STUDY OF SABR MODEL IN QUANTITATIVEFINANCE

by

Chenggeng Bi

Submitted in Partial Fulfillment

of the Requirements for the Degree of

Master of Science in Mathematics

New Mexico Institute of Mining and Technology

Socorro, New Mexico

May, 2008

ABSTRACT

In this thesis, we first introduce some basic models in quantitative

finance. Then we describe the method of heat kernel expansion to study the

fundamental solution of the heat equation for an elliptic second-order partial

differential operator. In particular, we use the recursion relations to find the

second coefficient of the short-time asymptotic expansion of the heat kernel.

The result is checked by asymptotic expansion of the exact heat kernel. At

last, we obtain the asymptotics of heat kernel until the second coefficient for

the SABR model.

ACKNOWLEDGMENT

I would like to express my gratitude to my advisor, Prof. Ivan

Avramidi for his guidance, enthusiasm and patience; without his encourage-

ment this study would not have been finished.

My special thanks goes to Tech CSSA and Mr. Charles Del Curto,

who help me greatly in my life in Socorro.

Finally, I am grateful to the whole Mathematics Department of New

Mexico Tech for providing me with excellent environment to study.

This thesis was typeset with LATEX1 by the author.

1LATEX document preparation system was developed by Leslie Lamport as a special versionof Donald Knuth’s TEX program for computer typesetting. TEX is a trademark of theAmerican Mathematical Society. The LATEX macro package for the New Mexico Institute ofMining and Technology thesis format was adapted from Gerald Arnold’s modification of theLATEX macro package for The University of Texas at Austin by Khe-Sing The.

ii

TABLE OF CONTENTS

1. INTRODUCTION 1

1.1 Basic Concepts of Finance . . . . . . . . . . . . . . . . . . . . . 1

1.2 Stochastic Differential Equations . . . . . . . . . . . . . . . . . 4

1.3 Models in Quantitative Finance . . . . . . . . . . . . . . . . . . 5

1.3.1 Black-Scholes Model . . . . . . . . . . . . . . . . . . . . 5

1.3.2 SABR Model . . . . . . . . . . . . . . . . . . . . . . . . 7

2. LINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUA-

TIONS 9

2.1 Background On Differential Geometry . . . . . . . . . . . . . . 9

2.1.1 Two-point Functions in Symmetric Spaces . . . . . . . . 12

2.2 Parabolic PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Asymptotic Expansion of the Heat Kernel . . . . . . . . . . . . 18

2.5 Mellin Transform of the Heat kernel . . . . . . . . . . . . . . . . 20

2.6 Recurrence Relations for Heat Kernel Coefficients . . . . . . . . 21

3. CALCULATION OF THE COEFFICIENTS b1 AND b2 FOR

SABR MODEL 22

3.1 Perturbation Theory for Heat Semigroups . . . . . . . . . . . . 22

3.2 Description of the SABR Model . . . . . . . . . . . . . . . . . . 24

3.3 Hyperbolic Poincare Plane . . . . . . . . . . . . . . . . . . . . . 28

iii

3.4 Exact Solution of Differential Recursion Relation . . . . . . . . 29

3.5 Asymptotic Formula for the SABR Model . . . . . . . . . . . . 33

3.6 Asymptotic Expansion of the Exact Heat Kernel . . . . . . . . . 37

4. CONCLUSION 40

REFERENCES 41

iv

This thesis is accepted on behalf of the faculty of the Institute by the following

committee:

Ivan G. Avramidi, Advisor

Chenggeng Bi Date

CHAPTER 1

INTRODUCTION

1.1 Basic Concepts of Finance

Here we provide a brief introduction of basic concepts and terminology

in finance, in order to understand the particular set of equations that arise in

quantitative finance. For more details see, for example, [7, 13].

Finance. Finance is an important branch of economics that studies the

money form of capital. What distinguishes finance from other branches of

economics is that it is primarily an empirical discipline, because vast quanti-

ties of financial data are generated every day, which makes finance closer to

natural science. Uncertainty and risk are of fundamental importance in finance.

Asset. An asset is defined as a probable future economic benefit obtained

or controlled by a particular entity as a result of a past transaction or event.

Assets can be classified into real assets and financial assets. A financial asset

is also called a security, and the specific form of a financial asset, be it a stock

or a bond, is called a financial instrument.

Equity. Equity, or common stocks and shares, represent a share in the own-

ership of a company. The value of a share may increse or decrese over time,

1

2

depending on the performance of the company, and hence the owner of equity

is exposed to the risks faced by the company.

Derivative Securities. Derivative securities are financial assets that are

derived from other financial assets. The payoff of a derivative security can

depend, for example, on the price of a stock or another derivative.

Options. Options are financial instruments that convey the right, but not

the obligation, to engage in a future transaction on some underlying security.

For example, buying a call option provides the right to buy a specified quantity

of a security at a set strike price at some time on or before expiration, while

buying a put option provides the right to sell. Upon the option holder’s choice

to exercise the option, the party who sold, or wrote, the option must fulfill the

terms of the contract. A European option is one which may only be exercised

at expiry. The expiry T of the option is determined at the time of writing the

contract, and we denote VT [xT , X] as the option payoff at expiry T given that

the asset price at expiry is xT . The strike price X is determined at the time of

writing of the contract. The simplest type of European option is known as a

vanilla option.

Arbitrage. Arbitrage is a term for gaining a risk-free (guaranteed) profit

by simultaneously entering into two or more financial transactions, be it in

the same market or in two or more different markets. Since one has risk-free

instruments, such as cash deposits, arbitrage means obtaining guaranteed risk-

free returns above the risk-less return that one can get from the money market.

3

A fundamental concept of finance is the principle of no arbitrage which states

that no risk-free financial instrument can yield a rate of return above that of

the risk-free rate. In other words, if one wants to harvest high returns one has

to take the commensurate high risks.

Hedging. In finance, a hedge is an investment that is taken out specifically to

reduce or cancel out the risk in another investment. Hedging is a strategy de-

signed to minimize exposure to an unwanted business risk, while still allowing

the business to profit from an investment activity. To hedge a financial instru-

ment, one needs to have at least one second instrument so that a cancellation

between the fluctuations of the two instruments can be attempted. The second

instrument clearly has to depend on the instrument one intends to hedge, since

only then can one expect a connection between their random fluctuations. For

example, to hedge a primary instrument, what is often required is a derivative

instrument, and vice versa.

