Calibration of Composite

Asymptotic Solutions to Heston

Option Prices

Frank Fung

Berkeley MFE

University of California

A thesis submitted for the degree of

MFE

March 2011

ii

Abstract

The variance process of the Heston volatility model has a singularity at

zero variance. When pricing derivatives using finite differencing under the

Heston model using an adapted mesh, the higher grid resolution near the

zero singularity could lead to very large eigenvalues in the discretization

matrix. We propose and study an alternative approach that takes advantage

of the singular perturbation nature in the Heston pricing PDE. We compute

the inner and outer solutions separately and construct a composite solution

as a weighted sum of the two by calibrating to the analytic Heston call

prices. Numerical test shows that this scheme provides satisfactory results

for widths of boundary layer that are not too small, as the deviation from

the analytic prices is reduced by a factor of 10 as compared to a single-mesh

benchmark, with only twice the CPU time.

iv

Acknowledgements

I owe my gratitude to Dr. Domingo Tavella for the guidance and insights

he has provided.

ii

Contents

List of Figures v

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Methodology 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Asymptotic Expansion of a Second-Order ODE Problem . . . . . . . . . 5

2.3 Asymptotic Expansion as Applied to Call Option Pricing Under Heston

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Composite Solution Construction Using Weighting Functions . . . . . . 10

2.5 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Numerical Results 13

3.1 Testing the Solution Patching on the Stylized Example . . . . . . . . . . 13

3.2 Calibration to Heston Call Prices . . . . . . . . . . . . . . . . . . . . . . 14

4 Conclusion 17

A Discretized PDE’s for Inner and Outer Problems 19

B Analytic Heston Call Prices 21

C Calibration Error 27

References 31

iii

CONTENTS

iv

List of Figures

1.1 Non-uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Exponential weighting functions . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Logistic weighting functions . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Composite versus exact solutions for the stylized example . . . . . . . . 13

3.2 Surface plot of composite solution . . . . . . . . . . . . . . . . . . . . . 14

3.3 Surface plot of the outer weighting function . . . . . . . . . . . . . . . . 14

3.4 Deviation from the analytic price . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Deviation from the analytic price . . . . . . . . . . . . . . . . . . . . . . 15

3.6 Effect of the width of the boundary layer . . . . . . . . . . . . . . . . . 16

v

LIST OF FIGURES

S - Stock price

v - Stock variance

r - Annualized risk-free rate

y - Annualized dividend yield

t - Physical time

W - Wiener process

X - Logarithm of stock price, log S

fx - Partial derivative of the function f

with respect to x

K - Strike of an option

T - Time to expiry of an option

δ - Scaling parameter for the inner layer

of the asymptotic expansion

ui0 - Zeroth-order term in the inner solu-

tion expansion

ui1 - First-order term in the inner solution

expansion

uo - Outer solution

u∗ - Analytic solution

ucom - Composite solution formed by the

inner and outer solutions

using - Solution computed using a single

grid ηi - Weighting function for the inner

solution

ηo - Weighting function for the outer so-

lution

vi

1

Introduction

1.1 Motivation

The Heston model [1] extends the Black-Scholes framework to option pricing under

stochastic volatility. Under the Heston model, the stock price process is specified as

the geometric Brownian process

dS = (r − y)Sdt+√vSdW 1 (1.1)

where r is the risk free rate and y is the dividend yield, while the variance process is

specified as the square root-process

dv = κ(θ − v)dt+ ξ√vdW 2 (1.2)

such that d⟨W 1,W 2

⟩= ρdt. Intuitively, κ is the mean-reverting rate, θ the long-term

average, and ξ√v the volatility of the variance process. Defining X = logS, the pricing

PDE of the Heston model is

1

2vuXX − 1

2vuX + v

1

2ξ2uvv + κ(θ − v)uv + vρξuXv + (r − y)uX − ut − ru = 0

(1.3)

One advantage of the Heston model is the availability of closed-form solution for vanilla

call option. However, as shown in [2], the zero variance level is a possible absorption

state depending on the Feller condition. The Feller condition, in the context of Eq.

(1.2), is said to be violated if2κθ

ξ2< 1 (1.4)

1

1. INTRODUCTION

A particular challenge arises when the Feller condition is violated. If the Feller

condition is violated, the zero state becomes accessible. If the boundary condition

imposed at v = 0 is such that the zero variance state is absorbing, there is leakage

in the probability mass of the variance distribution at v = 0 [3]. Although empirical

evidence shows that the violation of the Feller condition at least during certain periods

[4], we can argue from an economics point of view that prolonged, severe absorption

cannot occur at v = 0 because it leads to the unrealistic consequence of perpetually

deterministic stock price. The implication of Feller condition violation on analytic

pricing is that the analytic solution to a PDE is unique only up to a set of specified

boundary conditions, as shown in [5] in the context of zero coupon bond prices under the

CIR model. Fortunately, in numerical pricing using finite differencing under the Heston

model, the convection-only boundary condition spares us from the need to impose any

artificial condition. By imposing the convection-only boundary condition at v = 0 we

are solving a PDE that is exact at that point. However, we are still left with a technical

difficulty, namely the resolution of the PDE solution at small variances. One approach

is to use an adapted grid that has higher resolution at small variances [6, 7, 8, 9].

However, increasing the grid density would create values with larger magnitudes in

the eigenvalue spectrum, which in turns lead to poorer stability. Instead of adapted

mesh, an alternative approach is to price the instrument using two grids, as explained

as follows.

In this paper, we consider a finite difference approach to the boundary condition

problem using asymptotic expansion. We split the variance axis into two regimes.

In the outer regime, where the value of the variance is large (as compared to some

characteristic scale, as will be explained), we seek the outer solution that dominates in

this range; in the inner regime, where the value of the variance is small, we calculate the

inner solution that dominates there. A composite solution that is valid over the entire

range of variances is constructed as a weighted sum of the outer and inner solutions.

