Asymptotic Formulas in Analytically Tractable …Asymptotic Formulas in Analytically Tractable...
Transcript of Asymptotic Formulas in Analytically Tractable …Asymptotic Formulas in Analytically Tractable...
Asymptotic Formulas in Analytically Tractable
Stochastic Volatility Models
Archil Gulisashvili
Ohio University
(joint work with E. M. Stein)
Conference on small time asymptotics, perturbation theory and
heat kernel methods in mathematical finance
Vienna, Austria
February 10-12, 2009
1
Samuelson’s model of the stock price
In their celebrated work on pricing of options, Black and Scholes used
Samuelson’s model of the stock price. Samuelson suggested to de-
scribe the random behavior of the stock price by a diffusion process
Xt satisfying the following stochastic differential equation:
dXt = µXtdt + σXtdWt, X0 = x0,
where µ is a real constant, σ is a positive constant, and W is a stan-
dard Brownian motion. The constants µ ∈ R1 and σ > 0 are called
the drift and the volatility of the stock, respectively.
Explicit formula
The following formula holds for the stock price process in Samuel-
son’s model:
Xt = x0 exp
{(µ− 1
2σ2
)t + σWt
}.
The process X is called a geometric Brownian motion.
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Stock price distribution density
The distribution function of the stock price Xt is given by
r 7→ P (Xt < r) , 0 ≤ r <∞,
where P stands for the Wiener measure.
The distribution density of Xt (if it exists) is a function Dt on
[0,∞) such that
P (Xt < r) =
∫ r
0
Dt(u)du
for all r ≥ 0.
Stock price distribution in Samuelson’s model
The distribution density of the stock price process Xt in
Samuelson’s model can be computed explicitly. We have
Dt(x) =1√
2πtσxexp
−
(log x
x0−(µ− 1
2σ2)t)2
2tσ2
.
Such densities are called log-normal.
3
Black-Scholes model
The Black-Scholes formula for the price of a European call option at
t = 0 under the risk-free measure P∗ is the following:
CBS = x0N (d1) −Ke−rTN (d2) ,
where
d1 =log x0 − logK +
(r + 1
2σ2)T
σ√T
,
d2 = d1 − σ√T ,
and
N(z) =1√2π
∫ z
−∞exp
{−y
2
2
}dy.
Here T stands for the expiration date, K is the strike price, and
r denotes the interest rate.
4
Constant volatility assumption
It is assumed in the Black-Scholes model that the volatility of the
stock is constant. However, it has been established that the implied
volatility K 7→ I(K) for a European call option in uncorrelated
stochastic volatility models is decreasing on the interval [0, x0erT ]
and increasing on the interval [x0erT ,∞) (Renault-Touzi). This is
the so-called “volatility smile” effect. It contradicts the constant
volatility assumption.
Stochastic volatility models
To improve the Black-Scholes model, various stochastic volatility
models have been developed during the last decades, e.g., the Hull-
White model where the volatility is a geometric Brownian motion, the
Stein-Stein model where the absolute value of an Ornstein-Uhlenbeck
process is used as the volatility process, and the Heston model where
the volatility is a Cox-Ingersoll-Ross process (a Feller process).
General stochastic volatility models
{dXt = µXtdt + f (Yt)XtdWt
dYt = b (t, Yt) dt + σ (t, Yt) dZt
Here, µ ∈ R1; b and σ are continuous functions on [0, T ] × R
1; W
and Z are independent one-dimensional standard Brownian motions;
and f is a nonnegative function on R1. The process X plays the role
of the stock price process, while f (Yt), t ≥ 0, is the volatility process.
The initial conditions for the processes X and Y will be denoted by
x0 and y0, respectively. We also assume that the second equation in
the model above has a unique strong solution Y .
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Explicit formula
Under certain restrictions, the following formula holds for the stock
price process Xt:
Xt = x0 exp
{µt− 1
2
∫ t
0
f (Ys)2ds +
∫ t
0
f (Ys) dWs
}.