Efficient Market Hypothesis. The Efficient Market Hypothesis (EMH)

says that the entire history of information regarding an asset is reflected in

its price and that the market responds instantaneously to new information.

Thus the EMH implies that if any patterns do exist, they must be so small

that no systematic trading strategy can have a better risk/return profile than

the market portfolio. Hence according to the EMH, no profitable information

about future movements can be obtained by studying the past price series. So

the theoretical descriptions used in standard finance theory are typically built

around the assumption that changes in the prices of all securities, up to a drift,

4

are random.

Volatility. Suppose the value of an equity at time t is represented by S(t).

The change, dS, in the value of the equity in the time dt is random. That is

why the price S(t) of the security is treated as a random variable. The extent to

which the security S(t) is random is specified by a quantity called the volatility

of the security.

1.2 Stochastic Differential Equations

A stochastic differential equation (SDE) is an equation of the form

dX = b(X, t)dt+B(X, t)dW (1.1)

X(0) = X0 (1.2)

where b(X, t) is called the drift term, B(X, t) is called the noise intensity term

(or volatility function) and dW is a so-called Wiener process (or Brownian

motion).

For the derivative of function of a random variable, Ito calculus, not

the ordinary calculus is used, due to the singular nature of white noise. The

Ito formula is very important in the Ito calculus. In a practical sense it means

that for all our purposes we can replace

(dW )2 = dt . (1.3)

For a function V (t,W ) of t and W we have the following modified differential

dV (t,W ) =

(∂V

∂t+

1

2

∂2V

∂W 2

)dt+

∂V

∂WdW . (1.4)

5

This is usually called the Ito formula. For more details about Ito formula see

[9].

1.3 Models in Quantitative Finance

1.3.1 Black-Scholes Model

Let S(t) denote the price of a stock at time t ≥ 0. A standard model

assumes that the relative change of price, dSS

, evolves according to the SDE

dS

S= µdt+ σdW (1.5)

for certain constants µ > 0 and σ, called respectively the drift and the volatility

of the stock. Note that there is a random variable dW , which makes the price

nondeterministic.

In order to eliminate the random term we will make the following basic

assumptions. Let V (S, t) be the price of a European call option. Combining

(1.5) and (1.4), then using Ito’s formula to simplify, we get

dV =

(σS

∂V

∂S

)dW +

(µS

∂V

∂S+

1

2σ2S2∂

2V

∂S2+∂V

∂t

)dt . (1.6)

Here we need to use the hedging strategy in order to yield a risk-free

portfolio. Let us construct a portfolio Π comprising the option and a quantity

(−φ) of an underlying asset. The value of the portfolio is

Π = V − φS . (1.7)

The change in the value of the portfolio between t and t+ dt is

dΠ = dV − φdS . (1.8)

6

We can now substitute eq. (1.6) into eq. (1.8) to give

dΠ = σS

(∂V

∂S− φ

)dW +

(µS

∂V

∂S+

1

2σ2S2∂

2V

∂S2+∂V

∂t− µφS

)dt . (1.9)

Now, the idea is to specify φ in such a way to reduce all risk, that is,

to make the portfolio risk-free (at any time). To remove the stochastic term,

we should set

φ =∂V

∂S. (1.10)

Then we obtain

dΠ =

(1

2σ2S2∂

2V

∂S2+∂V

∂t

)dt , (1.11)

which is a deterministic equation for the change in the value of the portfolio at

each time t. In other words, the risk has been eliminated, yielding a zero-risk

portfolio.

Next, by the principle of no-arbitrage, Π must instantaneously earn

the risk-free bank rate r,

dΠ = rΠdt . (1.12)

Combining (1.11) and (1.12), we get

r

(V − S

∂V

∂S

)dt =

(1

2σ2S2∂

2V

∂S2+∂V

∂t

)dt . (1.13)

This gives us finally the equation

∂V

∂t+

1

2σ2S2∂

2V

∂S2+ rS

∂V

∂S− rV = 0 , (1.14)

which is the famous Black-Scholes equation.

7

1.3.2 SABR Model

We follow here [12]. Let us consider a European option on a forward

asset expiring T years from today. The forward rate process is assumed to

satisfy the stochatic differential equations

dF (t) = σ(t)C(F )dW1(t) (1.15)

dσ(t) = υσ(t)dW2(t) (1.16)

where υ is a constant parameter (volatility of volatility) and W1(t) and W2(t)

are Wiener processes with the constant correlation ρ, that is,

E[(dW1)2] = dt, E[(dW2)

2] = dt , (1.17)

E[(dW1)(dW2)] = ρdt . (1.18)

The parameter υ is assumed to be such that υ2T is small, since an asymptotic

expansion in υ2T will be used. Here the function C(F ) is supposed to be

positive monotone non-decreasing and smooth. It is extended to negative values

of the argument by

C(−F ) = −C(F ) . (1.19)

Let U(t, f, σ;T, F,Σ) be the price of Arrow-Debreu security whose

payoff at time T is given by a Dirac delta-function. For time 0 < t < T it

satisfies the following parabolic partial differential equation

(∂

∂t+ L

)U = 0 , (1.20)

where

L =1

2σ2

(C(f)2 ∂

2

∂f 2+ 2υρC(f)

∂2

∂f∂σ+ υ2 ∂

2

∂σ2

), (1.21)

8

with the terminal condition

U(T, f, σ;T, F,Σ) = δ(f − F )δ(σ − Σ) . (1.22)

The equation should also be supplemented by appropriate boundary conditions

at zero and at infinity. In particular, it is assumed that

limF,Σ→∞

U(t, f, σ;T, F,Σ) = 0 . (1.23)

In a very special case (called the normal SABR model)

C(f) = 1 , ρ = 0 , (1.24)

the operator L takes an especially simple form

L =1

2σ2

(∂2

∂f 2+ υ2 ∂

2

∂σ2

). (1.25)

If the funcion U(t, f, σ;T, F,Σ) is known, then the price of a European

call option struck at K and expiring time T is

C(t, f, σ) =

∫ ∞

−∞dF

∫ ∞

0

dΣU(t, f, σ;T, F,Σ)(F −K)+ . (1.26)

where (F −K)+ = max(F −K, 0) .

CHAPTER 2

LINEAR PARABOLIC PARTIAL DIFFERENTIALEQUATIONS

2.1 Background On Differential Geometry

In this section, we will survey some of the notions of differential ge-

ometry that are used in this work: manifolds, Riemann geometry, connections,

curvature, geodesics, and parallel transport. For further details about these

topics, see [8].