We calibrate the solution patching by pricing a vanilla call on the grid and requiring

that the error of the composite solution as compared to the closed-form Heston price

be minimized. Once we have calibrated the composite solution construction scheme,

the set of weighting functions can be used to price other instruments.

2

1.2 Literature Review

1.2 Literature Review

Singular perturbation problems have been studied extensively in many fields of engi-

neering, particularly in fluid mechanics where a boundary layer is involved [10, 11]. In

recent years, the method of solving singularly perturbed problems using asymptotic

techniques has also found its application in finance. One of such example can be found

in [12, 13, 14]. These works take advantage of the ’burstiness’ of volatility, i.e. the fact

that interest rate (or stock return) and its volatility vary on two different time scales,

to obtain a closed-form correction term that accounts for stochastic volatility in the

pricing of derivatives. The advantage of this approach is that it preserves mathematical

tractability. However, for every payoff one has to perform a new derivation, and it is

not guaranteed that the tractability is preserved for any arbitrary payoff.

Another approach to tackle singularly perturbed problem is by numerical matching

of asymptotic solutions. To facilitate the discussion, consider the situation when a

small parameter ε is multiplied to the highest-order derivative in a PDE, resulting

in a singularly perturbed problem. Since the PDE is changed substantially when ε

goes to zero (i.e. the second-order differential equation becomes a first-order one), the

convergence of the solution is not ε-uniform. Consider the following ODE problem that

we will revisit in Section 2.2:

εd2y

dx2+ b

dy

dx+ cy = 0

y(0) = 0

y(1) = 1,

(1.5)

This ODE reduces to

bdy

dx+ cy = 0 (1.6)

if ε goes to zero. As we can see in Figure 1.1, the solution to the singularly perturbed

problem converges to that of the reduced problem as ε approaches zero, except for the

point x = 0. This is in contrast with the regular perturbation problems, in which case

the solution converges ε-uniformly, and the solution that is valid for the entire range

can be expressed as a series

y(x) =

n−1∑i=0

εiYi(x) + O(εn) (1.7)

3

1. INTRODUCTION

For singular perturbation problems, one cannot use the same strategy as Eq. (1.7) of

seeking a single series that accurately describes the exact solution over the entire range

of x. Rather, one has to solve the problem in the outer regime and the inner regime

separately, then construct a composite solution out of the two solutions.

x

y(x)

Figure 1.1: Non-uniform convergence - The solution to the singularly perturbed

problem of Eq. (1.5). From top to bottom, the black dotted line, blue dotted line and red

solid line are the solutions to the problem with ε equals 0.2, 0.06 and 0.004 respectively.

Note that the solutions approaches that of the reduced problem of Eq. (1.6) except for at

x = 0.

We apply the technique of asymptotic expansion to seek an approximated solution to

the singularly perturbed problem. The paper is structured as follows. In Chapter 2 we

describe the methodology of the project. Section 2.2 introduces singular perturbation

using an ODE problem as an example. In Section 2.3, we apply the analysis to the

Heston pricing PDE. The outer and inner solutions are combined using the technique

described in Section 2.4. The detail of the numerical implementation can be found in

Section 2.5. Chapter 3 presents the numerical results with discussions. In Section 3.1

the calibration scheme is tested on the stylized ODE example. Section 3.2 shows the

main results with observation and discussion on the performance of the scheme under

various settings. Finally, we conclude the investigation in Chapter 4.

4

2

Methodology

2.1 Introduction

In this section we lay out the methodology for the investigation. We first motivate

the discussion by considering the asymptotic expansion of a second order ODE with

constant coefficients as a stylized example. We explain, using such a problem as an

example, how rescaling and solutions matching work. Then we analyze the Heston

problem with similar techniques, carefully noting how the asymptotic expansion of

Heston model deviates from the stylized example.

2.2 Asymptotic Expansion of a Second-Order ODE Prob-

lem

We study a stylized example problem that is adapted from [15]. We try to solve the

second order ODE

εd2y

dx2+ b

dy

dx+ cy = 0

y(x1) = α

y(x2) = β,

(2.1)

where ε is a small parameter, b and c are constants that do not depend on x, and

x2 > x1. As pointed out in Section 1.2, the order of the problem drops as ε → 0, which

leads to a singularity on the boundary. We handle the singularity by the following

procedure: (1) Seek the outer solution by solving the outer problem; (2) Seek the inner

5

2. METHODOLOGY

solution by solving the inner problem; (3) Create a composite solution that is valid over

the entire domain by matching the outer and inner solutions.

Outer Problem: We perform a regular expansion

yo = yo0(x) + εyo1(x) + ε2yo2(x) + · · · (2.2)

where yo is the outer solution. Substituting this regular expansion into Eq. (2.1), the

O(1) equation is

bdyo0dx

+ cyo0 = 0

yo0(x2) = β(2.3)

which has the solution

yo0(x) = βecx2/be−cx/b (2.4)

The O(ε) equation, using results from Eq. (2.4), is

bdyo1dx

+ cyo1 = −d2yo0dx2

= −c2

b2βecx2/be−cx/b

yo1(x2) = 0

(2.5)

where the boundary condition reflects the fact that the full outer solution at x2 is

dominated by yo0(x2). The solution to Eq. (2.5) is

yo1(x) =c2

b3βecx2/be−cx/b(1− x) (2.6)

Inner Problem: To expand the problem properly near the singularity, a rescaling is

required. Define

Z ≡ x

δ, (2.7)

where δ is the width of the boundary layer. With this change of variable Eq. (2.1)

becomesε

δ2d2y

dZ2+

b

δ

dy

dZ+ cy = 0

y(x1) = α

(2.8)

If δ is chosen so that O(ε) = O(δ), then

d2yi

dZ2+ b

dyi

dZ+ εcyi = 0

y(x1) = α

(2.9)

6

2.2 Asymptotic Expansion of a Second-Order ODE Problem

where yi denotes the inner solution. This rescaled problem can now be expanded as

yi = yi0(Z) + εyi1(Z) + ε2yi2(Z) + · · · (2.10)

Substituting this expansion into Eq. (2.1), the O(1) equation is

d2yi0dZ2

+ bdyi0dZ

= 0

yi0(x1) = α

(2.11)

which has the solution

yi0(Z) = α−B +Bebx1/δe−bZ (2.12)

where B is a constant to be determined. The O(ε) equation, using results from Eq.

