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

1

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 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:

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 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 Ito’s 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...

...Wiener Process Ito's Lemma Derivation of Black-Scholes Solving Black-Scholes
Introduction to Financial Derivatives
Understanding the Stock Pricing Model
22M:303:002
Understanding the Stock Pricing Model
22M:303:002
Wiener Process Ito's Lemma Derivation of Black-Scholes
Stock Pricing Model
Solving Black-Scholes
Recall our stochastic dierential equation to...

...Black-Scholes Option Pricing Model
Nathan Coelen
June 6, 2002
1
Introduction
Finance is one of the most rapidly changing and fastest growing areas in the
corporate business world. Because of this rapid change, modern ﬁnancial
instruments have become extremely complex. New mathematical models are
essential to implement and price these new ﬁnancial instruments. The world
of corporate ﬁnance once managed by business students is now controlled by...

...Continuous-Time Models
c 2009 by Martin Haugh
Fall 2009
Black-Scholes and the Volatility Surface
When we studied discrete-time models we used martingale pricing to derive the Black-Scholes formula for
European options. It was clear, however, that we could also have used a replicating strategy argument to derive
the formula. In this part of the course, we will use the replicating strategy argument in continuous time to
derive...

...Honours Project
Final Draft Derivation and Application of the Black-Scholes Equation for Option Pricing
Author: Yeheng XU
Supervisor: Dr. David Amundsen
April 30, 2012
Abstract In this project, I will first study the concept of a stochastic process, and discuss some properties of Brownian Motion. Then I generalize Brownian Motion to what it called an Itˆ process. The above concepts will be used to derive the Black-Scholes...

...Question: Discuss how an increase in the value of each of the determinants of the option price in the Black-Scholes option pricing model for European options is likely to change the price of a call option.
A derivative is a financial instrument that has a value determined by the price of something else, such as options. The crucial idea behind the derivation was to hedge perfectly the option by buying and selling the underlying asset in just the right way...

...Black-Scholes Option Pricing Formula
In their 1973 paper, The Pricing of Options and Corporate Liabilities, Fischer Black and Myron Scholes published an option valuation formula that today is known as the Black-Scholes model. It has become the standard method of pricing options.
The Black-Scholes model is a tool for equity options pricing. Options traders compare the prevailing...

...Case Study: Black-Scholes Implied Volatilities in Practice
The topic for this case study is to apply the Black-Scholes model to calculate the strike price of the F.X. options and estimate the implied volatilities in practice, finally delta-hedged strategy will be described in detail in order to hedge F.X. option.
The below formulas for Black-Scholes pricing are applied to the case study problems:...

...Forum Discussion Activities
Forum Discussion Week 3 – Question #1
Please post your response to ONE of the following questions in the Forum by Wednesday, midnight, of Week 3. Then please post at least three responses to other student’s postings by Saturday, midnight, of Week 3.
* You are the Vice President of a US based software company. You have been tasked with exploring the possibility of setting up a software development operation in India. You have heard that the rigid caste systems...

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