Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model

Transcription

1 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 1(23) Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model Martin Groth martijg@math.uio.no Ph.D. Workshop in Mathematical Finance Oslo, October 2006

2 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 2(23) The Barndorff-Nielsen - Shephard model Stochastic volatility model proposed by Barndorff-Nielsen - Shephard [BNS01] ds(t) = (µ + βσ 2 (t))s(t) dt + p σ 2 (t)s(t) db t, S(0) = s > 0 dσ 2 (t) = λy (t) dt + dl(λt), σ 2 (0) = y > 0 on the complete filtered probability space (Ω, F, F t, P) where {F t } t 0 is the completion of the filtration σ(b s, L λs ; s t).

3 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 3(23) Superposition of non-gaussian OU-processes Let w k, k = 1, 2,..., m, be positive weights summing to one, and define mx σ 2 (t) = w k Y k (t), (1) k=1 where dy k (t) = λ k Y k (t) dt + dl k (λ k t), (2) for independent background driving Lévy processes L k. The autocorrelation function for the stationary σ 2 (t) then becomes r(u) = mx ew k exp( λ k u ), k=1 thus allowing for much more flexibility in modelling long-range dependency in log-returns.

4 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 4(23) Volatility and variance swaps The realised volatility σ R (T ) over a period [0, T ] is defined as 1 T σ R (T ) = σ T 2 (s) ds. A volatility swap is a forward contract that pays to the holder the amount c (σ R (T ) Σ) where Σ is a fixed level of volatility and the contract period is [0, T ]. The constant c is a factor converting volatility surplus or deficit into money. 0

5 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 5(23) The price of a volatility swap The fixed level of volatility Σ is chosen so that the swap has a risk-neutral price equal to zero, that is, at time 0 t T, the fixed level is given as the conditional risk-neutral expectation (using the adaptedness of the fixed volatility level): Σ(t, T ) = E Q [σ R (T ) F t ] (3) where Q is an equivalent martingale measure. As can be seen, this is nothing but a forward contract written on realised volatility. As special cases, we obtain Σ(0, T ) = E Q [σ R (T )] Σ(T, T ) = σ R (T ).

6 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 6(23) Price of general contracts In a completely similar manner, we define a variance swap to have the price Σ 2 (t, T ) = E Q [ σ 2 R (T ) F t ]. (4) and more general, for γ > 1 [ ] Σ 2γ (t, T ) = E Q σ 2γ R (T ) F t. (5)

7 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 7(23) On the way to the Esscher transform Following Benth and Saltyte-Benth [BSB04], assume θ k (t), k = 1,..., m are real-valued measurable and bounded functions. Consider the stochastic process ( m ( Z θ t (t) = exp k=1 0 t )) θ k (s) dl k (λ k s) λ k ψ k (θ k (s)) ds, 0 where ψ k (x) are the log-moment generating functions of L k (t). Condition (L): There exist a constant κ > 0 such that the Lévy measure l k satisfies the integrability condition 1 e zκ l k (dz) <.

8 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 8(23) Constructing martingale measures The processes Z θ (t) are well-defined under natural exponential integrability conditions on the Lévy measures l k which we assume to hold. That is, they are well defined for t [0, T ] if condition (L) holds for κ = sup k=1,..,m,s [0,T ] θ k (s). Introduce the probability measure Q θ (A) = E[1 A Z θ (τ max )], where 1 A is the indicator function and τ max is a fixed time horizon including all the trading times.

9 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 9(23) The key formula Let z C and θ k : R + R, k = 1,..., m be real-valued measurable functions. Suppose condition (L) is satisfied and well defined for Re(z) < [ λ 1 k T (1 e λ k(t s) )] 1 κ for all k, where κ = sup k=1,..,m,s [0,T ] θ k (s). Then 0» mx Z! ds!1 T E θ e zσ2 R (T ) zω F t = k λ k ψ k k=1 t λ k T (1 e λ k (T s) ) + θ k (s) ψ k (θ k (s)) A m R T (t) + X 1 (1 e λ k (T t) )ω k Y k (t) AA. λ k=1 k

10 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 10(23) The main result; Swap prices Proposition For every γ > 1 and any c > 0 s.t. c < [ λ 1 k T (1 e λ k(t s) )] 1 κ for all k, where κ = sup k=1,..,m,s [0,T ] θ k (s), it holds Σ 2γ (t, T ) = exp Γ(γ + 1) 2πi z T Z c+i z (γ+1) Ψ θ (t, T, z) c i m!! tσ2 R (t) + X ω k Y k (t) (1 e λ k (T t) ) dz, λ k=1 k where Ψ θ (t, T, z) = exp mx Z T λ k k=1 t zωk «ψ k 1 e λ k (T s) + θ k (s) ψ k (θ k (s)) ds«!. λ k T

11 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 11(23) The proof Proof. We know from the theory of Laplace transforms that x γ = Γ(γ + 1) 2πi c+i c i z (γ+1) e zx dz, for any c > 0 and γ > 1. Thus, under the conditions of the Proposition making the moment generating function well-defined, we have Σ 2γ (t, T ) = Γ(γ + 1) 2πi c+i c i z (γ+1) E θ [ exp ( zσ 2 R (T ) ) F t ] dz. Applying the Key Formula gives the desired result.