(2.12), isd2yi1dZ2

+ bdyi1dZ

= −yi0(Z) = −(α−B)−Bebx1/δe−bZ

yi1(x1) = 0

(2.13)

with the solution

yi1(Z) = −C(1− e−bZ

)− α

bZ +

B

bZ(1− ebx1/δe−bZ

)(2.14)

where C is a constant to be determined.

Solutions Matching: We want to find a composite solution that is uniform for the

whole domain x ∈ [x1, x2]. Before we can form such a composite solution using both

outer and inner solutions, we must first determine the unresolved constants B and C

in Eq. (2.12) and (2.14). This is done by using the Van Dyke method [15, 16]. The

first step is to express the outer solution in terms of the scaled variable,

yo(Z) = βecx2/be−cδZ/b + δ

[c2

b3βecx2/be−cδZ/b(1− δZ)

](2.15)

Next, the inner solution is expressed in terms of the outer variable,

yi(x) = α−B +Bebx1/δe−bx/δ − δ

[C(1− e−bx/δ

)+

α

b

x

δ− B

b

x

δ

(1− ebx1/δe−bx/δ

)](2.16)

Van Dyke methods requires that the expansion of Eq. (2.15) up to the first order of δ,

yo(Z), be equal to the expansion of Eq. (2.16) up to the first order of δ, yi(x). Carrying

out the expansions and using Z = x/δ, it can be shown that

C =c2

b3βecx2/b

B = α− βecx2/b(2.17)

7

2. METHODOLOGY

The composite solution, which is uniform everywhere, is constructed by subtracting

the common component yo(Z) (or equivalently yi(x)) from the sum of the outer and

inner solutions, or

yc(x) = yo(x) + yi(x)− yo(Z)

= βecx2/be−cx/b + δ

[c2

b3βecx2/be−cx/b(1− x)

]

+ α−B +Bebx1/δe−bx/δ − δ

[C(1− e−bx/δ

)+

α

b

x

δ− B

b

x

δ

(1− ebx1/δe−bx/δ

)]

− βecx2/b(1− c

bx)− δ

c2

b3βecx2/b

(2.18)

2.3 Asymptotic Expansion as Applied to Call Option Pric-

ing Under Heston Model

To recap, with X = logS, the return and variance processes of the Heston model are

dX =

(r − 1

2v − y

)dt+

√vdW 1

dv = κ(θ − v)dt+ ξ√vdW 2

(2.19)

The pricing PDE of a call under the Heston model is

v

[1

2ξ2uvv

]+v [ρξuXv − κuv ]+

1

2v [uXX − uX ] +[κθuv]+[(r−y)uX−ut−ru] = 0 (2.20)

with u(X, v;T ) = (ST −K)+. Note how in Eq. (2.20) the terms are grouped into five

categories by square brackets:

1. Terms with second-order v derivatives and multiplied by v;

2. Terms with first-order v derivatives and multiplied by v;

3. Terms with zeroth-order v derivatives and multiplied by v;

4. Terms with first-order v derivatives and not multiplied by v; and

5. Terms with zeroth-order v derivatives and not multiplied by v.

8

2.3 Asymptotic Expansion as Applied to Call Option Pricing UnderHeston Model

Immediately we recognize a substantial difference from the stylized example. Here we

are dealing with a multivariable PDE with two spatial dimensions and one temporal

dimension. We would like to treat the variance v as the small parameter for power series

expansion. However, unlike ε in the stylized problem, v is a varying quantity that is not

necessarily small. Also, the solution matching procedure in the stylized example relies

on the two Dirichlet boundary conditions, while in finite difference derivative pricing

we do not have Dirichlet boundary conditions at zero variance. However, we do know

the exact price of the derivative at expiry since the price must equal the payoff on that

date. Hence in what follows we present a heuristic solution patching recipe by taking

advantage of the known derivative price at expiry.

Outer Problem: We propose to solve the full problem for the outer layer. The reason

for not having a separate outer problem that is distinct from the full problem is that

the variance v is not a small parameter in the outer layer. While we could in principle

introduce a change of variable so that the PDE is with respect to v−1 and the outer

series solution is in powers of v−1, this change of variable itself would have a singularity

at v = 0.

Inner Problem: For the inner problem, rescale the variance

W =v

δ(2.21)

Substituting the expansion

u = ui0 + vui1 + · · · (2.22)

into Eq. (2.20), it can be shown that the equation with spatial O(1) and temporal O(δ)

terms is1

2Wξ2

∂2ui0∂W 2

+ κθ∂ui0∂W

= −δ∂ui0∂t

(2.23)

The equation with spatial O(δ) and temporal O(δ2) terms is

1

2Wξ2

[W

∂2ui1∂W 2

+ 2∂ui1∂W

]+W

[ρξ

∂2ui0∂W∂X

− κ∂ui0∂W

]

+ κθ

[W

∂ui1∂W

+ ui1

]+

[(r − y)

∂ui0∂X

− rui0

]= −δ

∂ui1∂t

(2.24)

For the boundary conditions imposed on Eq. (2.23) and (2.24) in the numerical imple-

mentation, please refer to Section 2.5.

