A Comparison of Local Volatility and Implied Volatility

U.U.D.M. Project Report 2011:12

Examensarbete i matematik, 30 hp

Handledare och examinator: Johan Tysk

Juni 2011

Department of Mathematics

Uppsala University

A Comparison of Local Volatility and

Preface

Abstract

Acknowledgement

Contents

Chapter 1 Introduction---5

1.1 Motivation---5

1.2 Objectives---5

1.3 Chapter Review---6

Chapter 2. Background---8

2.1 The Local Volatility Models---8

2.2 The Dupire Model(Method)---10

Chapter 3 Implied Volatility Models---17

3.1 The Local Volatility Model---17

3.1.1 Option Pricing---17

3.1.2 Implied Volatilities and The Local Volatilities---22

3.1.3 Implied Volatilities and The Dupire Volatilities---41

3.1.4 Summary of Three Types of Volatilities---55

Chapter 4 Conclusions and Future Studies---58

Notation---60

Appendix A---61

Appendix B---89

Chapter 1. Introduction

1.1 Motivation

In general, volatility is a measure for variation of price of a certain financial instrument over time

in finance. There are many types of volatilities categorized by different standards. For example,

historical volatility is a type of volatility derived from time series based on the past market prices; a

constant volatility is an assumption of the nature of volatility that we usually make in deriving the

Black-Scholes formula for option prices. An implied volatility, however, is a type of volatility

derived from the market-quoted data of a market traded derivative, such as an option.

One of the most frequently used models, the Black-Scholes model which assumes a constant

volatility is used to derive the corresponding implied volatility for each quoted market price for

options. Indeed, the Black-Scholes model has been a great contribution to option pricing area

Nevertheless, there are still some facts that contradict the key assumptions in Black-Scholes model,

especially the constant volatility assumption. The evidence to this contradiction is a long-observed

pattern of implied volatilities, in which at-the money options tend to have lower implied volatilities

than in- or out-of-the-money options. This pattern is called "the volatility smile"(sometimes

referred to as "volatility skew") which was starting to show in American markets after the huge

stock market crash in 1987.

One explanation for this phenomenon is that in reality the volatility of an underlying asset is not

really a constant value throughout the lifespan of the derivative. That is why the volatility curve

plotted by the using of the values of implied volatility inferred by Black-Scholes model does not

appear to be horizontal, but displays a "volatility smile" in the plots. This, however, motivates us to

wonder whether such a model can be found, that gives a series of values of volatility close enough

to the volatility values in the volatility smile, i.e. , the implied volatility; or more specifically what

the difference between the volatility given by this alternative model and the corresponding implied

volatility inferred by Black-Scholes model is, if any.

Having this thought in mind, we can also apply this scheme of searching for suitable models to

testing among different types of models. Our demonstration in this paper uses the local volatility

model.

1.2 Objectives

Our objective here is to set up the pricing model for options using the stock price processes and

other conditions specified by the local volatility model, solve the option values for this model,

calculate the corresponding implied volatilities for this model, thus to achieve our goal of

comparing these two volatilities, the implied volatilities and the local volatilities.

Besides the local volatility given by the local volatility model, we also want to compare the implied

volatilities to another local volatility, the dupire volatility. The Dupire volatility is a way of

calculating volatility under the Dupire model, which treats the strike price

K

and the maturity

time

T

instead of the stock price

S

and current time point

t

as variables in the option value

function

V

(

K

,

T

;

S

,

t

)

. We will introduce this Dupire model and Dupire volatility in detail in

Chapter 2. This additional analysis would give us some additional points of views to this local

volatility model here.

1.3 Chapter Review

Chapter 1, Introduction, mainly talks about the theoretical and practical reasons that motivate us to

write about this topic on implied volatility models in this paper, and sets straight the objectives of

our research as well.

Chapter 2. Background

In this chapter, we briefly introduce the models we use in this paper.

2.1 The Local Volatility Models

In the 1970s, when Black-Scholes formula was initially derived, most people were convinced that

the volatility of a certain asset given the current circumstance was a constant number. Then, later

on,after the economic crash in 1987, people were starting to doubt the constant volatility

assumption. Especially after more and more evidence of volatility smile was collected, people tend

to believe that the implied volatilities can not remain constant during the whole time. They

probably have some dependent relationships with some other factors in the option pricing model as

well. One of such guesses is that, the implied volatility could be depending on the stock price

and time

. And if we study a model of price processes with a volatility that depends on the

)

(

t

S

t

stock price

S

(

t

)

and time

t

, we can try to explore the inner connection between the implied

volatility

σ

_imp

, and the local volatility

σ

(

S

(

t

),

t

)

. The volatility in such models depends on the

stock price

S

(

t

)

and time

t

. This is why we call these types of models the local volatility models,

whose volatilities are determined locally.

Hence, we take one example out of this category, and consider a case where the volatility is

decreasing with respect to the stock prices.

