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.

2

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 .

5

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.

6

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).

7

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.

8

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.

9

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.

10

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 .

11

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.

12

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.

13

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.

14

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→ ∞.

15

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→ ∞.

16

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.

17

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 <∞.

18

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 → ∞.

19

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.

20

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.

21

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.

22