9

2. METHODOLOGY

Composite Solution Construction: As mentioned earlier, to patch the outer and

inner solutions we make use of the fact that the price of the derivative at expiry is

know exactly. For both the outer and inner problems, we start with the same initial

conditions, which is the payoff, and march back in physical time. Once the solutions

at t are computed, we would like to use the two solutions to create a single continuous

and smooth composite solution that is valid for the entire range of variance. However,

we cannot directly follow the matching procedure outlined in Section 2.2 because of the

lack of Dirichlet boundary conditions on the variance dimension. On the other hand,

we do know the desired properties of the composite solution, such as continuity and

differentiability. Most importantly, the composite solution should have the same value

as ui as v → 0 and as uo as v → ∞. Hence we propose a heuristic solution patching

method to obtain the composite solution.

2.4 Composite Solution Construction Using Weighting Func-

tions

We construct the composite solution uc as a weighted sum of ui and uo. Define ηi(v)

and ηo(v) to be the weighting functions such that

uc(v) = ηi(v)ui(v) + ηo(v)u

o(v) (2.25)

The functions ηi,o should fulfill a set of criteria, listed and explained as below.

limv→∞ηi(v) = 0 and limv→0ηo(v) = 0:

At very large (small) values of variances, the composite solution should be dominated

by the outer (inner) solution.

ηi(ηo) decreases (increases) monotonically:

There are only two limits that are being considered, the small variance limit (which is

handled by the inner solution) and the large variance limit (which is handled by the

outer solution). Any local extremum of η(v) would signify a third, intermediate regime.

ηi,o are continuous:

This is a necessary condition for the composite solution to be continuous in v.

ηi,o are smooth:

This is a necessary condition for the composite solution to be differentiable with respect

to v.

10

2.4 Composite Solution Construction Using Weighting Functions

These criteria are fulfilled by many functions. We investigate two possibilities:

exponential functions and logistic functions. The exponential weighting functions are

ηi(v) = aexp(−αv)− exp(−αvmax)

1− exp(−αvmax)

ηo(v) = bexp (β(v − vmax))− exp (β(vmin − vmax))

1− exp (β(vmin − vmax))

(2.26)

where α and β are to be determined by fitting. Note that each value of X on the grid

has a unique set of fitting parameters. The exponential weighting functions each has

only two free parameters, and they are either convex (for positive α and β) or concave

(for negative α and β) over the whole range of v. The second class of function, the

logistic weighting functions, are

ηi(v) = ah(α+ βv)− h(α + βvmax)

h(α) − h(α+ βvmax)

ηo(v) = bh(γ) + h(γ + λv)

h(γ)− h(γ + λvmax)

(2.27)

where h(x) = 1/(1+exp(−x)). In the logistic weighting functions we allow for two free

parameters, and the functions need not be convex or concave over the whole range.

x

η(x)

Figure 2.1: Exponential weighting functions - Examples of the exponential weighting

functions (a = b = 1): red decreasing solid line is ηi with α = 5; red decreasing dotted line

is ηi with α = −5; blue increasing solid line is ηo with β = 5; blue increasing dotted line is

ηo with β = −5.

11

2. METHODOLOGY

x

η(x)

Figure 2.2: Logistic weighting functions - Examples of the logistic weighting functions

(a = b = 1): red decreasing solid line is ηi with γ = −10 and λ = 20; red decreasing dotted

line is ηi with γ = 10 and λ = 20; blue increasing solid line is ηo with γ = 10 and λ = −20;

blue increasing dotted line is ηo with γ = 10 and λ = 20.

2.5 Numerical Implementation

In this section we detail the numerical implementation of the method that is developed

in Section 2.3, with emphasis on some caveats. We set up a mesh at expiry, with prices

u(XT , v, T ) equal to the known payoff of a call. We also fix the value of δ. The range

of X on the mesh should be determined by discretion depending on the behavior of the

instrument. The discretized version of the PDE’s in Eq. (2.20), (2.23) and (2.24) can

be found in Appendix A.

The boundary conditions that are imposed are as follows. For ui0 and ui1, convection-

only condition is used for the upper variance boundary. For the lower variance boundary

where W = 0, the full equation coincides with the convection-only condition, hence we

solve the full PDE on the lower variance boundary. Convection-only condition for the

X derivatives is also used for both the upper and lower X boundaries, given that Xmax

and Xmin are chosen so that they are far enough from the strike. For solving uo, exactly

the same holds for the boundary conditions for X derivatives on the Xmax and Xmin

boundaries. For the variance dimension, convection-only condition is used on both the

upper and lower boundary.

We set the initial condition to be uo(T ) = u(T ), ui0(T ) = u(T ) and ui1 = 0, where

u(T ) is the payoff of the instrument at expiry. The outer and inner solutions are solved

individually by marching. However, since the PDE for ui1 involves a source term that

12

2.5 Numerical Implementation

depends on ui0, ui0(t) has to be solved at each time step before ui1 is solved. Suppose

at τ < T we have uo(τ), ui0(τ) and ui1(τ). If [vmin, vmax] of the outer solution and

δ× [Wmin,Wmax] do not coincide, extrapolation is required to extend the inner solution

computed from finite differencing to [vmin, vmax]. Assuming the variance axis contains

n grid points, the error minimization for seeking the weighting function parameters

amounts to solving

argminΓ

n∑k=1

[u∗(vk)− ucom(vk)]2 (2.28)

where u∗ is the analytic Heston call price, Γ stands for the set of free parameters to be

optimized (i.e. {a, b, α, β} for the exponential weighting functions and {a, b, α, β γ, λ}for the logistic weighting functions), and

ucom(vk) = ηi(vk,Γ)(ui0(vk) + vku

i1(vk)

)+ ηo(vk,Γ)u

o(vk) (2.29)

is the composite solution.