Definition 1 Suppose M is a Hausdorff topological space. If for any x ∈M ,

there exists a neighborhood U of x such that U is diffeomorphic to an open set

in Rn, then M is called an n-dimensional smooth manifold.

Definition 2 A first-order partial differential operator acting on smooth func-

tions over M is called a vector. The set of all vectors at the point x forms a

vector space called the tangent space TxM . In local coordinates a vector has the

form

Y = Y i∂i. (2.1)

Here, as usual, we denote ∂i = ∂/∂xi and we use the Einstein summation

convention so that the summation over repeated indices is understood.

9

10

Definition 3 A linear functional on the tangent space is called a covector.

The set of all covectors at the point x forms a vector space, called the cotangent

space T ∗xM . In local coordinates a covector has the form

α = αjdxj. (2.2)

Definition 4 A tensor of rank (p, q) at the point x is a multilinear functional

on the space TxM × · · · × TxM︸ ︷︷ ︸p

×T ∗xM × · · · × T ∗xM︸ ︷︷ ︸q

. The set of all tensors of

rank (p, q) forms a vector space T pq,xM = TxM ⊗ · · · ⊗ TxM︸ ︷︷ ︸

p

⊗T ∗xM ⊗ · · · ⊗ T ∗xM︸ ︷︷ ︸q

.

Definition 5 If a smooth manifold M is given a smooth, everywhere non-

degenerate symmetric covariant tensor field of rank (0, 2), then M is called a

generalized Riemannian manifold. If the tensor field is positive definite, then

M is called a Riemannian manifold.

The differential 2-form

ds2 = gij(x)dxidxj (2.3)

is independent of the choice of the local coordinate system xi and is called the

Riemannian metric.

An affine connection is defined by some coefficients Γijk. The affine

connection allows one to define the covariant derivative of arbitrary tensors. In

particular, for vectors vj and covectors αi we have

∇jvi = ∂jv

i + Γikjv

k , (2.4)

∇jαi = ∂jαi − Γkijαk . (2.5)

11

The torsion of the connection is the tensor T ijk of the type (1, 2) defined by

T ijk = Γi

jk − Γikj . (2.6)

A connection is called torsion-free (or symmetric) if the torsion is equal to zero.

A connection is said to be compatible with the metric if the covariant derivative

of the metric vanishes

∇igjk = ∂igjk − Γmjigmk − Γm

kigjm = 0 . (2.7)

A torsion-free metric-compatible connection is called Levi-Civita connection.

One can show that the Levi-Civita connection is unique.

Let (gij) be the matrix inverse to the matrix (gij). Then Christoffel

symbols are defined by

Γijk =

1

2gim(∂jgmk + ∂kgjm − ∂mgjk) . (2.8)

The Christoffel symbols determine the Levi-Civita connection.

Now let us define the quantities

Rijkl = ∂kΓ

ijl − ∂lΓ

ijk + ΓimkΓm

jl − ΓimlΓ

mjk . (2.9)

Then it is easy to show that these coefficients form a tensor of type (1, 3)

called Riemann curvature tensor. By using Riemann tensor one can define new

tensors, Ricci tensor

Rij = Rkikj , (2.10)

and scalar curvature

R = gijRij . (2.11)

12

The Riemann tensor has the following symmetry properties

Rijkl = −Rjikl = −Rijlk = Rklij (2.12)

Rijkl +Ri

klj +Riljk = 0 . (2.13)

By using these symmetry properties it is easy to show that the number of al-

gebraically independent components of Riemann tensor is n2(n2−1)12

. Thus for

two-dimensional manifolds Riemann tensor has only one independent compo-

nent, determined by the scalar curvature called Gauss curvature

K = R1212 =

1

2R . (2.14)

Then

Rijkl = Rδ

[i[kδ

j]l] . (2.15)

2.1.1 Two-point Functions in Symmetric Spaces

Definition 6 Suppose M is an n-dimensional Riemannian manifold. If a

parametrized curve C is a geodesic curve in M with respect to the Levi-Civita

connection, then C is called a geodesic of the Riemannian manifold M .

The coordinates of the geodesics x = x(τ) satisfy the non-linear second-order

ordinary differential equation

xi + Γijkx

kxj = 0 . (2.16)

A very important property is that the geodesic is the shortest curve between

any two sufficiently close points, x and x′. The distance between x and x′ along

the geodesics is called the geodesic distance, d(x, x′).

13

Let x′ be a fixed point in a manifold M . Consider a sufficiently small

neighborhood of this point and connect every other point x in this region with

the point x′ by a geodesic x = x(τ), with an affine parameter τ so that x(0) = x′

and x(t) = x. The Synge world function is defined as half of the square of the

geodesic distance

σ(x, x′) =1

2d2(x, x′) . (2.17)

This is a bi-scalar function that determines the local geometry of the manifold.

The derivatives of this function

σi = ∇iσ , σi′ = ∇i′σ . (2.18)

are exactly the tangent vectors to the geodesic at the points x and x′ respec-

tively pointing in opposite directions, that is

σi = −gij′σ

j′ . (2.19)

Then we define a two-point scalar ∆(x, x′) called Van Vleck-Morette determi-

nant

∆(x, x′) = g−1/2(x) det[−∇i∇j′σ(x, x′)]g−1/2(x′) . (2.20)

Let us define

D = σi∇i . (2.21)

These functions are known to satisfy the equations

σ =1

2σiσi , (2.22)

Dσi = σi , (2.23)

∆−1D∆ = (n−∇i∇iσ) . (2.24)

14

The coincidence limits of a two-point function is defined as

[f ] = limx→x′

f(x, x′) . (2.25)

In particular, one can show that [2, 4]

[σi] = [σi′ ] = 0 ,

[∇i∇jσ] = − [∇i∇j′σ] = gij ,

[∆] = 1 . (2.26)

To approximate a function in a neighborhood of a given point in a Riemannian

manifold, we will apply the covariant Taylor series method to the curved man-

ifold, since that method does not depend on the local coordinate system. For

the details of this method, see [2, 3, 4, 5]. Here we just quote some important

results for two-point functions in the case of constant curvature when

Rijkl = Λ(δi

kgjl − δilgjk) . (2.27)

The Van Vleck determinant is

∆ =

(sinh

√−2Λσ√−2Λσ

)−(n−1)

. (2.28)

The Laplacian of the world function σ is

∇i∇iσ = 1 + (n− 1)F . (2.29)

where

F =√−2Λσ coth(

√−2Λσ) . (2.30)

By using (2.29) and (2.30), we can derive the explicit form of the

Laplacian and operator D separately acting on a funcion depending on the

radius coordinate only.