Given the local volatility model under an EMM(equivalent martingale measure, we use the same

acronym in the following)

Q

as following,

,

(2.1)

dt

r

dW

S

t

S

dS

=

σ

(

,

)

⋅

⋅

_t

+

⋅

where we assume,

,

(2.2)

t t

S

S

)

1

(

=

σ

.

(2.3)

0

=

r

By (2.2) and (2.3), the original model (2.1) is degenerated into the following form:

.

(2.4)

t t t

S

dW

dS

=

⋅

Denote the option value function as

V

(

S

_t

,

t

)

.

Hence, it follows from Ito formula and equation (2.4) that,

2 2 2

)

(

2

1

dW

S

S

V

dS

S

V

dt

t

V

∂

∂

+

∂

∂

+

∂

∂

=

(2.5)

dS

S

V

dt

S

S

V

t

V

∂

∂

+

∂

∂

+

∂

∂

=

)

2

1

(

₂ 2

Then we consider delta-hedged portfolio,

.

(2.6)

S

S

V

V

∂

∂

+

−

=

π

Gven the martingale measure

Q

, thus under arbitrage-free condition, we will arrive at the

condition that,

.

(2.7)

dt

r

d

π

=

⋅

π

⋅

We re-write (2.6) in differential form that,

.

(2.8)

dS

S

V

dV

d

∂

∂

+

−

=

π

Compare (2.8) with (2.7), then insert (2.5), the corresponding partial differential equation (PDE)

for model (2.4) takes the form,

.

(2.9)

0

2

1

2 2

=

∂

∂

+

∂

∂

S

S

V

t

V

If we let

V

(

S

_t

,

t

)

,

V

_t

(

S

_t

,

t

)

and

V

_SS

(

S

_t

,

t

)

represent the option value, the first-order partial

derivative with respect to variable

t

, the second-order partial derivative with respect to variable

, respectively, (2.9) can be expressed in the following way,

t

S

.

(2.10)

0

)

,

(

2

1

)

,

(

S

t

+

SV

S

t

=

V

_{t t ss t}

Model (2.4) is known as one of the local volatility models, whose form can be included into the

SDEMRD Model category inside the matlab database.

Creating the Local Volatility Model from Mean-Reverting Drift (SDEMRD) Models

The SDEMRD class derives directly from the SDEDDO class. It provides an interface in which the

drift-rate function is expressed in mean-reverting drift form:

,

(2.11)

t t t t t

S

t

L

t

X

dt

D

t

X

V

t

dW

dX

=

(

)[

(

)

−

]

⋅

+

(

,

α()

)

⋅

(

)

where，

X

t

is an NVARS-by-1 state vector of process variables;

S

is an NVARS-by-NVARS matrix of mean reversion speeds;

D

is an NVARS-by-NVARS diagonal matrix, where each element along the main diagonal is the

corresponding element of the state vector raised to the corresponding power of

α;

V

is an NVARS-by-NBROWNS instantaneous volatility rate matrix;

dW

t

is an NBROWNS-by-1 Brownian motion vector.

SDEMRD objects provide a parametric alternative to the linear drift form by reparameterizing the

general linear drift such that:

.

(2.12)

)

(

)

(

),

(

)

(

)

(

t

S

t

L

t

B

t

S

t

A

=

=

−

Hence, we can create in matlab the model in

.

(2.4)

t t t

S

dW

dS

=

⋅

by inputing the following command in Matlab. SDEMRD objects display the familiar Speed and

Level parameters instead of A and B.

Table 2.1: The Local Volatility Model in Matlab

2.2 The Dupire Model(Method)

Frankly, this Dupire model is more of a method for calculating local volatilities than a pricing

model itself.

First of all, let us discuss some basic developments on the implied volatilities so far.

>> obj = sdemrd(0, 0, 0.5, 1)

% (Speed, Level, Alpha, Sigma)

obj =

Class SDEMRD: SDE with Mean-Reverting Drift

---Dimensions: State = 1, Brownian = 1

---StartTime: 0

StartState: 1

Correlation: 1

Drift: drift rate function F(t,X(t))

Diffusion: diffusion rate function G(t,X(t))

Simulation: simulation method/function simByEuler

Alpha: 0.5

One of the basic assumption in Black-Scholes is that the volatility of the underlying asset stays

constant during the entire time of option's lifespan. Hence, we can know from the Black-Scholes

formula for option prices, that, option prices has the following form

).

,

,

;

,

(

S

t

K

T

V

V

=

σ

If we quote from the market date, the option price

V =

V

₀

and underlying asset price

S =

S

0

of

an option with strike price

K =

K

₀

and maturity

T =

T

0

at time point

t =

t

0

, we can obtain an

equation from Black-Scholes formula for

σ

,

.

(2.13)

)

,

,

;

,

(

0 0 0 0 0

V

S

t

K

T

V

=

σ

From Black-Scholes formula, we can calculate the Greeks, in particular, vega,

.

0

)

(

)

(

₁

=

₂

>

=

∂

∂

=

− −

τ

φ

τ

φ

σ

ν

V

Se

qτ

d

Ke

rτ

d

Hence

σ =

σ

₀

can be uniquely determined by equation (2.13).