13

2. METHODOLOGY

14

3

Numerical Results

3.1 Testing the Solution Patching on the Stylized Exam-

ple

In Section 2.4 we introduced a method to construct the composite solution from the

inner and outer solutions. The ODE has an exact solution

yexact(x) =e−x/2 − e−2x/ε

e−1/2 − e−2/ε(3.1)

Since the exact solution as well as the inner and outer solutions (Eq. (2.16) and Eq.

(2.15)) to the stylized example introduced in Section 2.2 are known, we can test the

proposed patching recipe by applying it to the stylized example.

As we can see from Figure 3.1, the constructed composite solution approaches the

exact one as δ decreases. This is just as expected for an approximation scheme that is

correct to the first order of δ. For large values of δ, the O(δ2) residual term becomes

significant.

3.2 Calibration to Heston Call Prices

We calibrate the weighting functions with the techniques described in Section 2.5. The

CPU time is listed in Table 3.1, and it is compared to a single-mesh FDM run that

calculates using. The quantity using is the solution computed using a single mesh, and

with the boundary condition

−ut = κθuv + (r − y)uX − ru (3.2)

15

3. NUMERICAL RESULTS

x x x

y(x)

Figure 3.1: Composite versus exact solutions for the stylized example - Con-

structed composite solution (obtained using the exponential weighting functions) are com-

pared to the exact solutions for the stylized example. The value of δ, the scaling parameter,

for the left, central and right panels are 1.9, 0.4 and 0.01, respectively. The red dotted

curves are constructed using the method outlined in Section 2.4, while the black solid

curves are the exact solutions to the ODE.

imposed on the lower variance boundary v = 0 instead of the convection-only condition.

Grid T/Δt Grid Size CPU Time (s)

Single 200 31 ×31 106

Inner and Outer 200 31 ×31 182

Single 300 51 ×51 924

Inner and Outer 300 51 ×51 1536

Table 3.1: CPU time for various settings - The CPU time required for solving the

pricing problem under different settings. The second column, T/Δt, is the number of time

steps. For pricing using the scheme with inner and outer layers, the inner and outer grids

are of the same size.

Figure 3.2 shows the composite solution ucom that is constructed using this pro-

cedure, and Figure 3.3 shows the logistic outer weighting function that is produced

by the calibration. To evaluate the performance of calibration, we compare the fitted

composite solutions ucom(v) to the analytic prices u∗(v). The details regarding the

calculation of the analytic Heston call price, as well as a table containing prices that

are used for calibration, can be found in Appendix B. Figure 3.4 shows ucom − u∗ as a

function of v for specific spot prices. For comparison we have also plotted usingu∗ on

the same graphs.

Obviously, the composite solution matches the analytic solution, which it is designed

16

3.2 Calibration to Heston Call Prices

0.526

0.708

1.164

0

0.2

0.4−0.1

0

0.1

0.2

0.3

0.4

Sv

u com

Figure 3.2: Surface plot of composite solution - The graph shows the composite

solution ucom that is calibrated using the logistic weighting function. The call option

under consideration has expiry T = 1 and strike K = 0.95. Note that the tick interval

on the spot price axis is uneven because of log scale. The parameters for the stochastic

processes are r = 0.06, y = 0.02, κ = 0.5, θ = 0.0625, ξ = 0.24 and ρ = −0.75, and δ = 0.2.

2κθ/ξ2 = 1.085.

0.580

0.822

1.164

0

0.2

0.40

0.5

1

1.5

2

Sv

η

Figure 3.3: Surface plot of the outer weighting function - The logistic weighting

function ηo(v) as a surface plot for a range of spot prices. The call option under consider-

ation has expiry T = 1 and strike K = 0.95. The parameters for the stochastic processes

are r = 0.06, y = 0.02, κ = 0.5, θ = 0.0625, ξ = 0.24 and ρ = −0.75, and δ = 0.05.

2κθ/ξ2 = 1.085.

17

3. NUMERICAL RESULTS

0.552

0.782

1.107

0

0.2

0.4−0.02

−0.01

0

0.01

Sv

u com

− u

*

0.552

0.782

1.107

0

0.2

0.4−0.02

−0.01

0

0.01

Sv

u sing

− u

*Figure 3.4: Deviation from the analytic price - The graphs show the deviation

from the analytic price u∗ as a function of v. The left panel shows the surface for ucom,

while the right panel shows the surface for using. The call option under consideration

has expiry T = 1 and strike K = 0.95. The parameters for the stochastic processes are

r = 0.06, y = 0.02, κ = 0.5, θ = 0.0625, ξ = 0.24 and ρ = −0.75, and δ = 0.2 for ucom.

2κθ/ξ2 = 1.085.

to imitate, much better than using. If exponential weighting function is used instead

of logistic weighting function, the calibration performance is greatly compromised, as

shown in Figure 3.5. This can be explained by the fewer free parameters, as well as

the more rigid shape, of the exponential weighting function. The table that lists the

calibration error can be found in Appendix C.

Finally we note the effect of δ on the calibration performance. The parameter δ

is the width of the boundary layer that is chosen according to the magnitude of the

small perturbation parameter (see Section 2.2). Since the perturbation parameter in

the inner layer, v, is a changing quantity, the condition O(δ) ≈ O(v) cannot be fulfilled

if δ is chosen to be too small. Figure 3.6 illustrates this point.

18

3.2 Calibration to Heston Call Prices

0.552

0.782

1.107

0

0.2

0.4−0.2

−0.1

0

0.1

Sv

u com

− u

*

0.552

0.782

1.107

0

0.2

0.4−0.2

−0.1

0

0.1

Sv

u com

− u

*

Figure 3.5: Deviation from the analytic price - The graphs show the deviation

from the analytic price u∗ as a function of v. The left panel shows the surface calibrated

with logistic function, while the right panel shows the surface calibrated with exponential

function. The call option under consideration has expiry T = 1 and strike K = 0.95.

