) ψi m j=1 Pm i,j Y j y J j=1 Pm i,j y Compute the local volatility, e.g. from a FD approximation of the Dupire formula for calls with strike K i and maturity t m : σ LV (K i, t m ) 2 δ T C(K i, t m ) + rk i δ K C(K i, t m ) K 2 δk 2 C(K i, t m ) Now compute the leverage function at t m σ(s i, t m ) = σ LV (S i, t m ) ψ m i.

11 Assume the Heston parameters have been fitted by a rough calibration. Then the above algorithm fits the leverage function exactly (up to numerical errors) to all call prices. No theoretical convergence results. The calibrated model can then be used for hedging, and for pricing of exotics (e.g., forward start options).

12 Real-world FX calibration We use the calibrated Dupire local volatility σ LV and EUR-USD market data from Lecture 3. σ (s, 0) = σ LV (s,0) v0 T = 0 1. solve the forward Kolmogorov PDE for density p on [T, T + T ] with σ (s, [T, T + T ]) = σ (s, T ) 2. For each K in a grid E VT = 0 vp (K, v, T + T ) dv 0 p(k, v, T + T ) dv σ (K, T + T ) = σ LV (K, T + T ) EVT Set T = T + T and loop to 1. until reaching the last maturity.

13 Calibrated leverage function The maximum absolute calibration error is 0.02 % (i.e. for a volatility of 20 % the calibrated volatility can be 20 ± 0.02 % in the worst-case scenario)

14 Calibration fit Calibration fit: pure Heston model (SV) vs LSV model.

15 Model uncertainty Even if a model calibrates perfectly to observed vanilla prices, it normally will not fit exotics prices. Moreover, a hedging strategy based on the model Greeks will not give a perfect replication even of the vanilla prices the model is calibrated to. We refer to this as model uncertainty. The next two sections analyse pricing and hedging uncertainty.

16 Pricing under uncertainty A local volatility model, jump diffusion model, and (Heston) stochastic volatility model calibrated to 60 observed European S&P 500 calls for different strike/maturity pairs within 3 basis points σ = 0.10 σ (volatility) t (time in years) S (asset price) frequency jump size Parameter rate of long run volatility of correlation initial reversion variance volatility variance Value The value of an up-and-out barrier call with strike 90% and barrier 110% of the spot varies by 177 basis points.

17 Hedging under uncertainty El Karoui et al. assume the true volatility process σ t is unknown. A trader uses instead observations of calls C t to back out an implied volatility process σ t such that C BS (t, S t, σ t ) = C t. The trader uses this volatility to delta hedge by setting up a portfolio Π t = φ (1) t B t + φ (2) t S t, where φ (2) t = C BS ( σ). S Then the hedging error at T is Π T C(T, S T ) = 1 2 T 0 e r(t t) S 2 t Γ 2 t ( σ 2 t σ 2 t ) dt.

18 References A. Hirsa, Computational methods in finance, CRC Press, J. Guyon and P. Henry-Labordère, Non-linear option pricing, CRC Press, Y. Ren, D. Madan, and M. Qian Qian, Calibration and pricing with embedded local volatility models, Risk, N. El Karoui, M. Jeanblanc-Picqué, and S. Shreve, Robustness of the Black and Scholes formula, Mathematical Finance 8.2, , 1998.