15

We define

r =√

2σ , (2.31)

Then,

ri = ∇ir =σi√2σ

(2.32)

riri =

σiσi

2σ= 1 . (2.33)

Now by using (2.30), for any function f(r) we have

∇i∇if = ∇i

(ri∂f

∂r

)

=∂2f

∂r2+ (n− 1)F (r)

∂f

∂r. (2.34)

Then we consider D acting on f(r)

Df(r) = σi∇if = σi

(ri∂f

∂r

)

= r∂f

∂r. (2.35)

Thus

∇i∇if(r) =[∂2

r + (n− 1)√−Λ coth(

√−Λ r)∂r

]f(r) , (2.36)

Df(r) = r∂rf(r) . (2.37)

2.2 Parabolic PDE

Differential Equation. All models of quantitative finance lead to a linear

partial differential equation of the general form

(∂

∂t+ L

)V (t; x) = 0 , (2.38)

16

where L is an elliptic second-order partial differential operator of the form

L = L(t, x, ∂x) = −αij(t, x)∂2

∂xi∂xj+ βi(t, x)

∂

∂xi+ γ(t, x) , (2.39)

where the coefficients αij(t, x), βi(t, x) and γ(t, x) are given functions of n

variables xi, that we will call space variables, and t is a time variable.

Since such equations arise in studying heat conduction and diffusion,

we call this equation the heat equation. This equation has to be supplemented

with some initial (or terminal) conditions. By changing the sign of the time

variable and by shifting it if necessary without loss of generality we can always

assume that the time variable is positive, t > 0, and the initial condition is

posed at t = 0, i.e.

V (0;x) = f(x) , (2.40)

where f(x) is a given function of x.

Boundary Conditions. We will simply assume that the space variables

range in some open subset M of the Euclidean space Rn (with or without

boundary ∂M), which is a hypersurface in Rn. If the boundary is present,

then the above equation has to be supplemented also by some boundary con-

ditions. Even if there is no boundary one has to specify the behavior of the

unknown function at infinity. The choice of the boundary conditions depends

on the model under consideration. Here some of the space variables are stock

prices or volatilities thus should be positive. We will assume that the boundary

conditions have the form

BV (t, x) |∂M= 0 , (2.41)

17

where B is, in general, a first-order partial differential operator in space vari-

ables

B = υi(t, x)∂i + u(t, x) , (2.42)

where υi(t, x) and u(t, x) are some real functions of x and, in general, t, evalu-

ated at the boundary ∂M .

The classical boundary conditions are described as follows. The Dirich-

let boundary conditions simply set the value of the function equal to zero at

the boundary, i.e, the Dirichlet boundary operator is

BD = 1 . (2.43)

The conditions (2.40) and (2.41) completely determine the solution of eq.

(2.38).

2.3 Heat Kernel

The fundamental solution of the eq. (2.38) is a function U(t, x | t′, x′)that depends on two time variables, t and t′, and two sets of space variables, x

and x′. It is the solution of the differential equation

(∂t + L)U(t, x | t′, x′) = 0 , (2.44)

with initial condition in form of a Dirac delta-function

U(t′, x | t′, x′) = δ(x− x′) , (2.45)

and the boundary conditions

BU(t, x | t′, x′) |x∈∂M= 0 . (2.46)

18

Here δ(x− x′) is the n-dimensional delta-function

δ(x− x′) = δ(x1 − x′1) · · · δ(xn − x′n) . (2.47)

In the case when the operator L does not depend on time t, the fundamental

solution U(t, x; t′, x′) depends on just one time variable, t− t′, that is,

U(t, x | t′, x′) = U(t− t′;x, x′) , (2.48)

where U(t− t′;x, x′) satisfies the equation

(∂t + L)U(t;x, x′) = 0 , (2.49)

with the initial condition

U(0; x, x′) = δ(x− x′) . (2.50)

The function U(t, x | t′, x′) is called the heat kernel of the operator L.

2.4 Asymptotic Expansion of the Heat Kernel

We describe the asymptotic expansion of the heat kernel following

[2, 3, 4, 5].

As we have seen in the last section, the partial differential operators

in the finance models are of second order. Fortunately, every elliptic second-

order partial differential operator can be expressed in geometric terms, which

enables one to use powerful geometric methods in the study of analytic prob-

lems, like the heat kernel asymptotics. Let L be an elliptic second-order partial

differential operator of Laplace type. Then it must have the form [6]

L = −gij∇Ai ∇A

j +Q , (2.51)

19

where ∇Ai = ∇i + Ai, Ai(x) is some real vector field and Q = Q(x) is some

real function on M .

In this work we restrict ourselves for time-independent operator L.

It is well known that in Euclidean space Rn the heat kernel has the following

form

U(t; x, x′) = (4πt)−n/2 exp

(−|x− x′|2

4t

). (2.52)

Now on the curved manifold, we define

U(t;x, x′) = (4πt)−n/2P(x, x′)∆1/2(x, x′) exp

(−σ(x, x′)

2t

)Ω(t; x, x′) , (2.53)

where σ(x, x′) is the world function, ∆(x, x′) is the corresponding Van Vleck-

Morette determinant, and P(x, x′) is the two-point function, defined as

P(x, x′) = exp

(−

∫ t

0

dτ xi(τ)Ai(x(τ))

), (2.54)

where xi = dxi

dτ, Ai is a vector field and the integral is taken along the geodesic

x(τ) connecting the points x′ and x so that x(0) = x′ and x(t) = x. This

function satisfies the equation

σi(∇i +Ai)P = 0 , (2.55)

and the initial condition

[P ] = 1 . (2.56)

We will consider the case when the points x and x′ are sufficiently close to each

other so that all two-point functions are single-valued and well-defined.

As it is shown in [1, 4], the function Ω(t; x, x′) satifies the equation(∂

∂t+

1

tD + L

)Ω(t;x, x′) = 0 (2.57)

20

where

L = P−1∆−1/2L∆1/2P , (2.58)

and the initial conditions

Ω(0; x, x′) = 1 . (2.59)

2.5 Mellin Transform of the Heat kernel

In order to get an asymptotic expansion for Ω(t), Mellin transfrom

and Minackshisundaram-Pleijel expansion will be used. Let us follow [2].