The parameters for the stochastic processes are r = 0.06, y = 0.02, κ = 0.5, θ = 0.0625,

ξ = 0.24 and ρ = −0.75, and δ = 0.2. 2κθ/ξ2 = 1.085.

0.552

0.782

1.107

0

0.2

0.4−0.1

−0.05

0

0.05

0.1

Sv

u com

− u

*

0.552

0.782

1.107

0

0.2

0.4−0.1

−0.05

0

0.05

0.1

Sv 0.552

0.782

1.107

0

0.2

0.4−0.1

−0.05

0

0.05

0.1

Sv

Figure 3.6: Effect of the width of the boundary layer - The graphs show the

deviation from the analytic price u∗ as a function of v. The left panel shows the surface

with δ = 0.05, the central panel shows the surface with δ = 0.1, while the right panel shows

the surface with δ = 0.5. The call option under consideration has expiry T = 1 and strike

K = 0.95. The parameters for the stochastic processes are r = 0.06, y = 0.02, κ = 0.5,

θ = 0.0625, ξ = 0.24 and ρ = −0.75. 2κθ/ξ2 = 1.085.

19

3. NUMERICAL RESULTS

20

4

Conclusion

We studied a finite difference method to price options under the Heston volatility model.

Making use of the fact that the pricing PDE under Heston model contains highest-order

derivatives that are multiplied by v, we derived the inner problem that is valid in the

inner layer. The inner and outer problems are solved individually with FDM, and the

two solutions are combined, with either exponential or logistic weighting functions,

to form a composite solution by calibrating against the analytic Heston vanilla call

prices. The logistic weighting functions, with more degrees of freedom, outperform the

exponential counterparts. An Important observation is that the fitting scheme works

well for δ that is not too small, and the calibration performance for δ < 0.1. The

calibrated weighting functions can be used to price other instruments that do not have

closed-form solution under the Heston model.

The calibration residual errors for ITM spot prices are large, possibly because of

the difficulty in specifying a unique set of fitting parameters. An area for further

investigation is to improve the optimization process using genetic algorithm, which is

more effective in avoiding local minima.

21

4. CONCLUSION

22

Appendix A

Discretized PDE’s for Inner and

Outer Problems

Here we provide the discretized PDE’s for the interior points using central differencing

schemes. The indices i and j correspond to the variance and the log of stock price,

respectively. The variable t is the physical time. The discretized version of Eq. (2.20),

which is used in calculating the outer solution uo, is

−∂u

∂t=

[− v

ΔX2− ξ2v

Δv2− r

]uoi,j

+

[ξ2v

2Δv2+

κ(θ − v)

2Δv

]uoi+1,j +

[ξ2v

2Δv2+

κ(θ − v)

2Δv

]uoi−1,j

+

[v

2ΔX2+

(r − y − v

2

) 1

2ΔX

]uoi,j+1 +

[v

2ΔX2−

(r − y − v

2

) 1

2ΔX

]uoi,j−1

+

[ρξv

4ΔXΔv

]uoi+1,j+1 +

[− ρξv

4ΔXΔv

]uoi+1,j−1

+

[− ρξv

4ΔXΔv

]uoi−1,j+1 +

[ρξv

4ΔXΔv

]uoi−1,j−1

(A.1)

The discretized version of Eq. (2.23), which is used in calculating the inner solution

ui0, is

−δ∂u

∂t=

[− ξ2W

ΔW 2

](ui0)i,j +

[ξ2W

2ΔW 2+

κθ

2ΔW

](ui0)i+1,j +

[ξ2W

2ΔW 2− κθ

2ΔW

](ui0)i−1,j

(A.2)

23

A. DISCRETIZED PDE’S FOR INNER AND OUTER PROBLEMS

The discretized version of Eq. (2.24), which is used in calculating the inner solution

ui1, is

−δ∂u

∂t=

[−ξ2W 2

ΔW 2+ κθ

](ui1)i,j

+

[ξ2W 2

2ΔW 2+

(ξ2 + κθ

)W

2ΔW

](ui1)i+1,j +

[ξ2W 2

2ΔW 2−

(ξ2 + κθ

)W

2ΔW

](ui1)i−1,j + f

(ui0

)(A.3)

where f(ui0

)is a source term given by

f(ui0

)= [−r] (ui0)i,j +

[− κW

2ΔW

](ui0)i+1,j +

[κW

2ΔW

](ui0)i−1,j

+

[(r − y)

2ΔX

](ui0)i,j+1 +

[−(r − y)

2ΔX

](ui0)i,j−1

(A.4)

24

Appendix B

Analytic Heston Call Prices

To obtain the analytic option prices that are used for calibration, we implement the

closed-form solution of call price found in [1]. The integral in the closed-form formula

is calculated as a Riemann sum. The following tables list the prices that are obtained

as such.

