Analytical formulas for local volatility model with stochastic. Mohammed Miri

Transcription

1 Analytical formulas for local volatility model with stochastic rates Mohammed Miri Joint work with Eric Benhamou (Pricing Partners) and Emmanuel Gobet (Ecole Polytechnique Modeling and Managing Financial Risks Conference, 1-13 January, 211, Paris.

2 Summary Introduction Approximation Formula Numerical Application Bibliography

3 Summary Introduction Approximation Formula Numerical Application Bibliography

4 Motivation Long term callable path dependent equity options have generated new modeling challenges. The path dependency requires consistency in the equity asset diffusion. The early exercise on long period suggests to take in account interest rates risk. Several works has been done in the case of stochastic volatility with interest rates (Piterbarg 25, Balland 25, Andreasen 26 or Haastrecht et al 28). But, few have considered local volatility model plus stochastic rates (Benhamou et al 28).

5 No arbitrage relations Under the risk neutral probability Q, one has: ds t = r t dt + σ t dwt 1, S t r t = f (, t) where t γ(s, t).γ(s, t)ds + t γ(s, t)db s, σ t is the volatility (not nessecary deterministic). r t follows the HJM framework ( γ(t, T) = T Γ(t, T)).

6 The pricing of a European option with final payoff ϕ(s T ) can be reformulated in the forward measures as follows (see Geman et coauthors 95): E[e r sds ϕ(s T )] = B(, T)E T [ϕ(f T T )] where (F T t ) is a martingale under the forward measure QT. For path dependent options, we have to use as many volatility models as many the maturities used in the options.

7 Modeling the asset We define a model on the discounted price process: S d t = e t r sds S t which is a martingale (under Q). We assume that ds d t S d t = σ d (t, S d t )dw 1 t. Equivalently, we study the log discounted process X t = log(s d t ): dx t = σ(t, X t )dw 1 t σ2 2 (t, X t)dt, X = x, (1)

8 The interest rates framework t t r t = f (, t) γ(s, t).γ(s, t)ds + γ(s, t)db s. We consider Gaussian model for interest rates, by assuming that Γ, γ : R + R + R n are deterministic functions (n is the number of Gaussian factors). The Brownian motions W 1 and B = (B 1,, B n ) are correlated: d W 1, B i t = ρ S,r i,t dt 1 i n. Hence, the price to compute is now formulated as: A = E[e r sds h( r s ds + X T )]. (2)

9 Black Formula When the volatility σ is deterministic: The equivalent volatility σ Black of r sds + X T is deterministic and is defined by: (σ Black ) 2 T = [σ 2 (t, x ) + Γ(t, T) 2 2σ(t, x )ρ S,r t.γ(t, T)]dt. Hence, the call price in such model has a closed formula like the Black Scholes formula with volatility σ Black. In tis case, we note X B T X T. This model is the proxy of our approximation.

10 Summary Introduction Approximation Formula Numerical Application Bibliography

11 Assumptions Assumption (R 5 ). The function σ is bounded and of class C 5 w.r.t x. Its derivatives up to order 5 are bounded. Under (R 5 ), we set M =max( σ,, 5 xσ ), M 1 =max( 1 σ,, 5 xσ ). Assumption (E). The function σ does not vanish and its oscillation is bounded, meaning 1 σ σ inf C E where σ inf = inf (t,x) R + R σ(t, x). Assumption (Rho). The asset is not perfectly correlated (positively or negatively) to the interest rate: ρ S,r = sup ρ S,r t < 1. t [,T]

12 Table: Historical correlation between assets and short term interest rate EUR. Period: 23-Sep-27 to 22-Sep-9. Asset Historical correlation ADIDAS 18.32% BELGACOM 4.9% CARREFOUR 7.8% DAIMLER -.94% DANONE 7.23% LVMH 4.53% NOKIA 4.37% PHILIPS 5.23%

13 The approximation formula Keep in mind that the Greeks have the following definition: Greek l (Z) = l x= (E T(Z + x)) Theorem (Second order approximation price formula). Assume that the model fulfills (R 5 ), (E) and (Rho), and that the payoff h is a call-put option, we prove that: E[e rsds h( r s ds + X T )] = B(, T)(E T [h( + r s ds + X B T )] 3 α i,t Greek h i ( r s ds + X B T ) i=1 + Resid 2 ),

14 where α 1,T = (ρ S,r α 2,T = α 1,T α 3,T, α 3,T = t.γ(t, T)σ(t, x ) σ2 (t, x ) 2 a t σ(t, x )( a t =σ(t, x ) ρ S,r t.γ(t, T). and the error is estimated by: t a s 1 xσ(s, x )ds)dt, )( a s 1 xσ(s, x )ds)dt, t Resid 2 C( h (1) ( r s ds + X B T ) 2 + sup h (1) ( r s ds + vx T + (1 v)x B T ) 2) v [,1] M σ inf 1 ρ S,r 2 M 1 M 2 ( T) 3.

15 Example In the case of homogeneous volatility σ(t, x) = σ(x) Hull and White stochastic rate (volatilility of volatility ξ and mean reversion κ) one gets: α 1,T = e 2κT σ(x )σ (1) (x ) (2ρ 2 ξ 2 + 2e κt ρ(κσ(2κt + 1) + 2ρ(κT 1)ξ)ξ 4κ 4 + e 2κT ( σ 2 T 2 κ 4 + ρσ(κt(3κt 2) 2)ξκ + 2ρ 2 (κt 1) 2 ξ 2), α 2,T = α 1,T α 3,T, α 3,T = e 2κT σ(x )σ (1) (x ) ( ρξ + e κt ( σtκ 2 + ρtξκ ρξ )) 2 2κ 4.