Now consider the Mellin transformation of Ω(t)

bq(x, x′) =

1

Γ(−q)∫ ∞

0

dt t−q−1Ω(t;x, x′) , (2.60)

where Γ(−q) is introduced as a convenient scaling function. Under the above

assumptions this integral converges in the region Re q < 0. For Re q ≥ 0 the

function bq should be defined by analytic continuation.

Moreover, by making use of the asymptotic properties of the function

Ω(t;x, x′) it is not difficult to obtain the values of bq(x, x′) at the positive integer

points q = k, k = 0, 1, 2, ...,

bk(x, x′) =

(− ∂

∂t

)k

Ω(t;x, x′)∣∣∣t=0

. (2.61)

The asymptotic expansion of Ω(t;x, x′) as t→ 0

Ω(t;x, x′) ∼∞∑

k=0

(−t)k

k!bk(x, x

′) . (2.62)

The coefficients bk(x, x′) are some smooth functions that are usually called heat

kernel coefficents.

21

2.6 Recurrence Relations for Heat Kernel Coefficients

By substituting the ansatz (2.62) into the equation (2.57) we obtain

the recurrence relation for the coefficient bk(x, x′)

(1 +

1

kD

)bk(x, x

′) = Lbk−1(x, x′) . (2.63)

For k = 0 it gives

Db0 = 0 . (2.64)

By taking into account eq. (2.61) and the initial condition (2.59), we get the

initial condition of the recursion

b0(x, x′) = 1 . (2.65)

CHAPTER 3

CALCULATION OF THE COEFFICIENTS b1 AND b2FOR SABR MODEL

3.1 Perturbation Theory for Heat Semigroups

We follow below [6]. Consider the heat equation

(∂t + A)U(t) = 0 (3.1)

with the initial condition

U(0) = I , (3.2)

where A is an operator in a Hilbert space, I is the identity operator. Then the

operator

U(t) = exp(−tA) =∞∑0

(−1)k

k!tkAk (3.3)

can be defined. It is easy to show that the operator U(t) satisfies the semigroup

property: for any t1, t2 > 0

U(t1 + t2) = U(t1)U(t2) . (3.4)

Now suppose that the operator A = A(s) depends on a parameter s

such that the operator A(s) does not necessarily commute for different values

of the parameter s. Then the heat semi-group varies according to the Duhamel

formula

∂sU(t) = −∫ t

0

dτ U(t− τ)[∂sA]U(τ) . (3.5)

22

23

Suppose that the operator A(s) is linear in s.

A(s) = A0 + sA1 , (3.6)

where A0 is an operator with a well defined heat semi-group U0(t) = exp(−tA0).

Then by treating s as a small parameter and using the Duhamel

formula we obtain Taylor series for the heat semi-group U(t) = exp[−t(A0 +

sA1)]

U(t) = U0(t) +∞∑

k=1

(−1)ksk

∫ t

0

dτk

∫ τk

0

dτk−1 · · ·∫ τ2

0

dτ1

× U0(t− τk)A1U0(τk − τk−1) · · ·U0(τ2 − τ1)A1U0(τ1) . (3.7)

This expansion is called Volterra series.

Let us define an operator AdA that acts on operators as

AdAB = [A,B] . (3.8)

The k-th power of this operator defines k-fold commutators

(AdA)kB = [A, [A, · · · , [A,B] · · · ]]︸ ︷︷ ︸k

. (3.9)

Now we consider an operator-valued function

F (t) = etABe−tA . (3.10)

By differentiating it with respect to t we obtain the differential equation

∂tF = [A,F ] = AdAF , (3.11)

with the initial condition

F (0) = B , (3.12)

24

The solution of this equation is

F (t) = exp[tAdA]B . (3.13)

Thus, we obtain the following expansion

etABe−tA =∞∑

k=0

tk

k!(AdA)kB

= B + t[A,B] +1

2t2[A, [A,B]] +O(t3) . (3.14)

This expansion is particularly useful when the commutators of the operators

A and B are small.

Now we define an operator

V (t) = etA0A1e−tA0

= A1 + t[A0, A1] +1

2t2[A0, [A0, A1]] +O(t3) . (3.15)

Then the Volterra series can be written as

U(t) =

I +

∞∑

k=1

(−1)ksk

∫ t

0

dτk

∫ τk

0

dτk−1 · · ·∫ τ2

0

dτ1

× V (τk − t)V (τk−1 − t) · · ·V (τ1 − t)

U0(t) , (3.16)

Since V (τ) is a power series in τ , we get an expansion as t→ 0

U(t) =

1− stA1 +

t2

2(s2A2

1 + s[A0, A1]) +O(t3)

U0(t) . (3.17)

3.2 Description of the SABR Model

We have described the SABR model in the first chapter and derived

the partial differential equation(∂

∂t+ L

)U = 0 , (3.18)

25

where

L =1

2σ2

(C(f)2 ∂

2

∂f 2+ 2υρC(f)

∂2

∂f∂σ+ υ2 ∂

2

∂σ2

), (3.19)

with the terminal condition

U(T, f, σ;T, F,Σ) = δ(f − F )δ(σ − Σ) . (3.20)

The idea now is to apply the perturbation method described above to

compute the heat kernel U . Next, we change variables to convert this problem

to the usual heat equation setting

τ = T − t , x1 = x = f , x2 = y =σ

υ. (3.21)

Then the equation becomes

(∂τ + L)U(τ ;x, x′, y, y′) = 0 , (3.22)

where

L = −υ2

2y2[C2(x)∂2

x + 2ρC(x)∂x∂y + ∂2y ] . (3.23)

The operator L defines a Riemannian metric gij with components

g11 =υ2

2y2C2 , (3.24)

g12 =υ2

2ρy2C , (3.25)

g22 =υ2

2y2 . (3.26)

The covariant components of the metric are obtained by inverting the matrix

26

(gij)

g11 =2

υ2(1− ρ2)

1

y2C2, (3.27)

g12 = − 2ρ

υ2(1− ρ2)

1

y2C, (3.28)

g22 =2

υ2(1− ρ2)

1

y2. (3.29)

The Riemannian volume element is determined now by the determinant of the

metric gij

g = det gij =4

υ4(1− ρ2)