Distribution densities
We will denote by Dt the distribution density of the stock price
Xt, and by mt the mixing distribution density associated with the
volatility process f (Yt), that is, the distribution density of the ran-
dom variable
αt =
{1
t
∫ t
0
f (Ys)2ds
}12
.
Special models. The Hull-White model
The stock price process X and the volatility process Y in the Hull-
White model satisfy the following system of stochastic differential
equations:
{dXt = µXtdt + YtXtdWt
dYt = νYtdt + ξYtdZt.
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Special models. The Heston model
The stock price process X and the volatility process Y in the Heston
model satisfy the following system of stochastic differential equations:
{dXt = µXtdt +
√YtXtdWt
dYt = (a + bYt) dt + c√YtdZt.
We consider the case where µ ∈ R, a ≥ 0, b ≤ 0, and c > 0. It
is often assumed that the volatility equation in the model above is
written in a mean-reverting form. Then the Heston model becomes
{dXt = µXtdt +
√YtXtdWt
dYt = r (m− Yt) dt + c√YtdZt,
where r ≥ 0, m ≥ 0, and c > 0. The volatility process in the Heston
model is called an Cox-Ingersoll-Ross process (a CIR process).
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Special models. The Stein-Stein model
The stock price process X and the volatility process Y in the Stein-
Stein model satisfy the following system of stochastic differential
equations:
{dXt = µXtdt + |Yt|XtdWt
dYt = q (m− Yt) dt + σdZt.
An important special case is the following: µ ∈ R, q ≥ 0, m = 0,
and σ > 0. Then the Stein-Stein model becomes
{dXt = µXtdt + |Yt|XtdWt
dYt = −qYtdt + σdZt.
The solution to the second stochastic differential equation in the
model above is an Ornstein-Uhlenbeck process, for which the long
run mean equals zero. Such Ornstein-Uhlenbeck processes and CIR-
processes and can be dealt with in a similar way, using Bessel pro-
cesses.
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Explicit formula
The following formula holds for the stock price process in the
general stochastic volatility model described above:
Xt = x0 exp
{µt− 1
2
∫ t
0
f (Ys)2ds +
∫ t
0
f (Ys) dWs
}.
Integral representation
Dt(x) =1
x0eµt
∫ ∞
0
L
(t, y,
x
x0eµt
)mt(y)dy,
where L is the log-normal density defined by
L(t, y, v) =1√
2πtyvexp
−
(log v + ty2
2
)2
2ty2
.
It follows that
Dt(x) =
√x0eµt√2πt
x−32
∫ ∞
0
y−1mt(y) exp
{−[
1
2ty2log2 x
x0eµt+ty2
8
]}dy.
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Symmetry property
Dt
(x0e
µtx)
= x−3Dt
(x0e
µt
x
).
Asymptotic behavior of the mixing distribution
The Hull-White model. A special case:
mt
(y;
1
2, 1, 1
)
= c1yc2 (log y)c3 exp
{− 1
2t
(log y +
1
2log log y
)2}
(1 + O
((log y)−
12
)), y → ∞.
Formulas for the constants:
c1 =1√πt
2−12t exp
{−(log 2)2
2t
}, c2 = −1 − 1 − 2 log 2
2t,
and c3 = −1 + 2 log 2
4t.
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The law of the time integral. A special case:
m(0)T denotes the distribution density of the following time integral
of a geometric Brownian motion:
A(ρ)T =
∫ T
0
exp {2 (ρu + Zu)} du.
The next formula characterizes the asymptotic behavior of this den-
sity:
m(0)T (y) = c12
−1−c3T−c2+12 y
c2−12
(log
y
T
)c3
exp
{− 1
2T
(log
√y
T+
1
2log log
√y
T
)2}
(1 + O
((log y)−
12
)), y → ∞,
where c1, c2, and c3 are as above with t replaced by T .