25

B. ANALYTIC HESTON CALL PRICES

v/S 0.74411 0.78203 0.82188 0.86375 0.90777

0.00000 7.1745 × 10−5 0.00047078 0.0030314 0.013112 0.034289

0.0080000 0.00039172 0.0018335 0.0071906 0.020330 0.042635

0.016000 0.0011527 0.0041280 0.011957 0.027048 0.049984

0.024000 0.0024573 0.0070908 0.016858 0.033252 0.056575

0.032000 0.0042647 0.010433 0.021680 0.038990 0.062575

0.040000 0.0064571 0.013950 0.026337 0.044326 0.068104

0.048000 0.0089095 0.017517 0.030802 0.049319 0.073248

0.056000 0.011521 0.021063 0.035076 0.054017 0.078074

0.064000 0.014220 0.024553 0.039166 0.058462 0.082630

0.072000 0.016956 0.027966 0.043089 0.062687 0.086955

0.080000 0.019699 0.031295 0.046857 0.066720 0.091080

0.088000 0.022428 0.034537 0.050484 0.070582 0.095029

0.096000 0.025129 0.037695 0.053983 0.074293 0.098822

0.10400 0.027796 0.040770 0.057366 0.077869 0.10248

0.11200 0.030423 0.043766 0.060641 0.081321 0.10600

0.12000 0.033008 0.046688 0.063818 0.084663 0.10942

0.12800 0.035550 0.049539 0.066904 0.087902 0.11273

0.13600 0.038049 0.052323 0.069907 0.091048 0.11594

0.14400 0.040506 0.055044 0.072831 0.094109 0.11907

0.15200 0.042920 0.057705 0.075684 0.097089 0.12212

0.16000 0.045294 0.060311 0.078468 0.099996 0.12509

0.16800 0.047628 0.062862 0.081189 0.10283 0.12799

0.17600 0.049924 0.065363 0.083851 0.10561 0.13082

0.18400 0.052183 0.067817 0.086457 0.10832 0.13359

0.19200 0.054406 0.070224 0.089010 0.11098 0.13631

Table B.1: Analytic Heston call prices - The call option under consideration has

expiry T = 1 and strike K = 0.95. The parameters for the stochastic processes are

r = 0.06, y = 0.02, κ = 0.5, θ = 0.0625, ξ = 0.24 and ρ = −0.75. The first row of the table

is the spot price, while the first column of the table is the spot variance.

26

v/S 0.74411 0.78203 0.82188 0.86375 0.90777

0.20000 0.056595 0.072588 0.091513 0.11358 0.13897

0.20800 0.058751 0.074911 0.093969 0.11613 0.14158

0.21600 0.060874 0.077195 0.096380 0.11863 0.14413

0.22400 0.062967 0.079440 0.098749 0.12109 0.14665

0.23200 0.065029 0.081650 0.10108 0.12351 0.14912

0.24000 0.067063 0.083826 0.10337 0.12588 0.15154

0.24800 0.069069 0.085968 0.10562 0.12821 0.15393

0.25600 0.071049 0.088078 0.10784 0.13051 0.15628

0.26400 0.073002 0.090159 0.11002 0.13277 0.15859

0.27200 0.074930 0.092209 0.11217 0.13500 0.16087

0.28000 0.076834 0.094232 0.11429 0.13719 0.16311

0.28800 0.078715 0.096228 0.11638 0.13936 0.16532

0.29600 0.080573 0.098197 0.11844 0.14149 0.16751

0.30400 0.082408 0.10014 0.12048 0.14359 0.16966

0.31200 0.084223 0.10206 0.12248 0.14567 0.17178

0.32000 0.086017 0.10396 0.12446 0.14771 0.17387

0.32800 0.087790 0.10583 0.12642 0.14974 0.17594

0.33600 0.089544 0.10768 0.12835 0.15173 0.17798

0.34400 0.091280 0.10951 0.13026 0.15370 0.18000

0.35200 0.092996 0.11132 0.13215 0.15565 0.18199

0.36000 0.094695 0.11311 0.13401 0.15758 0.18396

0.36800 0.096376 0.11488 0.13586 0.15948 0.18591

0.37600 0.098041 0.11663 0.13768 0.16136 0.18784

0.38400 0.099689 0.11836 0.13949 0.16323 0.18974

0.39200 0.10132 0.12008 0.14127 0.16507 0.19163

Table B.2: Analytic Heston call prices - Continued.

27

B. ANALYTIC HESTON CALL PRICES

v/S 0.95402 1.0026 1.0537 1.1074 1.1638

0.00000 0.065267 0.10375 0.14802 0.19699 0.24999

0.0080000 0.073158 0.11048 0.15342 0.20115 0.25312

0.016000 0.080135 0.11658 0.15849 0.20521 0.25629

0.024000 0.086429 0.12220 0.16328 0.20916 0.25946

0.032000 0.092193 0.12742 0.16783 0.21301 0.26263

0.040000 0.097534 0.13233 0.17218 0.21675 0.26578

0.048000 0.10253 0.13698 0.17635 0.22039 0.26890

0.056000 0.10724 0.14139 0.18036 0.22395 0.27198

0.064000 0.11170 0.14561 0.18423 0.22741 0.27502

0.072000 0.11595 0.14965 0.18797 0.23080 0.27802

0.080000 0.12002 0.15354 0.19159 0.23410 0.28098

0.088000 0.12392 0.15729 0.19511 0.23733 0.28390

0.096000 0.12768 0.16092 0.19853 0.24050 0.28677

0.10400 0.13131 0.16443 0.20186 0.24359 0.28960

0.11200 0.13482 0.16784 0.20511 0.24663 0.29239

0.12000 0.13822 0.17116 0.20828 0.24960 0.29514

0.12800 0.14152 0.17438 0.21137 0.25252 0.29785

0.13600 0.14473 0.17753 0.21440 0.25538 0.30052

0.14400 0.14786 0.18060 0.21736 0.25820 0.30315

0.15200 0.15091 0.18360 0.22026 0.26096 0.30575

0.16000 0.15389 0.18653 0.22310 0.26368 0.30832

0.16800 0.15680 0.18940 0.22589 0.26635 0.31085

0.17600 0.15965 0.19222 0.22863 0.26898 0.31334

0.18400 0.16243 0.19497 0.23132 0.27157 0.31581

0.19200 0.16516 0.19768 0.23397 0.27413 0.31824

Table B.3: Analytic Heston call prices - Continued.