16 Plan of the proof Expand the model X T around the proxy model X B T. Perform a Taylor expansion for the payoff h around the proxy model by assuming h smooth enough. Estimate the corrections as a Greeks using Malliavin calculus technique. Upper bound the errors using a suitable choice of the Brownian motion used for the Malliavin differentiation and the estimates of the inverse of the Malliavin covariance. use a regularisation method in order to prove the approximation formula for call-put option.

17 Proxy model How to expand the model X T around the proxy model X B T? Suitable parameterisation: dxt ɛ = ɛ(σ(t, X ɛ )dw t σ2 (t, X ɛ ) dt), X ɛ 2 = x so that X ɛ t ɛ= + (Xɛ t ) ɛ ɛ= = X B t and X ɛ t ɛ=1 = X t. Hence using a Taylor approximation for the model: X t = X B t + 2 (X ɛ t ) 2 ɛ 2 ɛ= +

18 Perform a Taylor expansion for the payoff h around the proxy model (Xt B ): E[e T rsds h( r s ds + X T )] = E[e T rsds h( r s ds + X B T + 2 (X ɛ T ) 2 ɛ 2 ɛ= + )] = E[e rsds h( + E[e rsds h (1) ( + Resid 2 = B(, T)E T [h( + E[e rsds h (1) ( + Resid 2 r s ds + X B T )] r s ds + X B T ) 2 (X ɛ T ) 2 ɛ ɛ=] 2 r s ds + X B T )] r s ds + X B T ) 2 (X ɛ T ) 2 ɛ ɛ=] 2

19 Greeks identification E[e T rsds h (1) ( r s ds + X B T ) 2 (X ɛ T ) 2 ɛ 2 ɛ=] = B(, T)( 3 α i,t Greek (i) ( r h s ds + X B T ) How we can achieve that? This technique can be seen as an inverse procedure used in the literature about integration by parts formula and Malliavin calculus (Fournie et al 99). i=1

20 Estimates of the error In Resid 2, there is terms which constains the second derivative h (2) of the payoff function while the payoff h of interest is only one time differentiable? Lemma Assume (E), (Rho) and (R k+1 ) for a given k 1. Let Z belong to p 1 D k,p. For any v [, 1], there exists a random variable Z v in any Ł k p (p 1) such that for any function l C (R), we have E T [l (k) ( r s ds + vx T + (1 v)x B T )Z] = E T[l(v r s ds + X T + (1 v)x B T )Zv k ]. Moreover, we have Z v k p C Z k,2p ( 1 ρ S,r 2 σ inf T) k, uniformly in v and the constants C is an increasing constant on the bounds of the model.

21 Extensions The error of estimation is analyzed for other payoffs (smooth, digitals): the more the payoff is smooth, the more the error is small. Extensions to third order formula. Extension to stochastic convenience yield. This can be seen as an extension to Gibson Schwartz model to handle local volatility functions for example: ds t S t = (r t y t )dt + σdw 1 t, dy t = κ(α t y t )dt + ξ t dw 2 t, d W 1, W 2 t = ρ t dt.

22 Summary Introduction Approximation Formula Numerical Application Bibliography

23 Numerical application We consider the one factor Hull and White model for interest rates, the CEV diffusion for the spot and constant correlation ρ. Then, γ(t, T) = ξe κ(t t), σ(t, x) = νe (β 1)x. As a benchmark, we use Monte Carlo methods with a variance reduction technique (3 1 6 simulations using Euler scheme with 5 time steps per year). Parameters: β =.8, ν =.2, ξ =.7%, κ = 1%, ρ = 15% and x =.

24 Numerical Application Table: Implied Black-Scholes volatilities for the second order formula, the third order formula and the Monte Carlo simulations for maturity T = 1Y Relative Strikes 3% 6% 1% 16% 22% Second Order formula 22.99% 22.16% 21.25% 2.38% 19.77% Third Order formula 23.54% 22.25% 21.27% 2.4% 19.84% MC with control variate 23.66% 22.32% 21.34% 2.47% 19.91% MC % 22.18% 21.28% 2.43% 19.87% MC % 22.47% 21.4% 2.51% 19.94% MC- and MC+ are the bounds of the 95%-confidence interval of the Monte Carlo estimator

25 Conclusion Efficient modelisation for the hybrid model(local volatility plus stochastic rates). Non asymptotic estimates expressed by all the model parameters and analysed according to the payoff smoothness. Accurate and fast analytical formulas for the price of European options.

26 Summary Introduction Approximation Formula Numerical Application Bibliography

27 Bibliography J. Andreasen. Closed form pricing of FX options under stochastic interest rates and volatility. Global Derivatives Conference,Paris,26. Benhamou E., Gobet E. and Miri M. Analytical formulas for local volatility model with stochastic rates. Forthcoming in Quantitative Finance. Geman, H., Karoui, N. El, and Rochet, J.C. Changes of numeraire, changes of probability measure and option pricing. Journal of Applied Probability, Vol 32, (1995). Gibson, R. and Schwartz, E. S. Stochastic convenience yield and the pricing of oil contingent claims. Journal of Finance 199. Piterbarg, V. V. A multi-currency model with FX volatility skew. ssrn working paper.

28 Thank you for your attention!