Since the volatility

σ

of the underlying asset is constant by assumption of the Black-Scholes

model. Then, theoretically the implied volatility

σ =

σ

₀

derived from (2.13) should be a constant,

i.e., independent of the strike price

K

₀

and maturity

T

₀

chosen here. However, in reality, this is

contradicted by the existences of volatility smile and volatility skew. In fact,

the implied volatility

inferred from option prices with different strike prices and expiration dates is a function of

σ

,

[1].

T

K ,

σ =

σ

(

K

,

T

)

The dependence on strike prices can be shown by the following figure 2.1 and figure. 2.2, given a

fixed maturity time

T =

T

₀

and a fixed initial price

S =

S

0

at time point

t =

t

0

. The curve in

figure 2.1 is called the volatility smile, the curve in figure 2.2 is called the volatility skew.

Figure 2.1: Volatility Smile

Figure 2.2: Volatility Skew

To explore the characteristics of implied volatility in a more mathematical way, let us discuss the

model analytically.

Under risk-neutral measure, the underlying asset price process is

,

(2.14)

t

dW

t

S

dt

q

r

S

dS

)

,

(

)

(

−

+

σ

=

where

r

is the risk-free interest rate,

q

is the dividend yield,

S

is the asset prices,

{

W

_t

}

0≤_t≤_T

is a Brownian motion(Wiener process),

σ

is the asset's volatility that depends on asset prices

and time

.

S

t

Thus, by using the same approach as in section 2.1, we obtain the PDE for this option under

Black-Scholes model,

.

(2.15)

0

)

(

)

,

(

2

1

2 2 2 2

=

−

∂

∂

−

+

∂

∂

+

∂

∂

rV

S

V

S

q

r

S

V

S

t

S

t

V

σ

Adding the terminal and boundary conditions to equation (2.15), we can estabilish the following

value problem for option price, in particular, an European call option price.

Definition 2.1

G

(

S

,

t

;

ξ

,

T

)

is called the fundamental solution of the Black-Scholes equation, if

it satisfies the following terminal value problem to the Black-Scholes equation:

⎪

⎩

⎪

⎨

⎧

−

=

=

−

∂

∂

−

+

∂

∂

+

∂

∂

=

)

17

.

2

(

),

(

)

,

(

)

16

.

2

(

,

0

)

(

2

2 2 2 2

ξ

δ

σ

S

T

S

V

rV

S

V

S

q

r

S

V

S

t

v

Lv

where

0

<

S

<

∞

,

0

<

ξ

<

∞

,

0

<

t

<

T

,

δ

(

x

)

is the Dirac function..

□

Problem 2.2

Let

V

=

V

(

S

,

t

;

σ

,

K

,

T

)

be a call option price, satisfying the following terminal

value problem:

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

∞

<

≤

−

=

≤

≤

∞

≤

≤

=

−

∂

∂

−

+

∂

∂

+

∂

∂

+

)

19

.

2

(

)

0

(

.

)

(

)

,

(

)

0

,

0

(

)

18

.

2

(

,

0

)

(

)

,

(

2

1

2 2 2 2

S

K

S

T

S

V

T

t

S

rV

S

V

S

q

r

S

V

S

t

S

t

V

σ

Suppose

S =

S

*

at

t

=

t

*

,

(

0

≤

t

*

<

T

₁

)

is given as the boundary condition,

)

,

0

(

)

,

(

)

,

,

;

,

(

1 2 * *

T

T

T

K

T

K

F

T

K

t

S

V

σ

=

<

<

∞

≤

≤

find

σ

=

σ

(

S

,

t

),

(

0

≤

S

<

∞

,

T

₁

≤

t

≤

T

₂

)

.

□

Theorem 2.3

If the fundamental solution

G

(

S

,

t

;

ξ

,

η

)

is regarded as a function of

ξ ,

η

, then

it is the fundamental solution of the adjoint equation of the Black-Scholes equation. That is, let

,

)

,

;

,

(

)

,

(

ξ

η

G

S

t

ξ

η

v

=

then

v

(

ξ

,

η

)

satisfies

⎪

⎩

⎪

⎨

⎧

−

=

=

−

∂

∂

−

−

∂

∂

+

∂

∂

−

=

∗

)

21

.

2

(

),

(

)

,

(

)

20

.

2

(

,

0

)

(

)

(

)

(

2

2 2 2 2

S

t

v

rv

v

q

r

v

v

v

L

ξ

δ

ξ

ξ

ξ

ξ

ξ

σ

η

where

0

,

0

,

1

.

□

η

ξ

<

∞

<

<

∞

<

<

S

t

Corollary 2.4

Theorem 2.1 indicates, if the fundamental solution of equation (2.18) is

, then

)

,

;

,

(

*

t

S

G

ξ

η

□

).

,

;

,

(

)

,

;

,

(

S

t

G

*

S

t

G

ξ

η

=

ξ

η

The proof of above theorem 2.1 and corollary 2.2 can be referred to Lishang Jiang(1994)[1].

