Appendix 27A:An Alternative

Method to Derive The Black-Scholes

Option Pricing Model

(Related to 27.5)

ByCheng Few LeeJoseph Finnerty

John LeeAlice C Lee

Donald Wort

• Perhaps it is unclear why it is assumed that investors have risk-neutral preferences when the usual assumption in finance courses is that investors are risk averse.

• It is feasible to make this simplistic assumption because investors are able to create riskless portfolios by combining call options with their underlying securities.

• Since the creation of a riskless hedge places no restrictions on investor preferences other than nonsatiation, the valuation of the option and its underlying asset will be independent of investor risk preferences.

• Therefore, a call option will trade at the same price in risk-neutral economy as it will in a risk-averse or risk-preferent economy.

Appendix 27A: An Alternative Method to Derive The Black-Scholes Option Pricing Model

2

Appendix 27A.1: Assumptions and

the Present Value of the Expected

Terminal Option Price

ByCheng Few LeeJoseph Finnerty

John LeeAlice C Lee

Donald Wort

• In the risk-neutral assumptions of Cox and Ross (1976) and Rubinstein (1976), today’s option price can be determined by discounting the expected value of the terminal option price by the riskless rate of interest.

• Today’s call option price is:(27A.1)

• where

Appendix 27A.1: Assumptions and the Present Value of the Expected Terminal Option Price

exp Max ,0tC rt S X

the market value of the call option;

riskless rate of interest;

time to expiration;

the market value of the underlying stock at time ; and

exercise or striking price.t

C

r

t

S t

X

4

• Assuming that the call option expires in the money, then the present value of the expected terminal option is equal to the present value of the difference between the expected terminal stock price and the exercise price as:

(27A.2)

where is the log-normal density function of• To evaluate the integral in (27A.2), rewrite it as the difference between

two integrals:(27A.3)

Appendix 27A.1: Assumptions and the Present Value of the Expected Terminal Option Price

th StS

exp ,0

exp

t

t t tx

C rt E Max S X

rt S X h S dS

exp

exp exp 1

t t t t tx x

x t

C rt S h S dS X h S dS

E S rt X rt H X

the partial expectation of , truncated from below at ; andx t tE S S x the probability that tH X S X

5

• The terminal stock price can be rewritten as the product of the current price and the t-period log-normally distributed price ratio. So, Equation (27A.3) can also be rewritten as:

(27A.4)

where

Appendix 27A.1: Assumptions and the Present Value of the Expected Terminal Option Price

exp

exp exp 1

t t t t t

x s x s

tx S

S S dS S dSC rt S g X g

S S S S S

S XS rt E X rt G

S S

log normal density function of ;

the partial expextation of , truncated from below at ;

the probability that .

tt

tx S t

t

Sg S SS

SE S S x S

S

XG S S X S

S

6

Appendix 27A.2: Present Value of the Partial Expectation

of the Terminal Stock Price

ByCheng Few LeeJoseph Finnerty

John LeeAlice C Lee

Donald Wort

Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price

• The right-hand side of Equation (27A.4) is evaluated by considering the two integrals separately.

• The first integral, , can be solved by assuming the return on the underlying stock follows a stationary random walk:

(27A.5)

• Following Garven (1986), it can be transformed into a density function of a normally distributed variable according to the relationship as:

(27A.6)

exp x S tS rt E S S

exptS KtS

ln tS KtS

Kt exptS S Kt

t tS Sg f KtS S

8

Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price• These transformations allow the first integral in Equation

(27A.4) to be rewritten as:

• Substitution yields:

(27A.7)• Equation (27A.7)’s integrand can be simplified by adding the

terms in the two exponents, multiplying and dividing the result by

ln

exp exp exp tx S x S

SS rt E S rt f Kt Kt t dK

S

1 2 22 2122 expK K Kf Kt t Kt t

1 22

2 212ln

exp exp 2

exp exp

tx S K

K Kx S

SS rt E S rt t

S

Kt Kt t t t dK

212exp K t

9

Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price

First, expand the term and factor out t so that:

