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
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).
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.
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
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 ).
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)
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) <.
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.
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 = exp @ 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 0 0 11 exp @ z @tσ 2 m R T (t) + X 1 (1 e λ k (T t) )ω k Y k (t) AA. λ k=1 k
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
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.
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)
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.
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.
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 + 1 + 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.!
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.
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).
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.
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 233.0 5.612 5.331 10 4 0.0370 Table: Estimated parameters for the NIG-distribution
Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 20(23) The Ornstein-Uhlenbeck processes λ ω OU 1 0.9127 0.9224 OU 2 0.0262 0.0776 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 1.59 10 4 and for the numerical tests we then let Y 1 (t) = 1.66 10 4 and Y 2 (t) = 7.5 10 5.
Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 21(23) The variance swap results x 10 6 14 12 10 abs. error 8 6 4 2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 sigmar 2 Figure: Absolute error between the explicit and FFT-solution of the variance swap price as a function of σ R.
Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 22(23) The volatility swap results 0.035 FFT solution Brockhaus and Long approximation 0.035 FFT solution Brockhaus and Long approximation 0.03 0.03 0.025 0.025 Swap price 0.02 0.015 Swap price 0.02 0.015 0.01 0.01 0.005 0.005 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Yearly volatility 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 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
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, 1999. 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, 63:167 241, 2001. 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):177 192, 2004. Peter Carr and Dilip B. Madan. Option valuation using the Fast Fourier transform. J. Computational Finance, 2:61 73, 1998.