Then, let us move on to discuss the Dupire method in detail.

We denote an European call option price as

, define the second derivative of the option prices with respect to strike prices

)

,

;

,

(

S

t

K

T

V

V =

.

(2.22)

)

,

;

,

(

2 2

T

K

t

S

G

K

V

=

∂

∂

By equation (2.22) and (2.23),

G

satisfies the system that

⎪

⎩

⎪

⎨

⎧

−

=

=

−

∂

∂

−

+

∂

∂

+

∂

∂

)

24

.

2

(

,

)

(

)

,

(

)

23

.

2

(

,

0

)

(

)

,

(

2

1

2 2 2 2

K

S

T

S

G

rG

S

G

S

q

r

S

G

S

t

S

t

G

δ

σ

where

δ

(

S −

K

)

is the Dirac function. We know that

δ

(

−

x

)

=

δ

(

x

)

, thus (2.24) can be written

as,

.

(2.25)

)

(

)

(

)

,

(

S

T

S

K

K

S

G

=

δ

−

=

δ

−

Then by Definition 2.1, we know that

G

(

S

,

t

;

K

,

T

)

is the fundamental solution to equation

(2.18). By Theorem 2.3,

G

(

S

,

t

;

K

,

T

)

is the fundamental solution, as a function of

K ,

T

(

S,

t

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

∞

<

≤

−

=

<

∞

<

≤

=

−

∂

∂

−

−

∂

∂

+

∂

∂

−

)

27

.

2

(

)

0

(

.

)

(

)

,

;

,

(

)

26

.

2

(

)

,

0

(

,

0

)

(

)

(

)

)

,

(

(

2

1

2 2 2 2

K

S

K

T

K

t

S

G

T

t

K

rG

KG

K

q

r

G

K

T

K

K

T

G

δ

σ

We substitute (2.22) into (2.26), (2.27), then integrate both sides twice with respect to K in interval

. Since we know that,

]

,

[

K

∞

i) given a certain

S

, if

K

→

∞

, for a call option, the following items will all tend to 0, i.e.,

,

0

)

(

,

,

,

,

2 2 2 2

→

∂

∂

∂

∂

∂

∂

G

K

K

K

G

K

G

K

K

V

K

V

σ

σ

ii)

ξ

δ

η

η

η

δ

η

η

ξ

S

d

K

S

d

d

K K

(

−

)

=

∫

(

−

)

(

−

)

∫

∞

∫

∞ ∞

,

)

(

)

(

)

(

0 + + ∞

−

=

−

−

=

_∫

K

S

d

S

K

δ

η

η

η

iii)

(

,

;

,

)

₂

,

2

K

V

d

V

d

T

t

S

G

K K

_∂

∂

−

=

∂

∂

=

∫

∞

∫

∞

ξ

ξ

ξ

ξ

iv)

V

(

S

,

t

;

,

T

)

d

V

(

S

,

t

;

K

,

T

),

K

_∂

=

−

∂

∫

∞

ξ

ξ

ξ

v)

(

,

;

,

)

₂

,

2

V

K

V

K

d

V

d

T

t

x

G

K K

_∂

+

∂

−

=

∂

∂

=

_∫

∫

∞ ∞

ξ

ξ

ξ

ξ

ξ

ξ

vi)

(

(

,

)

)

(

,

)

₂

.

2 2 2 2 2 2 2

K

V

K

T

K

d

G

T

d

K

_∂

∂

=

∂

∂

∫

∞

∫

∞

σ

η

η

η

σ

η

ξ

ξ

Thus, we can transform the system of (2.33), (2.34) based on

G

(

S

,

t

;

K

,

T

)

into the following

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

∞

<

≤

−

=

=

<

∞

<

≤

=

−

∂

∂

−

−

∂

∂

+

∂

∂

−

+

)

29

.

2

(

)

0

(

.

)

(

)

,

;

,

(

)

28

.

2

(

)

,

0

(

,

0

)

(

)

,

(

2

1

2 2 2 2

K

K

S

t

T

K

t

S

V

T

t

K

qV

K

V

K

q

r

K

V

T

K

K

T

V

σ

From equation (2.28), we obtain the explicit expression for implied volatility

This idea of Dupire's of calculating volatility seems to be simple and nice in theory.

However, when it becomes to the reality, when traders want to apply this into real market, the first

obstacle we must overcome is calculating the derivatives of option price, i.e.,

₂

.

2

,

,

K

V

K

V

T

V

∂

∂

∂

∂

∂

∂

And in fact, there is no simple analytical way to do it but to resort to some numerical approach, for

example, finite difference method, etc. Nevertheless, as we are about to see in chapter 3 section 2,

the numerical approach is not good enough for calculating this Dupire volatility, as a slight amount

change in option value would lead to some significant change in the value of derivatives, thus the

volatility value. We can almost say that using (2.30) to calculate implied volatility is

ill-posed

Chapter 3. Implied Volatility Models

In this chapter, we compare two different types of volatilities, the local volatility and Dupire

volatility, with implied volatilities under the structure of local volatility model.