Next, factor out t so:

Now combine the two exponents:

Now, multiply and divide this result by to get:

Next, rearrange and combine terms to get:

(27A.8)

2

KKt t

2 212exp exp K KKt Kt t t

2 2 212exp exp 2 K K KKt t K K

2 2 2 212exp ( 2 2 )K K K Kt K K K

212exp K t

2 2 2 4 4 212exp ( 2 2 )K K K K K Kt K K K

22 4 2 212

22 2 21 1

22

exp 2

exp exp

K K K K K K

K K K K K

t K

t Kt t t

10

Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock PriceIn Equation (27A.8),Therefore, Equation (27A.7) becomes:

(27A.9)

• Since the equilibrium rate of return in a risk-neutral economy is the riskless rate:

• So Equation (27A.9) becomes(27A.10)

212exp K K tt E S S

21 22 2 212ln

exp

exp 2 exp ( ) exp

tx S

tK K K Kx S

SS rt E

S

SS E rt t Kt Kt t t

S

exp exp exptSS E rt S rt rt SS

21 22 2 212ln

exp

2 exp

tx S

K K K Kx S

SS rt E

S

S t Kt t t t dK

11

Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price• To complete the simplification of this part of the Black–Scholes

formula, define a standard normal random variable y:

• The lower limit of integration becomes:

• Further simplify the integrand by noting that the assumption of a risk-neural economy implies:

• Hence,

2 2 1 2K K Ky Kt t t

2 1 2 ,K K KKt t t y 1 2 Kt dK t dy

2 1 2ln K K Kx S t t

212exp exp ,K K t rt 21

2K K t rt

2 21 12 2K K Kt r t

12

Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price• The lower limit of integration is now:

• Substituting this into Equation (27A.10) and making the transformation to y yields:

• Since y is a standard normal random variable (distribution is symmetric around zero) the limits of integration can be exchanged:

(27A.11)

2 1 2112ln K KS x r t t d

1

2 1 212 exp exp 2 t

x S d

SS rt E S y dy

S

1 1 2212

1

exp exp 2

dt

x S

SS rt E S y dy

S

S N d

13

Appendix 27A.3: Present Value of the Exercise Price under

Uncertainty

ByCheng Few LeeJoseph Finnerty

John LeeAlice C Lee

Donald Wort

Appendix 27A.3: Present Value of the Exercise Price under Uncertainty• Start by making the logarithmic transformation:

• The differential can be written as:

• Therefore,

(27A.12)

• The integrand is now simplified by following the same procedure used in simplifying the previous integral. Define a standard normal random variable Z:

ln tS KtS

exp tSd Kt t dKS

ln

1 2 22 212ln

exp 1 exp

exp 2 exp

X S

K K KX S

X rt G X S X rt f Kt t dK

X rt t Kt t t t dK

1 2K

K

Kt tZ

t

15

Appendix 27A.3: Present Value of the Exercise Price under Uncertainty• Making the transformation from Kt to Z means the lower

limit of integration becomes:

• Again, note that the assumption of a risk-neutral economy implies:

• Taking the natural logarithm of both sides yields:

• Therefore, the lower limit of integration becomes:

1 2

ln K

K

X S t

t

212exp expK K t rt

212 ,K K t rt 21

2K Kt r t

21

2 1 21 21 2

ln K

KK

S x r td t d

t

16

(27A.13)

• Substituting, Equations (27A.11) and (27A.13) into Equation (27A.4) completes the derivation of the Black–Scholes formula:

(27A.14)

• This appendix provides a simple derivation of the Black–Scholes call-option pricing formula.

• Under an assumption of risk neutrality the Black–Scholes formula was derived using only differential and integral calculus and a basic knowledge of normal and log-normal distributions.

Appendix 27A.3: Present Value of the Exercise Price under Uncertainty• Substitution yields:

2

2

1 2212

1 22122

exp 1 exp exp 2

exp exp 2 exp

d

d

x rt G X S x rt Z dZ

x rt Z dZ x rt N d

1 2 exp C S N d X rt N d

17