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The Hull-White model. General case:
mt (y; ν, ξ, y0) = c1yc2(log y)c3
exp
{− 1
2tξ2
(log
y
y0+
1
2log log
y
y0
)2}
(1 + O
((log y)−
12
)), y → ∞.
Formulas for the constants:
c1 =1
ξ√πt
2− 1
2tξ2y
2 log 2−1
2tξ2−α
0 exp
{−(log 2)2
2tξ2
}exp
{−α
2ξ2t
2
},
c2 = α− 1 +1 − 2 log 2
2tξ2,
c3 =α
2− 1 + 2 log 2
4tξ2,
where
α =2ν − ξ2
2ξ2.
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The law of the time integral. General case:
The following formula holds for the distribution density of the time
integral of a geometric Brownian motion:
m(ρ)T (y) = C12
−1−C3T−C2+12 y
C2−12
(log( yT
))C3
exp
{− 1
2T
(log
√y
T+
1
2log log
√y
T
)2}
(1 +O
((log y)−
12
)), y → ∞.
Constants:
C1 =1√πT
2−1
2T exp
{−(log 2)2
2T
}exp
{−ρ
2T
2
},
C2 = ρ− 1 +1 − 2 log 2
2T,
C3 =ρ
2− 1 + 2 log 2
4T.
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Asymptotic behavior of the stock price density
Hull-White model. Special case:
Dt
(x; 0,
1
2, 1, 1, 1
)= c12
c2−1−2c32 t−
c2+12 x−2
(log x)c2−1
2
(log
[2 log x
t
])c3
exp
− 1
2t
(log
√2 log x
t+
1
2log log
√2 log x
t
)2
(1 + O
((log log x)−
12
)), x→ ∞,
where the constants c1, c2, and c3 are as above.
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Hull-White model. General case:
Let −∞ < µ <∞, −∞ < ν <∞, ξ > 0, x0 > 0,
y0 > 0, and t > 0. Then
Dt
(x0e
µtx;µ, ν, ξ, x0, y0
)=
c1
x0eµt2
c2−1−2c32 t−
c2+12 x−2
(log x)c2−1
2
(log
[2 log x
t
])c3
exp
− 1
2tξ2
(log
[1
y0
√2 log x
t
]+
1
2log log
[1
y0
√2 log x
t
])2
(1 +O
((log logx)−
12
))
as x→ ∞.
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Asymptotic behavior of the mixing distribution
Heston model:
For a ≥ 0, b ≤ 0, and c > 0, there exist A > 0, B > 0, and C > 0
such that
mt (y; a, b, c, y0) = Ay−1
2+2a
c2 eBye−Cy2(1 + O
(y−
12
)), y → ∞.
Formulas for the constants in the case b = 0:
A =2
14+ a
c2y14− a
c2
0
c√t
exp
{4y0
c2t
},
B =2√
2y0π
c2t, and C =
π2
2c2t.
Asymptotic behavior of the stock price distribution
Heston model.
For a ≥ 0, b ≤ 0, and c > 0, there exist A1 > 0, A2 > 0, and
A3 > 0 such that for every δ with 0 < δ < 12,
Dt
(x0e
µtx)
= A1(log x)−3
4+ a
c2eA2
√log xx−A3
(1 + O
((log x)−δ
)), x→ ∞.
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Asymptotic behavior of the stock price distribution
Stein-Stein model.
For q ≥ 0, m = 0, and σ > 0, there exist B1 > 0, B2 > 0, and
B3 > 0 such that for every δ with 0 < δ < 12,
Dt
(x0e
µtx)
= B1(log x)−12eB2
√log xx−B3
(1 +O
((log x)−δ
)), x→ ∞.
Conclusions. Fat tails.