3.1 The Local Volatility Model

As we establish in Section 2.1 that, given an asset's price process under an EMM

Q

with the risk

free interest rate

r

=

0

that

,

(3.1)

dW

S

dS

=

⋅

we will have the option pricing problem for an European call option as

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

∞

→

→

−

→

−

=

=

≤

≤

=

+

+

)

5

.

3

(

.

1

)

,

(

,

)

,

(

)

4

.

3

(

,

)

(

)

,

(

)

3

.

3

(

,

0

)

,

0

(

)

2

.

3

(

)

0

(

0

)

,

(

2

1

)

,

(

S

as

t

S

V

K

S

t

S

V

K

S

T

S

V

t

V

T

t

t

S

SV

t

S

V

S ss t

3.1.1 Option Pricing

Since there is no simple analytical solution for the system (3.2)-(3.5), we then have to resort to the

numerical way to solve the option terminal value problem for this system.

We use software Matlab in this paper to solve numerical problems.

After a closer examination, we realize that we have a terminal boundary value problem here instead

of an initial one, hence in order to use the built-in initial boundary value solver function in Matlab,

we have to substitute some variables in the problem to shift the terminal boundary problem to an

initial boundary problem in order fit this problem into the solving range of the built-in function.

If we denote the time-to-maturity as

τ

=

T

-

t

, then it becomes obvious that if any one of these

three variables(

τ

,

t,

T

) is fixed, the other two will either move in the same direction or in the

opposite ones. Thus, for every given

τ

, we have difference between

T

and

t

is fixed, written

in the differential form, i.e.,

.

(3.6)

dt

dT

=

Thus, system (3.2)-(3.5) can be transformed into the following system,

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

∞

→

→

−

→

−

=

=

=

≤

≤

=

=

+

−

+

)

10

.

3

(

.

1

)

,

(

,

)

,

(

)

9

.

3

(

,

)

(

)

,

(

)

8

.

3

(

,

0

)

,

0

(

)

7

.

3

(

)

0

(

0

)

,

(

2

1

)

,

(

0 0

S

as

T

S

V

K

S

T

S

V

K

S

t

T

S

V

T

V

t

T

t

T

S

SV

T

S

V

S T ss T

Then, we can apply the built-in function

pdepe

in Matlab to solve the above problem.

pdepe

is a function that solves initial-boundary value problems for parabolic-elliptic Partial

Differential Equations (PDEs) in one-dimension.

pdepe

solves PDEs of the form:

.

(3.11)

⎜⎜

⎝

⎛

⎟

⎠

⎞

∂

∂

+

⎜⎜

⎝

⎛

⎟

⎠

⎞

∂

∂

∂

∂

=

∂

∂

∂

∂

−

x

u

u

t

x

s

x

u

u

t

x

f

x

x

x

t

u

x

u

u

t

x

c

(

,

,

,

)

m m

(

,

,

,

)

,

,

,

The PDE holds for

t

₀

≤

t

≤

t

_f

and

a

≤

x

≤

b

. The interval

[

a

,

b

]

must be finite.

m

can be 0,

1, or 2, corresponding to slab, cylindrical, or spherical symmetry, respectively. If

m

>

0

, then

a

must be non-negative.

In (3.11),

(

,

,

,

)

is a flux term and

is a source term. The coupling of the

x

u

u

t

x

f

∂

∂

)

,

,

,

(

x

u

u

t

x

s

∂

∂

partial derivatives with respect to time is restricted to multiplication by a diagonal matrix

. The diagonal elements of this matrix

are either identically zero or

)

,

,

,

(

x

u

u

t

x

c

∂

∂

)

,

,

,

(

x

u

u

t

x

c

∂

∂

positive. An element that is identically zero corresponds to an elliptic equation and otherwise to a

parabolic equation, and there must be at least one parabolic equation. An element of

c

that

corresponds to a parabolic equation can vanish at isolated values of

x

if those values of

x

are

mesh points. Discontinuities in

c

and/or

s

due to material interfaces are permitted provided that

a mesh point is placed at each interface[2].

For

t =

t

₀

and all

x

, the solution components satisfy initial conditions of the form

.

(3.12)

)

(

)

,

(

x

t

0

u

0

x

u

=

For all

t

and either

x

=

a

or

x

=

b

, the solution components satisfy boundary conditions of

the form

.

(3.13)

0

)

,

,

,

(

)

,

(

)

,

,

(

=

∂

∂

+

x

u

u

t

x

f

t

x

q

u

t

x

p

Particularly, in our PDE (3.2) here, if we denote in (3.2),

S

as

x

,

T

as

t

,

V

(

S

,

T

)

as

, then (3.7)-(3.10) become

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

∞

→

→

→

−

=

=

=

≤

≤

=

⋅

=

+

)

17

.

3

(

.

1

)

,

(

,

)

,

(

)

16

.