28

v/S 0.95402 1.0026 1.0537 1.1074 1.1638

0.20000 0.16784 0.20033 0.23657 0.27664 0.32064

0.20800 0.17047 0.20294 0.23912 0.27912 0.32301

0.21600 0.17305 0.20551 0.24164 0.28156 0.32536

0.22400 0.17558 0.20803 0.24412 0.28397 0.32768

0.23200 0.17807 0.21051 0.24656 0.28635 0.32997

0.24000 0.18052 0.21295 0.24897 0.28869 0.33223

0.24800 0.18293 0.21536 0.25134 0.29101 0.33447

0.25600 0.18530 0.21772 0.25368 0.29330 0.33669

0.26400 0.18764 0.22006 0.25599 0.29556 0.33888

0.27200 0.18994 0.22236 0.25827 0.29779 0.34104

0.28000 0.19221 0.22463 0.26052 0.30000 0.34319

0.28800 0.19445 0.22687 0.26274 0.30218 0.34531

0.29600 0.19665 0.22908 0.26493 0.30433 0.34741

0.30400 0.19883 0.23126 0.26710 0.30646 0.34949

0.31200 0.20098 0.23342 0.26924 0.30857 0.35155

0.32000 0.20310 0.23555 0.27135 0.31066 0.35359

0.32800 0.20520 0.23765 0.27345 0.31272 0.35562

0.33600 0.20727 0.23973 0.27552 0.31477 0.35762

0.34400 0.20931 0.24178 0.27756 0.31679 0.35960

0.35200 0.21133 0.24381 0.27959 0.31879 0.36157

0.36000 0.21333 0.24582 0.28159 0.32078 0.36352

0.36800 0.21530 0.24781 0.28357 0.32274 0.36545

0.37600 0.21726 0.24977 0.28554 0.32469 0.36736

0.38400 0.21919 0.25172 0.28748 0.32661 0.36926

0.39200 0.22110 0.25365 0.28940 0.32852 0.37114

Table B.4: Analytic Heston call prices - Continued.

29

B. ANALYTIC HESTON CALL PRICES

30

Appendix C

Calibration Error

The deviation of the composite solution from the analytic price, expressed in basis

points, is shown in the following tables.

31

C. CALIBRATION ERROR

v/S 0.78203 0.82188 0.86375 0.90777

0.040000 −0.94527 2.2916 0.75275 0.39342

0.048000 0.62891 1.9825 0.82558 0.48253

0.056000 1.5799 1.8242 0.87596 0.53606

0.064000 2.1381 1.7379 0.90801 0.56933

0.072000 2.4512 1.6844 0.92654 0.59015

0.080000 2.6128 1.6452 0.93513 0.60260

0.088000 2.6816 1.6113 0.93614 0.60891

0.096000 2.6934 1.5782 0.93101 0.61030

0.10400 2.6696 1.5436 0.92056 0.60749

0.11200 2.6229 1.5059 0.90528 0.60087

0.12000 2.5607 1.4644 0.88539 0.59066

0.12800 2.4868 1.4184 0.86100 0.57699

0.13600 2.4037 1.3675 0.83213 0.55989

0.14400 2.3122 1.3114 0.79874 0.53936

0.15200 2.2130 1.2497 0.76079 0.51537

0.16000 2.1062 1.1823 0.71821 0.48788

0.16800 1.9917 1.1090 0.67094 0.45686

0.17600 1.8697 1.0297 0.61892 0.42226

0.18400 1.7400 0.94427 0.56212 0.38405

0.19200 1.6025 0.85267 0.50053 0.34224

Table C.1: Deviation from the analytic price - The call option under consideration

has expiry T = 1 and strike K = 0.95. The parameters for the stochastic processes are

r = 0.06, y = 0.02, κ = 0.5, θ = 0.0625, ξ = 0.24 and ρ = −0.75. The first row of the table

is the spot price, while the first column of the table is the spot variance.

32

v/S 0.95402 1.0026 1.0537 1.1074

0.20000 0.10502 0.029638 0.019838 −0.0033900

0.20800 0.071314 0.0063635 0.0037769 −0.014424

0.21600 0.035224 −0.018527 −0.013317 −0.026064

0.22400 −0.0030484 −0.044838 −0.031264 −0.038150

0.23200 −0.043211 −0.072302 −0.049823 −0.050474

0.24000 −0.084862 −0.10056 −0.068685 −0.062775

0.24800 −0.12747 −0.12915 −0.087465 −0.074742

0.25600 −0.17038 −0.15751 −0.10570 −0.086010

0.26400 −0.21274 −0.18497 −0.12286 −0.096175

0.27200 −0.25358 −0.21073 −0.13834 −0.10480

0.28000 −0.29174 −0.23392 −0.15151 −0.11144

0.28800 −0.32593 −0.25360 −0.16169 −0.11565

0.29600 −0.35476 −0.26879 −0.16824 −0.11705

0.30400 −0.37681 −0.27853 −0.17058 −0.11533

0.31200 −0.39067 −0.28196 −0.16824 −0.11032

0.32000 −0.39508 −0.27839 −0.16093 −0.10203

0.32800 −0.38901 −0.26736 −0.14864 −0.090707

0.33600 −0.37179 −0.24881 −0.13164 −0.076918

0.34400 −0.34326 −0.22307 −0.11067 −0.061584

0.35200 −0.30391 −0.19110 −0.086917 −0.046073

0.36000 −0.25497 −0.15449 −0.062196 −0.032269

0.36800 −0.19861 −0.11565 −0.038995 −0.022651

0.37600 −0.13808 −0.077896 −0.020590 −0.020370

0.38400 −0.077841 −0.045593 −0.011150 −0.029342

Table C.2: Deviation from the analytic price - Continued.

33

C. CALIBRATION ERROR

34

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