Valuing volatility and variance swaps for a non-Gaussian Ornstein

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

Valuing volatility and variance swaps for a non-Gaussian Ornstein-Uhlenbeck stochastic volatility model Martin Groth [email protected]

Ph.D. Workshop in Mathematical Finance Oslo, October 2006

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

The Barndorff-Nielsen - Shephard model

Stochastic volatility model proposed by Barndorff-Nielsen Shephard [BNS01] dS(t)

=

(µ + βσ 2 (t))S(t) dt +

dσ 2 (t)

=

−λY (t) dt + dL(λt),

p σ 2 (t)S(t) dBt ,

S(0) = s > 0

σ 2 (0) = y > 0

on the complete filtered probability space (Ω, F, Ft , P) where {Ft }t≥0 is the completion of the filtration σ(Bs , Lλs ; s ≤ t).

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

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Superposition of non-Gaussian OU-processes Let wk , k = 1, 2, . . . , m, be positive weights summing to one, and define m σ 2 (t) =

X

wk Yk (t),

(1)

k=1

where dYk (t) = −λk Yk (t) dt + dLk (λk t),

for independent background driving L´evy processes Lk . The autocorrelation function for the stationary σ 2 (t) then becomes r (u) =

m X

ek exp(−λk |u|), w

k=1

thus allowing for much more flexibility in modelling long-range dependency in log-returns.

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

Volatility and variance swaps The realised volatility σR (T ) over a period [0, T ] is defined as s Z 1 T 2 σR (T ) = σ (s) ds. T 0 A volatility swap is a forward contract that pays to the holder the amount c (σR (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.

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

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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 ) = EQ [σR (T ) | Ft ]

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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 ) = EQ [σR (T )] Σ(T , T ) = σR (T ).

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

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Price of general contracts

In a completely similar manner, we define a variance swap to have the price   Σ2 (t, T ) = EQ σR2 (T ) | Ft . (4) and more general, for γ > −1 h i Σ2γ (t, T ) = EQ σR2γ (T ) | Ft .

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

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On the way to the Esscher transform Following Benth and Saltyte-Benth [BSB04], assume θk (t), k = 1, . . . , m are real-valued measurable and bounded functions. Consider the stochastic process θ

Z (t) = exp

m Z X k=1

t

Z θk (s) dLk (λk s) −

0

t

λk ψk (θk (s)) ds



0

where ψk (x) are the log-moment generating functions of Lk (t). Condition (L): There exist a constant κ > 0 such that the L´evy measure `k satisfies the integrability condition Z ∞ e zκ `k (dz) < ∞. 1

,

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

Constructing martingale measures The processes Z θ (t) are well-defined under natural exponential integrability conditions on the L´evy measures `k which we assume to hold. That is, they are well defined for t ∈ [0, T ] if condition (L) holds for κ = supk=1,..,m,s∈[0,T ] |θk (s)|. Introduce the probability measure Q θ (A) = E[1A Z θ (τmax )], where 1A is the indicator function and τmax is a fixed time horizon including all the trading times.

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

The key formula Let z ∈ C and θk : R+ −→ R, k = 1, . . . , m be real-valued measurable functions. Suppose condition (L) is satisfied and well λ−1

defined for |Re(z)| < [ Tk (1 − e −λk (T −s) )]−1 κ for all k, where κ = supk=1,..,m,s∈[0,T ] |θk (s)|. Then 0 – » m X zσ 2 (T ) Eθ e R | Ft = exp @ λk

Z T t

k=1

0 × exp @

z T

ψk

0 @tσ 2 (t) +

zωk λk T

(1 − e

m X 1

R

k=1

λk

−λk (T −s)

! ) + θk (s)

!1 − ψk (θk (s)) ds A 11

(1 − e

−λk (T −t)

)ωk Yk (t)AA .

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

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The main result; Swap prices Proposition λ−1

For every γ > −1 and any c > 0 s.t. c < [ Tk (1 − e −λk (T −s) )]−1 κ for all k, where κ = supk=1,..,m,s∈[0,T ] |θk (s)|, it holds Σ2γ (t, T ) =

Γ(γ + 1) 2πi

× exp

c+i∞

Z

z −(γ+1) Ψθ (t, T , z)

c−i∞

z T

tσR2 (t)

„Z

T

m X ωk Yk (t) + (1 − e−λk (T −t) ) λk k=1

!! dz ,

where Ψθ (t, T , z) = exp

m X k=1

λk

„ ψk

t

« « ” zωk “ 1 − e−λk (T −s) + θk (s) − ψk (θk (s)) ds λk T

! .