3

(

,

)

(

)

,

(

)

15

.

3

(

,

0

)

,

0

(

)

14

.

3

(

)

0

(

2

1

0 0

x

as

t

x

u

x

t

x

u

K

x

t

t

x

u

t

u

t

t

t

u

x

u

x T xx t

In fact, from a mere observation in the real market, we know that underlying stock price

_S

₌

₁₀₀

is quite high for an option with strike price

K

=

10

. Then we can replace the infinity requirement

of limits in equation (3.17) by setting stock price to

_S

₌

₁₀₀

, given a strike price

K

=

10

. Then,

(3.10) and (3.17) become

,

(3.18)

100

,

1

)

,

(

,

-)

,

(

S

T

=

S

K

V

S

T

=

where

S

=

V

_S

.

(3.19)

100

,

1

)

,

(

,

-)

,

(

x

t

=

x

K

u

x

t

=

where

x

=

u

_x

Then (3.14)-(3.17) take the new forms of (3.20)-(3.23),

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

=

=

=

−

=

=

=

≤

≤

=

⋅

=

+

)

23

.

3

(

.

100

,

1

)

,

(

,

-)

,

(

)

22

.

3

(

,

)

(

)

,

(

)

21

.

3

(

,

0

)

,

0

(

)

20

.

3

(

)

0

(

2

1

0 0

x

where

t

x

u

K

x

t

x

u

K

x

t

t

x

u

t

u

t

t

t

u

x

u

x T xx t

Now, let us rearrange (3.20) in the following form

.

(3.24)

⎟

⎠

⎞

⎜

⎝

⎛

−

⋅

⋅

∂

∂

=

∂

∂

−

)

(

2

1

0 0

u

u

x

x

x

x

t

u

x

Comparing (3.24) to (3.11), i.e.

,

(3.11)

⎜⎜

⎝

⎛

⎟

⎠

⎞

∂

∂

+

⎜⎜

⎝

⎛

⎟

⎠

⎞

∂

∂

∂

∂

=

∂

∂

∂

∂

−

x

u

u

t

x

s

x

u

u

t

x

f

x

x

x

t

u

x

u

u

t

x

c

(

,

,

,

)

m m

(

,

,

,

)

,

,

,

we find out that.

,

(3.25)

0

=

m

,

(3.26)

1

)

,

,

,

(

=

∂

∂

x

u

u

t

x

c

,

(3.27)

)

(

2

1

)

,

,

,

(

x

u

u

x

u

u

t

x

f

=

⋅

x

−

∂

∂

.

(3.28)

0

)

,

,

,

(

=

∂

∂

x

u

u

t

x

s

,

(3.13)

0

)

,

,

,

(

)

,

(

)

,

,

(

=

∂

∂

+

x

u

u

t

x

f

t

x

q

u

t

x

p

is equivalent to finding pairs of values of function

p

(

x

,

t

,

u

)

and function

q

(

x

,

t

)

, which satisfies

the form in (3.13), given the flux function

(

)

by (3.27).

2

1

)

,

,

,

(

x

u

u

x

u

u

t

x

f

=

⋅

_x

−

∂

∂

If we substitute (3.27) into (3.13), we have

.

(3.29)

0

)

(

2

1

)

,

(

)

,

,

(

x

t

u

+

q

x

t

⋅

x

⋅

u

−

u

=

p

_x

And the boundary conditions (3.21) and (3.23) are

⎪

⎩

⎪

⎨

⎧

=

=

=

=

)

31

.

3

(

.

100

,

1

)

,

(

,

-)

,

(

)

30

.

3

(

,

0

)

,

0

(

x

where

t

x

u

K

x

t

x

u

t

u

x

We insert (3.30) into (3.29) at

x

=

0

, then

.

(3.32)

0

)

0

(

2

1

)

,

0

(

)

,

,

0

(

t

u

+

q

t

⋅

x

⋅

u

_x

−

=

p

For (3.32) to hold, one option is to put

p

(

0

,

t

,

u

)

and

q

(

0

,

t

)

to 0,

i.e.,

⎩

⎨

⎧

=

=

)

34

.

3

(

.

0

)

,

0

(

)

33

.

3

(

,

0

)

,

,

0

(

t

q

u

t

p

Similarly, we insert (3.31) into (3.29) at

x

=

100

, then

.

(3.35)

0

|

)

(

2

1

)

,

100

(

)

,

,

100

(

t

u

+

q

t

⋅

x

⋅

u

_x

−

u

_x=100

=

p

We simplify (3.35), obtain

.

0

)

(

2

1

)

,

100

(

)

,

,

100

(

|

))

(

1

(

2

1

)

,

100

(

)

,

,

100

(

|

)

(

2

1

)

,

100

(

)

,

,

100

(

100 100

=

⋅

+

=

−

−

⋅

⋅

+

=

−

⋅

⋅

+

= =

K

t

q

u

t

p

K

x

x

t

q

u

t

p

u

u

x

t

q

u

t

p

x x x

This is to say,

.

