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
martijg@math.uio.no
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 Lévy 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 ]
(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 ) = 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 Lévy
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 Lévy 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
(7)
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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