Hull-White: The distribution density Dt of the stock price in the
Hull-White model decays extremely slowly. This is the so-called
“fat tail” effect. The function x 7→ Dt(x) behaves near infinity like1x2 times certain logarithmic factors. In addition, the function
x 7→ Dt(x) behaves near zero like 1x
times logarithmic factors
making the density integrable near zero. In a sense, the stock price
distribution density in the Hull-White model has the slowest decay
among stock price distribution densities in similar stochastic
volatility models.
Heston: The density Dt(x) behaves at infinity roughly like the
function x−A3 and at zero as the function xA3−3. The number A3 is
strictly greater than 2.
Stein-Stein: The density Dt(x) behaves at infinity roughly like
the function x−B3 and at zero like the function xB3−3. The number
B3 is strictly greater than 2.
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Pricing functions for European call options
Stochastic volatility model:{dXt = rXtdt + f (Yt)XtdW
∗t
dYt = b (t, Yt) dt + σ (t, Yt) dZ∗t
under a risk-free measure P∗.
The price of a European call option at t = 0 is given by the following
formula:
C(K) = E∗ [e−rT (XT −K)+
].
It is clear that
C(K) = e−rT∫ ∞
K
xDT (x)dx − e−rTK
∫ ∞
K
DT (x)dx.
Implied volatility
The implied volatility in an option pricing model is the volatility in
the Black-Scholes model for which the corresponding Black-Scholes
price of the option is equal to its price in the model under consid-
eration. More precisely, for K > 0 the implied volatility I(K) is
determined from the equality
CBS(K, I(K)) = C(K).
The implied volatility can be considered as a function of the log-strike
k = log K
x0erT . In terms of k, the definition is as follows:
I(k) = I(K), −∞ < k <∞, 0 < K <∞.
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Asymptotic behavior of the implied volatility
Suppose there exist positive increasing continuous functions ψ and
φ such that
limK→∞
ψ(K) = limK→∞
φ(K) = ∞
and
C(K) ≈ ψ(K)
φ(K)exp
{−φ(K)2
2
}.
Then the following asymptotic formula holds:
I(K) =1√T
(√2 logK + φ(K)2 − φ(K)
)+O
(ψ(K)
φ(K)
)
as K → ∞.
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Hull-White model
Let ψ be a positive increasing continuous function such that
limK→∞
ψ(K) = ∞. Then
I(k) =
√2√Tk
12 − 1
2Tξlog k − 1
2Tξlog log k
+ (α + 2A + 2C)ξ − 1
2Tξ
log log k
log k+O
(ψ(k)
log k
)
as k → ∞, where
α =2ν − ξ2
2ξ2, A =
1 − 2 log 2
4Tξ2,
and
C =2 log y0 + logT
2Tξ2.
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Heston model
Let ψ be a positive increasing continuous function such that
limK→∞
ψ(K) = ∞. Then
I(k) = β1k12 + β2 + β3
log k
k12
+O
(ψ(k)
k12
)
as k → ∞, where
β1 =
√2√T
(√A3 − 1 −
√A3 − 2
),
β2 =A2√2T
(1√
A3 − 2− 1√
A3 − 1
),
β3 =1√2T
(1
4− a
c2
)(1√
A3 − 1− 1√
A3 − 2
),
and A3 is the constant that appears in the asymptotic formula for
the distribution density of the stock price in the Heston model.
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Stein-Stein model
Let ψ be a positive increasing continuous function such that
limK→∞
ψ(K) = ∞. Then
I(k) = γ1k12 + γ2 + O
(ψ(k)
k12
)
as k → ∞, where
γ1 =
√2√T
(√B3 − 1 −
√B3 − 2
),
γ2 =B2√2T
(1√
B3 − 2− 1√
B3 − 1
),
and B3 is the constant that appears in the asymptotic formula for
the distribution density of the stock price in the Stein-Stein model.
Remark
The previous theorems contain asymptotic formulas with error es-
timates for the implied volatility in analytically tractable stochastic
volatility models. These formulas are sharper than the asymptotic
formulas which can be derived from general results due to Lee, Friz,
and Benaim.
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