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

The proof Proof. We know from the theory of Laplace transforms that Γ(γ + 1) x = 2πi γ

Z

c+i∞

z −(γ+1) ezx dz ,

c−i∞

for any c > 0 and γ > −1. Thus, under the conditions of the Proposition making the moment generating function well-defined, we have Γ(γ + 1) Σ2γ (t, T ) = 2πi

Z

c+i∞


 z −(γ+1) Eθ exp zσR2 (T ) | Ft dz .

c−i∞

Applying the Key Formula gives the desired result.

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

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Explicit solution for variance swaps

Proposition The variance swap has a price given by the following expression: m  X t 2 ωk  Σ2 (t, T ) = σR (t) + 1 − e −λk (T −t) Yk (t)+ T T λk k=1 Z m h i X ωk T 0 ψk (θk (s))(1 − e −λk (T −s) ) ds . + T t k=1

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

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 (Σ2γ (τ, T )) at exercise time τ > t is given by C (t) = e −r (τ −t) Eθ [f (Σ2γ (τ, T )) | Ft ], where Σ2γ (τ, T ) in the above proposition, with T > τ .

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

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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 e) exp(−αK C (t) = π

Z

∞

e−iv K Φ(v ) dv e

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0

where Z

∞

Φ(v ) = −∞

    e e e + e. eiv K Eθ e−r (τ −t) eαK e Σ2 (τ,T ) − eK | Ft dK

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

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The function Φ

Φ(v ) =

e−r (τ −t) (α + 1)(α + 1 + iv ) × exp

(1 + α + iv )

× exp

(1 + α + iv )

× exp

m X k=1

τ

Z λk t

! m ” X ωk Yk (t) “ τ + (1 − τ )e−λk (τ −t) − e−λk (T −t) λk T k=1

!! Z T m X τ 2 ωk ψk0 (θk (s))(1 − e−λk (T −s) ) ds σR (t) + T T τ k=1 ! „ “ ”« ωk −λk (τ −s) −λk (T −s) ψk (1 + α + iv ) τ + (1 − τ )e −e ds λk T

where we recall ψk (·) to be the log-moment generating functions of the subordinators Lk .

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

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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 ) =

Σ4 (t, T ) − 2Σ2 (0, T )Σ2 (t, T ) + Σ22 (0, T ) 1p Σ2 (t, T ) − Σ2 (0, T )+ p +R(t, T ) , 3/2 2 2 Σ2 (0, T ) 8Σ (0, T ) 2

where 1 R(t, T ) = Eθ 32

"

# ` 2 ´3 σR (T ) − Σ2 (0, T ) ` ` ´´5/2 | Ft , Σ2 (0, T ) + Θ σR2 (T ) − Σ2 (0, T )

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

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FFT

The fast Fourier method is a computationally efficient way to do the discrete Fourier transform ω(k) =

N X

e −i

2π (j−1)(k−1) N

x(j), for k = 1, . . . , N,

j=1

when N is a power of 2, reducing the number of multiplications from order N 2 to N ln2 (N).

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

Some numerical considerations

As we see from the formula we actually need to discretise σ e2 := σR2 × 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.

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

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 233.0

β 5.612

µ −5.331 × 10−4

δ 0.0370

Table: Estimated parameters for the NIG-distribution

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

The Ornstein-Uhlenbeck processes

OU1 OU2

λ 0.9127 0.0262

ω 0.9224 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 Y1 (t) = 1.66 × 10−4 and Y2 (t) = 7.5 × 10−5 .

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

The variance swap results −6

x 10 14

12

abs. error

10

8

6

4

2

0 0

0.05

0.1

0.15

0.2

0.25 0.3 sigmaR2

0.35

0.4

0.45

0.5

Figure: Absolute error between the explicit and FFT-solution of the variance swap price as a function of σR .

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

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The volatility swap results 0.035

0.035 FFT−solution Brockhaus and Long approximation 0.03

0.025

0.025

0.02

Swap price

Swap price

FFT−solution Brockhaus and Long approximation 0.03

0.015

0.02

0.015

0.01

0.01

0.005

0.005

0

0

0.1

0.2

0.3

0.4 0.5 Yearly volatility

0.6

0.7

0.8

0

0

0.1

0.2

0.3

0.4 0.5 Yearly volatility

0.6

0.7

0.8

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

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.

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