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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
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42
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