12 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 12(23) Explicit solution for variance swaps Proposition The variance swap has a price given by the following expression: Σ 2 (t, T ) = t T σ2 R (t) + m + m k=1 [ ωk T k=1 T t ω ( ) k 1 e λ k(t t) Y k (t)+ T λ k ψ k (θ k(s))(1 e λ k(t s) ) ds ]. (6)

13 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 13(23) Options Let f be a real-valued measurable function with at most linear growth. Then the fair price C(t) at time t of an option price paying f (Σ 2γ (τ, T )) at exercise time τ > t is given by C(t) = e r(τ t) E θ [f (Σ 2γ (τ, T )) F t ], where Σ 2γ (τ, T ) in the above proposition, with T > τ.

14 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 14(23) Using the Carr & Madan approach From Carr and Madan [CM98], after introducing an exponential damping to get a square integrable function we can represent the price of the option as where Φ(v) = C(t) = exp( α K) π 0 e iv ek Φ(v) dv (7) e iv ek E θ [ e r(τ t) e α e K ( e Σ 2(τ,T ) e e K ) + Ft ] d K.

15 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 15(23) The function Φ Φ(v) = e r(τ t) (α + 1)(α iv) exp (1 + α + iv) exp (1 + α + iv) exp mx Z τ λ k k=1 t mx k=1 ω k Y k (t) λ k T τ m T σ2 R (t) + X ω k T k=1 τ + (1 τ)e λ k (τ t) e λ k (T t)! Z!! T ψ k (θ k(s))(1 e λ k (T s) ) ds τ ψ k ωk λ k T (1 + α + iv) τ + (1 τ)e λ k (τ s) e λ k (T s) «ds where we recall ψ k ( ) to be the log-moment generating functions of the subordinators L k.!

16 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 16(23) The Brockhaus and Long approximation Brockhaus and Long [BL99] used a second-order Taylor expansion to derive swap price dynamics. Using their approach we get for BNS-model that the volatility swap price dynamics can be expressed by Σ(t, T ) = 1 p Σ2 (0, T )+ Σ 2(t, T ) 2 2 p Σ 2 (0, T ) Σ 4(t, T ) 2Σ 2 (0, T )Σ 2 (t, T ) + Σ 2 2 (0, T ) 8Σ 3/2 +R(t, T ), 2 (0, T ) where " # R(t, T ) = 1 `σ2 32 E R (T ) Σ 2 (0, T ) 3 θ `Σ2 (0, T ) + Θ `σr 2 (T ) Σ 2(0, T ) 5/2 Ft, and Θ is a random variable such that 0 < Θ < 1.

17 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 17(23) FFT The fast Fourier method is a computationally efficient way to do the discrete Fourier transform ω(k) = N e i 2π N (j 1)(k 1) x(j), for k = 1,..., N, (8) j=1 when N is a power of 2, reducing the number of multiplications from order N 2 to N ln 2 (N).

18 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 18(23) Some numerical considerations As we see from the formula we actually need to discretise σ 2 := σr 2 t/t, hence we get a time scaling of the output variable. Since FFT are restricted by sampling constraints this have the undesirable consequence that if t is small compared to T we get few data points in the domain of interest.

19 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 19(23) NIG and AstraZenica We consider the inverse Gaussian distribution, and in this case the log-moment generating function is ψ(θ) = θδ(γ 2 2θ) 1/2. α β µ δ Table: Estimated parameters for the NIG-distribution

20 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 20(23) The Ornstein-Uhlenbeck processes λ ω OU OU Table: Estimated parameters for the decay rates and weights of the OU-processes Left unknown are estimates of the current level of variance for both processes. With the parameters in Table 1 we get that the variance of the NIG distribution is and for the numerical tests we then let Y 1 (t) = and Y 2 (t) =

21 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 21(23) The variance swap results x abs. error sigmar 2 Figure: Absolute error between the explicit and FFT-solution of the variance swap price as a function of σ R.

22 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 22(23) The volatility swap results FFT solution Brockhaus and Long approximation FFT solution Brockhaus and Long approximation Swap price Swap price Yearly volatility Yearly volatility Figure: Comparison between the Brockhaus and Long approximation and the FFT-solution for the volatility swap price as a function of yearly volatility. Left:t = 1, T = 31, Right: t = 31, T = 61

23 Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 23(23) O. Brockhaus and D. Long. Volatility swaps made simple. RISK magazine, 2(1):92 95, Ole E. Barndorff-Nielsen and Neil Shepard. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. the Royal Statistical Society, 63: , Fred Espen Benth and Jurate Saltyte-Benth. The normal inverse gaussian distribution and spot price modelling in energy markets. Intern. J. Theor. Appl. Finance, 7(2): , Peter Carr and Dilip B. Madan. Option valuation using the Fast Fourier transform. J. Computational Finance, 2:61 73, 1998.