1

y4C2. (3.30)

We also note the following useful combination

g1/2g11 =1√

1− ρ2C , (3.31)

g1/2g12 =ρ√

1− ρ2, (3.32)

g1/2g22 =1√

1− ρ2

1

C. (3.33)

The Christoffel symbols are

Γ111 = −C

′

C+

ρ

1− ρ2

1

yC(3.34)

Γ211 =

ρ

1− ρ2

1

yC2(3.35)

Γ122 =

ρ

1− ρ2

C

y(3.36)

Γ222 = −1− 2ρ2

1− ρ2

1

y(3.37)

Γ112 = − 1

1− ρ2

1

y(3.38)

Γ212 = − ρ

1− ρ2

1

yC. (3.39)

27

By using these equations one can show that the Riemann tensor

(which has only one nontrivial component in two dimensions) takes the form

R1212 = −υ

2

2. (3.40)

This is nothing but Gaussian curvature. The scalar curvature is

R = −υ2 . (3.41)

Since the curvature is constant and negative, this metric defines the geometry

of the hyperbolic plane. Since the curvature does not depend on the function

C(x) at all, the arbitrariness of the function C just reflects the possibility

of making an arbitrary change of coordinates (diffeomorphism). It does not

change the geometry, which remains the geometry of the hyperbolic plane,

a space of constant negative curvature. Therefore, our metric has negative

constant curvature and is nothing but the hyperbolic plane H2 in some non-

trival coordinates.

Now, we rewrite the operator in the form

L = L0 + L1 (3.42)

where L0 is the scalar Laplacian,

L0 = −∇i∇i , (3.43)

and L1 is a first order operator,

L1 =υ2

2y2C(x)C ′(x)∂x . (3.44)

Then by treating the operator L1 as a perturbation, we get

U(t; x, y, x′, y′) =

1− tL1 +

t2

2

(L2

1 + [L0, L1])

+O(t3)

U0(t;x, x

′) . (3.45)

where U0 is the heat kernel for L0.

28

3.3 Hyperbolic Poincare Plane

In order to obtain the relation of x and y coordinates to the standard

geodesic coordinates, we will find the equations of geodesics.

The Hyperbolic Poincae Plane is the upper half plane H2 = (x, y) :

y > 0 with the Poincare metric tensor

ds2 =dx2 + dy2

y2. (3.46)

So the metric tensor in Poincare plane is given by

h =1

y2

(1 00 1

). (3.47)

It is known that the geodesic distance between the points (x, y), (x′, y′) on H2

is given by

cosh d(x, y;x′, y′) = 1 +(x− x′)2 + (y − y′)2

2yy′. (3.48)

By (3.27), (3.28) and (3.29), the metric tensor in the state space S2

associated with the SABR model is

g =2

υ2(1− ρ2)y2C2

(1 −ρC(x)

−ρC(x) C(x)2

). (3.49)

Now we will show that this metric is diffeomorphic to the metric on H2. Let

us define a map φp : S2 → H2 by

φp(x, y) =

(1

υ√

1− ρ2

(∫ x

p

du

C(u)− ρy

),y

υ

), (3.50)

where p is an arbtrary constant. The Jacobian of φp is

J =

1

υ√

1− ρ2C(x)− ρ

υ√

1− ρ2

01

υ

. (3.51)

29

It is easy to show that JThJ = g, which means that for the SABR model, the

second-order differential operator L corresponds to the same geometry as the

Poincare plane.

Thus, we have an explicit formula for the geodesic distance r(x, y;x′, y′)

in (x, y) and (x′, y′) coordinates

cosh r(x, y; x′, y′) = cosh d(φp(x, y), φp(x′, y′))

= 1 +µ2 − 2ρ(y − y′)µ+ (y − y′)2

2(1− ρ2)yy′, (3.52)

where µ =∫ x

x′du

C(u).

3.4 Exact Solution of Differential Recursion Relation

Now we will show how to find the heat kernel of the Laplacian operator

L in the n-dimensional hypebolic space with the negative constant curvature

Λ = −$2 . (3.53)

For pure Laplacian L0, the heat kernel coefficients depends only on

the geodesic distance r. When applied to radial functions, the operator D and

the Laplacian are

Df(r) = r∂rf(r) (3.54)

Lf(r) = − [∂2

r + (n− 1)$ coth($r)∂r

]f(r) . (3.55)

The recursion relations can be written as

(1 +

1

kr∂r

)bk = Lbk−1 , (3.56)

30

where

L = ∆−1/2L∆1/2 (3.57)

with the initial condition b0 = 1. These relations can be easily integrated to

get

bk(r) = k1

rk

∫ r

0

dr′r′k−1Lr′bk−1(r

′) . (3.58)

Since the first coefficient is known exactly, b0 = 1, we can compute the

coefficient b1 simply by integrating the derivative of the Van Vleck determinant

b1 =n− 1

4ρ2

(n− 3)

[coth2($r)− 1

$2r2

]+ 2

(3.59)

Notice that when r = 0 this gives the coincidence limit

[b1] =n(n− 1)

6$2 . (3.60)

Since the scalar curvature is now

R = n(n− 1)Λ = −n(n− 1)$2 , (3.61)

this coincides with the coincidence limit see, for example, for a general case

b1 = −1

6R . (3.62)

Let

∆12 = eφ , (3.63)

where s = $r. Then

φ = −n− 1

2ln

sinh s

s, (3.64)

L = $2e−φ[∂2s + (n− 1) coth s∂s]e

φ . (3.65)

31

Note that

e−φ∂seφ = ∂s + φ′ (3.66)

e−φ∂2se

φ = ∂2s + 2φ′∂s + φ′′ + φ′2 (3.67)

where prime denotes the derivative with respect to s.

Next, we find an identity between φ′2, φ′′ and φ′

φ′2 =n− 1

2

(φ′′ +

n− 1

2+

2φ′

s

)(3.68)

Therefore, we can simplify L as

L = −$2

[∂2

s +n− 1

s∂s +

3− n

2φ′′ − (n− 1)2

4

](3.69)

Thus,

b2(r) =2

r2

∫ $r

0

ds s L b1(s)

=2$2

r2

∫ $r

0

ds s Ln− 1

4

[(n− 1)− n− 3

s2(s coth s− 1)

]

=$4

2r2

∫ $r

0

ds s

−

[3− n

2φ′′ − (n− 1)2

4

](n− 1)2

− 2(n− 3)

[∂2

s +n− 1

s∂s +

3− n

2φ′′ − (n− 1)2

4

]φ′

s

(3.70)

To compute the integral, we integrate by parts to get

b2(r) =$4

2r2

[−(n− 1)2(3− n)

2sφ′ +

(n− 1)4

8s2

]

− 2(n− 3)φ′′ − 2(n− 3)φ′

s+

(n− 3)2

2φ′2

]∣∣∣∣$r

0

(3.71)

Now we need to evaluate the values at s = $r and s = 0. Obviously, at s = $r

the value is very easy to get, so the only problem is how to evaluate the value

at s = 0.

