Proof Of Black Scholes Formula 2w5x1h

Mean of a log normal random variable: Theorem 1: Suppose Y = ln X is a normal distribution with mean m and variance v, then X has mean exp( m + v /2 ) Proof: The density function of Y= ln X

Therefore the density function of X is given by

Using the change of variable x = exp(y), dx = exp(y) dy, We have

= Note that the integral inside is just the density function of a normal random variable with mean (m-v) and variance v. By definition, the integral evaluates to be 1.

Proof of Black Scholes Formula Theorem 2: Assume the stock price following the following PDE

Then the option price

for a call option with payoff

1

is given by

Proof: By Ito’s lemma,

If form a portfolio P

Applying Ito’s lemma

Since the portfolio has no risk, by no arbitrage, it must earn the risk free rate,

Therefore we have

Rearranging the we have the Black Scholes PDE

With the boundary condition

To solve this PDE, we need the Feynman-Kac theorem: Assume that f is a solution to the boundary value problem:

Then f has the representation:

2

Where S satisfies the following stochastic differential equation

Proof: Suppose that

is the solution to the PDE. Let

Applying the Ito’s lemma

Since the last term involves only second order only,

Collecting we have got

As the first term is simply the PDE, it is zero. Therefore

Integrating from 0 to T

Taking expectation on both side,

Since the integral is a limiting sum of independent Brownian motions increments, i.e. =0 it is zero. Recall that W has independent and stationary increment with a zero mean, i.e. is normally distributed with zero mean. 3

Therefore In other words

End of Proof.

By the Feynman Kac Theorem, the solution to the Black Scholes PDE is given by

Where S follows

Consider Z = ln S, by Ito’s lemma,

Integrate both side from 0 to T, We have

Recall that

has a normal distribution with mean 0, and variance T,

with mean To simplify our notation, let

has a normal distribution

and variance , let g(X) be the density function of X.

Theorem 3: if ln(X) is a normal distributed random variable and the standard deviation is , then

Where

4

Proof: From Theorem 1, the mean of ln X is

,

Define . Q is a standard normal random variable with mean 0 and standard deviation 1. Hence the density function of Q is given by

Since

pply change of variable

Or

Let’s consider the first integrand

Note that the last expression is nothing more than the density function of a normal random variable with mean and variance 1, i.e.

By definition, and apply change of variable again,

5

By definition, the second integrand is

End of Proof. Since

has a normal distribution with mean

and variance

1,

Applying Theorem 3,

6

, from theorem

Final Exam question 1: Question 1 (total: 25%) Prove the Black Scholes formula a) (2%) If the price of a stock follows the SDE

State the pricing formula of the option price

for a call option with payoff

b) (5%) derive the Black Scholes PDE. c) (5%) State and prove the Feynman Kac theorem. d) (5%) If Y=ln X is a normal distribution with mean m and variance v, show that the mean of X is exp(m +v/2) e) (8%) By applying the Feynman Kac theorem to the Black Scholes PDE, derive the equation you state in part a)

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