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B L A C K S C H O L E S A N D N U M E R I C A L T E C H N I Q U E S

Numerical Methods in Finance (Implementing Market Models)

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Agenda

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The Black Scholes PDE

An Asset Pricing Model1

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The Black Scholes Formula 15

Hedging 17

Numerical Techniques 21

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Lecture Objectives

Asset Pricing Models

Understand how the GBM process can describe a typicalinvestment asset performance

Describe the components of the GBM

The Black-Scholes PDE Demonstrate how the stochastic process of an asset and an

option on an asset can be combined

Discuss the advantages of this riskless portfolio

Hedging Discuss the importance of portfolio hedging

Numerical Techniques Discuss why these are required analytical tools in addition to

closed form solutions

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An Asset Pricing Model

We can make some assumptions about asset price

behaviour over time Stocks

Commodities

Interest Rates

House Prices

A variable whose value changes randomly over time issaid to follow a stochastic process

A Stochastic Differential Equation (SDE) is adifferential equation in which one or more of the

terms is a stochastic process, thus resulting in asolution which is itself a stochastic process

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One such asset price behaviour model assumes that

the asset follows a Geometric Brownian Motion (GBM)

Assuming a non-dividend paying asset, the GBM is

governed by the following SDE

Where and are constants

dS represents a change in asset price over a small

time interval dt

dz is a random component

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SdzSdtdS += Eq 1.2.1

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Divide Eq 1.2.1 across by S

Now we can say that the change in the asset priceover time is governed by two components

The first component,dt,says the asset will changebydt over the time interval dt is known as the drift and is deterministic

The second component is made up of a randomchange dz multiplied by referrs to the volatilityof the asset

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dzdtS

dS += Eq 1.2.2

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dz is a Weiner Process

dz is normally distributed with a mean zero and a variance of

dt (I.e. a standard deviation of sqrt(dt))

Values of dz are independent

Eq 1.2.1 and 1.2.2 are example of Ito processes. I.e.

the drift and volatility only depend on the currentvalue of the asset (S) and time (t)

A more general form of Eq 1.2.2 is then

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dztSdttSdS ),(),( += Eq 1.2.3

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Weak-Form Efficiency share prices fully reflect all

information contained in past price movements I.e. Past performance is not an indicator of future performance

and the current price of the asset reflects all available

information

A Markov Process is one that depends only upon thepresent state and not on any past states

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Now consider a derivative security. The payoff on the

derivative security is dependent on the price of theasset described.

We can describe a riskless portfolio that results in a

Partial Differential Equation (PDE) that governs the

price of a derivative security

The construction of this riskless portfolio containing

assets and a derivative asset was the basis for the

construction of the famous Black-Scholes Formula

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The Black-Scholes PDE

Options are financial instruments that convey the

right, but not the obligation, to engage in a futuretransaction on some underlying security

For example, a European call option provides the right to buy a

specified amount of a security at a set strike price at

expiration

The function C(t,S) describes the value of a derivative

whose value, C, depends on the underlying asset, S,

and time

Given that S has a drift component and a volatilitycomponent, we can anticipate these characteristics in

the derivative.

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Its Lemma governs the process followed by

functions of stochastic variables

Given that S obeys Eq 1.2.3, Its Lemma tells us thatthe process followed by C(t,S) is given by

Notice that the process for C has a drift and volatility

component, the same as that of Eq 1.2.1, asanticipated

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dzSS

CdtSS

C

t

CSS

CdC

+

+

+

= 22

2

2

21

Eq 1.2.4

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Substituting Eq 1.2.3 and Eq 1.2.4 into Eq 1.2.5

eliminates the random term

The portfolio is therefore riskless and must grow at

the riskfree rate of interest, r

This leads to the Black-Scholes PDE

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02

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222 =

+

+

rCS

CS

S

CrS

t

C Eq 1.2.6

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We can solve the Black-Scholes PDE knowing the

boundary conditions of the option.

For a European Call Option, the value of the option at

maturity is given by

And for a put option, the value is given by

The subscript T denotes the value at maturity

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)0,max( KSC TT =

)0,max(TT

SKC =

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The Black-Scholes Formula

We can continue to solve the Black-Scholes PDE to

obtain the famous Black-Scholes formula

Where c is the option value on a non-dividend payingasset and

Also, is the cumulative probability distributionfor the standardnormal distribution

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( ) ( )21 dNKedSNcrT=

( ) ( )

( ) ( )Td

T

TrKSd

T

TrKSd

=+

=

++=

1

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2//ln

;2//ln

( )N

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Hedging

We have seen how a portfolio of options and their

underlying assets can be combined to create a risklessportfolio

Clearly, the ratio of options to assets is crucial in

determining the risk of the portfolio

A short position in 1 call option requires a long

position in C/S of the underlying asset

The quantity C/S is called the option delta

The option delta is the rate of change of the option

price relative the change in the underlying asset price

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Hedging

Mathematically, delta is defined as

Managing a portfolio of options, requires risk

management skills. We need to effectively balance

the number of shares against the number of options

to ensure we are within our risk limits

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Hedging

We also need to hedge against

Gamma Risk

Rapid changes in delta

Vega Risk

Changes in the volatility

Theta Risk

Changes in maturity

Rho Risk

Changes in Interest Rates

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Numerical Techniques

The Black-Scholes Formula is a Closed Solution

This means is it simple, easy to use and analytically

tractable

It is limited to certain simple, vanilla instruments

For more complicated options, such as American

Options, we need more numerical techniques

Some options are path dependent. These also require

numerical techniques

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Numerical Techniques

Numerical Techniques also allow us to value options

on the performance of multiple underlying assets E.g. Energy versus Oil Prices

They also allow more realistic assumptions such as

Stochastic volatility

Stochastic interest rates Jumps

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Recommended Texts

Required/Recommended

Clewlow, L. and Strickland, C. (1996) Implementing derivativemodels, 1st ed., John Wiley and Sons Ltd.

Chapter 1

Additional/Useful

Hull, J. (2009) Options, futures and other derivatives, 7th ed.,

Prentice HallChapters 12 and 13

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