32

By using the Taylor expansion

sinh s = s+s3

6+

s5

120+O(s7)

coth s =1

s+s

3− s3

45+O(s5) .

we obtain

lims→0

φ = 0 ,

lims→0

φ′ = 0 ,

lims→0

φ′2 = 0 ,

lims→0

φ′′ = −n− 1

6.

Thus, we get b2 in the form

b2 =(n− 1)4

16$4

+1

24(n− 1)(n− 3)(3n2 − 10n+ 23)

$2

r2

+1

16(n− 1)(n− 3)(n− 5)(n− 7)

1

r4

− 1

8(n− 1)3(n− 3)

$3 coth($r)

r

+1

16(n+ 1)(n− 1)(n− 3)(n− 5)

$2 coth2($r)

r2

− 1

8(n− 1)(n− 3)2(n− 5)

$ coth($r)

r3. (3.72)

When r = 0 this gives the coincidence limit

[b2] =n

180(n− 1)(5n2 − 7n+ 6)$4 . (3.73)

Due to (3.61), we can write it as

[b2] = −5n2 − 7n+ 6

180n(n− 1)R2 . (3.74)

33

For further reference we specify the value of b2 in the case when n = 2

b2 =1

16$4 +

1

8

$3 coth($r)

r+

3

8

$ coth($r)

r3+

9

16

$2 coth2($r)

r2

− 5

8

$2

r2− 15

16

1

r4. (3.75)

Next, notice that for pure Laplacian L0, the vector field A is 0, then

by using (2.54), we obtain

P(x, x′) = 1 . (3.76)

Thus by using (2.28) and (2.53), the heat kernel for pure Laplacian

L0 on the two-dimensional hyperbolic space is approximated by

U0(t;x, x′, y, y′) =

1

4πt

√$r

sinh($r)exp

(−r

2

4t

)

×

1− t

4r2

[$2r2 +$r coth($r)− 1

]

+t2

32r4

[$4r4 + 2$3r3 coth($r) + 6$r coth($r)

+ 9$2r2 coth2($r)− 10$2r2 − 15]+O(t3)

. (3.77)

3.5 Asymptotic Formula for the SABR Model

By substituting the above into (3.45), we can compute the heat kernel

for the SABR model.

34

For simplicity, let

φ = −υ2

2C(x)2

ϕ = −υ2ρC(x)

ψ = −υ2

2

ω =υ2

2C(x)C ′(x) .

(3.78)

By using (3.19), (3.44) and L0 = L− L1, we obtain

L1U0 = y2ωrxU′0

L21 U0 = y4

[ω2r2

xU′′0 + (ω2rxx + ωω′rx)U

′0

]

[L0, L1]U0 = [L,L1]U0

= (−y4ωφ′ + 2y4φω′ + 2y3ϕω)(r2xU

′′0 + rxxU

′0)

+ (−y4ωϕ′ + y4ϕω′ + 4y3ψω)(rxryU′′0 + ryxU

′0)

+ (y4φω′′ + 2y3ϕω′ + 2y2ψω)rxU′0 , (3.79)

where the subscripts x and y denote the derivatives with respect to x and y,

for instance,

rx =µ− ρy + ρy′

(1− ρ2)C(x)yy′

ry =y2 − y′2 − µ2 − 2ρµy′

2(1− ρ2)y2y′

ryx =∂2r

∂x∂y

= − µ+ ρy′

(1− ρ2)y2y′C(x). (3.80)

U ′0 and U ′′0 denote the derivatives with respect to r.

35

Next, by using (3.77) and notify that for the SABR model $ = υ2

we

compute

U ′0 =1

4πt

√$r

sinh($r)exp

(−r

2

4t

)− r

2t+

(3

8r+$2r

8− 3$

8coth($r)

)

+

(3$3

32coth($r) +

15$2

64rcoth2($r)− 3$

32r2coth($r)− 7$2

32r

− 9

64r3− $4r

64

)t+O(t2)

, (3.81)

and

U ′′0 =1

4πt

√$r

sinh($r)exp

(−r

2

4t

)r2

4t2+

(7$

16r coth($r)− $2

16r2 − 15

16

)1

t

+$4

128r2 − 15

128r2− 5$2

64− 21$

64rcoth($r) +

57$2

128coth2($r)

− 7$3

64r coth($r) +O(t)

. (3.82)

At last, by using equations (3.45), (3.79), (3.81), (3.82) and substituting the

variables (3.78), we obtain the following asymptotic formula for the SABR

model

U(t;x, x′, y, y′) =1

4πt

√$r

sinh($r)exp

(−r

2

4t

) (a0 + a1t+ a2t

2 +O(t3))

(3.83)

where

a0 = 1 +r

4υ2CC ′y2rxα+

r2

32υ4C2C ′2y4r2

xα+r2

8(Pr2

x +Qryrx)α , (3.84)

a1 =1

4r2− $2

4− $ coth($r)

4r+υ2

4y2CC ′rrx

+y4

8υ4CC ′[CC ′rxx + (C ′2 + CC ′′)rx]

+1

8

(υ4

4y4C2C ′2r2

x + Pr2x +Qrxry

) [ρr coth(ρr)− 2rx − 2rβ − 1

]

− r

4(Prxx +Qryx +Rrx) , (3.85)

36

a2 =$4

32+$3 coth($r)

16r+

3$ coth($r)

16r3+

9$2 coth2($r)

32r2

− 5$2

16r2− 15

32r4− y2

2υ2CC ′rxβ

+1

4

(y4

4υ4C2C ′2r2

x + Pr2x +Qrxry

) (2αx +

α

r− υα coth(υr)− rβ

)