(3.36)

0

2

1

)

,

100

(

)

,

,

100

(

t

u

+

q

t

⋅

K

=

p

For (3.36) to hold, we can simply choose a pair of values of

p

(

100

,

t

,

u

)

and

q

(

100

,

t

)

,

⎪

⎩

⎪

⎨

⎧

=

=

)

38

.

3

(

.

1

)

,

100

(

)

37

.

3

(

,

2

1

-)

,

,

100

(

t

q

K

u

t

p

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

=

=

=

=

)

42

.

3

(

.

1

)

,

100

(

)

41

.

3

(

,

2

1

-)

,

,

100

(

)

40

.

3

(

,

0

)

,

0

(

)

39

.

3

(

,

0

)

,

,

0

(

t

q

K

u

t

p

t

q

u

t

p

After specifying all the conditions and function forms, we are ready to gather together all the

thoughts stated above to write them into a program file

pdex_u.m

(which is included in Appendix A,

Table A.1)in Matlab. We set the values for each one of the variables and parameters, the initial

value of maturity

T

=

t

₀

=

0

, strike price

K

=

10

, risk free interest rate

r

=

0

, dividend yield

, using 201 mesh points in the option price range from 0 to 100 and 51 mesh points in the

0

=

q

maturity range from 0 to 5, to simulate the numerical option value at each price level. The option

value curves, option values plotted against asset prices, under different time-to-maturity periods

τ

are shown in Figure 3.1(the more complete series of curves of option value at different levels of

maturity is included in Appendix A). The option price surface with respect to the

time-to-maturity

τ

and asset price is shown in Figure 3.2.

Figure 3.2: The Option Value Surface for European Call Options

As shown in figure 3.1 and figure 3.2, without any unexpected outcome , the option value curve

and surface under this local volatility model have no substantial difference to those of a vanilla

European call option under a generic Black-Scholes model. The longer the period of

time-to-maturity, the more valuable the call options; the higher the stock/asset price, the closer to payoff

the option values at maturity.

In the following subsection, we try to find out the internal connection between the implied

volatilities and the local volatilities.

3.1.2. Implied Volatilities and The Local Volatilities

The existence of the implied volatility can be observed from the corresponding relationship

between the option price

V

and the implied volatility

σ

_imp

. This is true by the formulation of the

Black-Scholes formula for option pricing. The problem of the uniqueness of the implied volatility

can be solved by the monotonicity of the option price

with respect to the maturity time

imp

σ

V

.

For a call option value

, where

are variables.

are

T

V

Call

=

V

(

S

,

T

;

σ

,

K

,

t

)

S,

T

σ

,

K,

t

parameters. We know that the one of the Greeks in Black-Scholes formula for call options, vega,

[3]. Hence, given any value set of

, we will find a unique

0

-

<

∂

∂

=

∂

∂

=

t

V

T

V

ν

(

V

,

S

,

T

;

K

,

t

)

value for

σ

, which is called the implied volatility, denoted as

σ

_imp

. For example,

σ

imp

=

σ

0

, for

an input set of

(

V

,

S

,

T

;

K

,

t

)

=

(

V

0

,

S

0

,

T

0

;

K

,

t

0

)

.

Therefore, we can regard

σ

as a function of

S,

T

, where

K ,

t

are parameters,

V

is also

quoted from market price, i.e.

σ =

σ

(

S

,

T

;

K

,

t

,

V

)

. While at the same time, the local volatility

denoted as

σ

_loc

can be easily observed from the price processes of this local volatility model, that

, for each mesh point in the price axis. Thus, the distance between two corresponding

S

loc

1

=

σ

volatilities can be easily calculated. The program for implied volatilities' calculation

pdex_imp.m

is

included in Appendix A Table A.2.

Figure 3.3: The Implied Volatility Curves(Plotted against the Stock Price

S

)

Figure 3.3 is the implied volatility curve plotted against the stock prices at three different time

points. As we can see in figure 3.3, the implied volatility of the option is quite large (In fact, when

the stock price is close to 0, the implied volatility tends to infinity. We will discuss this in detail at

this end of Section 3.1.2) at those points where the stock prices

S

are close to 0, and as the stock

price

S

goes up, the implied volatilities gradually fall back to a relatively low and stable level.

The implied volatility decreases at a decreasing speed as the stock price increase. From an

economic point view, if the stock prices drop to a level close to 0, then the options based on the

same stock will be extremely risky, thus the indicator of riskiness will be extremely large, i.e.

as

. On the contrary, the higher the stock price

, the less risky the call

,

∞

→

imp

σ

S

→

0

S

not rewarded at an interest

r

=

0

for taking the systematic risk, this means that the systematic risk

is 0. Hence, in our special case here(the risk-free interest rate is 0), when the stock price tends to

infinity, the implied volatility tends to 0. The term structure of the implied volatility is shown in

figure 3.4.

