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Proof of Black Scholes Formula

1) The document discusses the mean of a log normal random variable and proves that if Y=ln(X) is normally distributed with mean m and variance v, then X has mean exp(m + v/2). 2) It then proves Black-Scholes formula, showing that if the stock price follows a particular SDE, the option price is given by a Black-Scholes type formula. 3) It states and proves the Feynman-Kac theorem, which is then used to solve the Black-Scholes PDE and derive the option pricing formula.

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Michael Yuk
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684 views7 pages

Proof of Black Scholes Formula

1) The document discusses the mean of a log normal random variable and proves that if Y=ln(X) is normally distributed with mean m and variance v, then X has mean exp(m + v/2). 2) It then proves Black-Scholes formula, showing that if the stock price follows a particular SDE, the option price is given by a Black-Scholes type formula. 3) It states and proves the Feynman-Kac theorem, which is then used to solve the Black-Scholes PDE and derive the option pricing formula.

Uploaded by

Michael Yuk
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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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

is given by

Proof: By Itos lemma,

If form a portfolio P

Applying Itos lemma

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

Therefore we have

Rearranging the terms 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:

Where S satisfies the following stochastic differential equation

Proof: Suppose that is the solution to the PDE. Let

Applying the Itos lemma

Since the last term involves only second order terms only,

Collecting terms 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 Itos lemma,

Integrate both side from 0 to T, We have

Recall that with mean

has a normal distribution with mean 0, and variance T, and variance , let g(X) be the density function of X.

has a normal distribution

To simplify our notation, let

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

Where

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

Lets 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,

By definition, the second integrand is

End of Proof. Since 1, has a normal distribution with mean and variance , from theorem

Applying Theorem 3,

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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