+1

2

(y4

4υ4C2C ′2rxx +

y4

4υ2CC ′(C ′2 + CC ′′)rx

+ Prxx +Qryx +Rrx

)α , (3.86)

and we already know $ is

$ =υ√2, (3.87)

and α, β are functions of r defined by

α =3

8r+$2r

8− 3$

8coth($r) (3.88)

β =3$3

32coth($r) +

15$2

64rcoth2($r)− 3$

32r2coth($r)− 7$2

32r

− 9

64r3− $4r

64, (3.89)

P , Q, R are functions of x, y defined by

P = −y4

2υ4C3C ′′ − y3ρυ4C2C ′ (3.90)

Q = −y4

2υ4ρC2C ′′ − y3υ4CC ′ (3.91)

R = −3

4y4υ4C ′C ′′C2 − y4

4υ4C3C ′′′ − y3υ4ρCC ′2

− y3υ4ρC2C ′′ − y2

2υ4CC ′ . (3.92)

37

3.6 Asymptotic Expansion of the Exact Heat Kernel

In order to check the results for b1 and b2, we now study the heat

kernel U0 for L0 when the case n = 2, which is known exactly according to [11].

U0(t;x, x′) =

$e−$2t/4√

2

4π3/2√t

exp

(−r

2

4t

) ∞∫

0

e−νdν√cosh(ρ

√4tν + r2)− cosh($r)

.

(3.93)

First of all, we rewrite it in the form

U0(t;x, x′) =

1

4πt

√$r

sinh($r)exp

(−r

2

4t

)Ω0(t, r) , (3.94)

where

Ω0(t, r) = $e−$2t/4√

2tπ−12

√sinh$r

$r

∞∫

0

e−νdν√cosh(ρ

√4tν + r2)− cosh($r)

.

(3.95)

Expanding the integrand in powers of t yields

cosh

[$r

(1 +

4tν

r2

) 12

]− cosh($r)

− 12

= (sinh($r))−12x−

12

(1 +

1

2x coth($r) +

1

6x2 +O(t3)

)− 12

,(3.96)

where

x = $r

(1 +

4tν

r2

) 12

−$r

=2$tν

r− 2$t2ν2

r3+

4$t3ν3

r5+O(t4) . (3.97)

38

Then we expand(1 + 1

2x coth($r) + 1

6x2

)− 12 and x−

12 in power of t to get

(1 +

1

2x coth($r) +

1

6x2

)− 12

= 1− 1

2coth($r)

$tν

r+

1

2coth($r)

$2t2ν2

r2+

3

8coth2($r)

$2t2ν2

r2

−1

3

$2t2ν2

r2+O($6t3ν3) (3.98)

and

x−12 =

(2$tν

r

)− 12(

1− 2tν

r2+

2t2ν2

r4+O(t3)

)− 12

=

(2$tν

r

)− 12(

1 +1

2

tν

r2− 5

8

t2ν2

r4+O(t3)

). (3.99)

Substituting these results into Ω0 and integrate over ν, we get

Ω0(t, r) = 1 +1

4

(1

$2r2− coth($r)

$r

)$2t

+3

4

(1

2

coth($r)

$3r3+

3

8

coth2($r)

$2r2− 1

3

1

$2r2− 1

4

coth($r)

$3r3− 5

8

1

$4r4

)$4t2

+O($6t3) . (3.100)

Now we expand

e−$2t/4 = 1− $2t

4+$4t2

32+O(t3) . (3.101)

Finally, we obtain

Ω0(t, r) = 1− b1t+b22t2 +O(t3) , (3.102)

where

b1 =1

4$2 +

$ coth($r)

4r− 1

4r2

b2 =1

16$4 +

1

8

$3 coth($r)

r+

3

8

$ coth($r)

r3+

9

16

$2 coth2($r)

r2

− 5

8

$2

r2− 15

16

1

r4. (3.103)

39

By comparing this to (3.59) and (3.75) we see that our results for b1 and b2

obtained by direct solution of the differential recursion equation are correct, at

least in two dimensions, n = 2.

CHAPTER 4

CONCLUSION

Let us summarize the main results of this thesis. First, making use of

the differential relation, we compute the second coefficient b2 for the heat kernel

expansion in n-dimensional hyperbolic space. In particular, we obtain the heat

kernel coefficients b1, b2 on two-dimensional hyperbolic plane H2. Then this

result is checked by asymptotic expansion of the exact heat kernel.

Since for the SABR model, the second-order differential operator also

corresponds to the geometry of the hyperbolic plane, we obtain the heat kernel

expansion until the second coefficient by the perturbation of the operator. This

is a new asymptotic formula up to the second-order for the SABR model, which

is more precise than the original Hagan formula.

40

REFERENCES

[1] I. G. Avramidi, The covariant technique for calculation of the heat kernelasymptotic expansion, Physics Letters B, 238 (1990) 92-97.

[2] I. G. Avramidi, A covariant technique for the calculation of the one-loopeffective action, Nuclear Physics B, 355 (1991) 712–754; Erratum: NuclearPhysics B, 509 (1998) 557–558.

[3] I. G. Avramidi, Covariant techniques for computation of the heat kernel,Reviews in Mathematical Physics, 11 (1999) 947–980.

[4] I. G. Avramidi, Heat Kernel and Quantum Gravity, Lecture Notesin Physics, New Series m: Monographs, LNP:m64 (Berlin-New York:Springer-Verlag 2000).

[5] I. G. Avramidi, Heat kernel in quantum field theory, Nuclear Physics Proc.Suppl., 104 (2002) 3–32.

[6] I. G. Avramidi, Analytic and Geometric Methods for Heat Kernel Appli-cations in Finance, Preprint, NATIXIS, Paris, 2007

[7] E. Baaquie, Quantum Finance, Cambridge University Press, 2007

[8] S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Differential Geometry,World Scientific, 2000

[9] L. Evans, An Introduction to Stochastic Differential Equations, LectureNotes, UC Berkeley

[10] J. Gatheral and M. Lynch, Lecture 1: Stochastic Volatility and LocalVolatility, Preprint

[11] P. Hagan, A. Lesniewski and D. Woodward, Probability distribution in theSABR model of stochastic volatility, Preprint

[12] P. S. Hagan, D. Kumar, A. S. Lesniewski and D. E. Woodward, Managingsmile risk, Preprint

41

42

[13] N. F. Johnson, P. Jefferies and P. K. Hui, Financial Market Complexity,Oxford University Press, 2003

[14] M. E. Taylor, Partial Differential Equations, vol. I, II, Springer, New York,1991