Figure 3.4: The Implied Volatility Curves(Plotted against the Time-to-maturity

τ

)

Figure 3.5: The Local Volatility Curve

S

loc

1

=

σ

Figure 3.5 shows the local volatility curve that given by

which only depends on the

S

loc

1

=

σ

stock price

S

. And, we know that

=

→

∞

as

, as well as

S

loc

1

σ

S

→

0

=

1

→

0

S

loc

σ

as

S

→

∞

.

From the illustration of above figure 3.3-3.5, we find out that the implied volatility

σ

_imp

and the

local volatility

σ

_loc

almost have the same tendency of change. Then, we are more curious to find

out exactly how far away they are from each other.

The Distance between Implied Volatilities and Local Volatilities

volatility has the form

, then we can find out the distance between

and

by

S

loc

1

=

σ

σ

_imp

σ

_loc

distance function

.

S

d

=

σ

_imp

−

σ

_loc

=

σ

_imp

−

1

We put our theory here into practice by program file

pdex_dis_imp_loc.m

written in matlab(this

program is include in Appendix A). All the parameters and indicators that need to be specified are

gathered in the following table 3.1.

Table 3.1: The Initial Variable Set-up for Program

pdex_dis_imp_loc.m

The plots of this section is shown in the following figure 3.6 and figure 3.7.

Price(stock/asset price)

201 mesh points, from 0 to 100.

Strike(option strike price)

10

Rate(risk-free interest rate)

0

Time(time-to-maturity)

51 mesh points, from 0 to 5

Value(option value)

51×201 values, calculated in Section 3.1.1

Limit(the upper bound for volatility searching

interval)

10 times

Yield(dividend yield)

0

Tolerance(calculation accuracy)

10

-16

Figure 3.6: The Comparison of Implied Volatility and Local Volatility (

K

=

10

)

Figure 3.6 is a demonstration of how the distance between the implied volatility and the local

volatility changes as the stock price increases. At first, the the local volatility curve is above the

implied volatility curve, then as the stock price increases, the local volatility decreases more rapidly,

Figure 3.7: The Absolute Difference between Implied Volatility and the Local Volatility Curve

(Plotted Against

,

)

imp loc

σ

σ

−

S

K

=

10

Figure 3.8: The Absolute Difference between Implied Volatility and the Local Volatility Curve

(Plotted Against

)

imp loc

σ

σ

−

T

In order to show how the distance between the implied volatility and the local volatility changes as

time goes by, we plot figure 3.8. Notice in the distance function that

,

S

d

=

σ

_imp

−

σ

_loc

=

σ

_imp

−

1

the only time-sensitive factor in it is the implied volatility

σ

_imp

. Hence, at the parts of curves

where the maturity time

T

is far away, the distance curves in figure 3.8 reveal some similar

nature to implied volatility curves in figure 3.4, as they both tend to stay at a relative stable level,

almost parallel to the time-to-maturity axis. Another interesting fact can be observed in figure 3.8 as

well is that: when the stock price

S

is below the strike price

K

, the distance curve shifts

The Limits of Implied Volatilities and Local Volatilities at

S=0.

One vague statement that we have not really explained in this section is that we say the implied

volatility is very large for those points at which the stock prices

S

are close to 0.

Although the plots of the implied volatility curves have indicated that the implied volatilities would

more than likely to go to infinity when the stock price tends to 0. However, we still need more

concrete evidence to prove our speculation here.

We know from initial condition of option pricing that when the stock price falls back to 0, the

option value is also 0, meaning that the ownership of this asset is worthless. Then it makes no sense

to talk about the implied volatility of the option value. Thus, we choose a small neighbourhood of 0

on the stock price axis

S

with its left side end open. For example, we choose

S

=

(

0

,

10

−10

]

. By

usage of the option pricing scheme described in Section 3.1.1, we calculate the option values within

this small interval. We modify our previous program for option pricing by equally choosing 201

mesh points on interval

S

=

[

0

,

10

−10

]

and adding a single point

S

=

100

to the collection of

mesh points. In this way, we can both achieve the pricing for option prices at small stock price

points and keep our boundary conditions unchanged. This altered program for option pricing is

named as

pdex_u_small_s.m

, which is included in Appendix A. Then, we use the same scheme for

implied volatility calculations as before. The program file of implied volatility computation for

small

S

,

pdex_imp_small_s.m

is included in Appendix A as well. The more detailed initial

parameter set-up for program

pdex_imp_small_s.m

is displayed in the following table 3.2.

Table 3.2: The Initial Variable Set-up for Program

pdex_imp__small_s.m

The plots of this program are shown in figure 3.9-3.12. Figure 3.9 depicts the implied volatility

curves plotted against the stock prices at different time-to-maturities. However, according to figure

Price(stock/asset price)

202 mesh points, 100 points from 0 to 10

-10

_,

and 1.

Strike(option strike price)

10

Rate(risk-free interest rate)

0

Time(time-to-maturity)

51 mesh points, from 0 to 5

Value(option value)

51×202 values

Limit(the upper bound for volatility searching

interval)

10

9

_times

Yield(dividend yield)

0

Tolerance(calculation accuracy)

10

-18