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 model stock prices:
dS = σ dX + µ dt S
where
µ is known as the asset's drift , a measure of the average rate of growth of the asset price, σ is the volatility of the stock, it measures the standard deviation of an asset's returns, and
dX is a random sample drawn from a normal distribution with mean zero.
Both µ and σ are measured on a 'per year' basis.
Understanding the Stock Pricing Model 22M:303:002
Wiener Process Ito's Lemma Derivation of Black-Scholes
Ecient Market Hypothesis
Solving Black-Scholes
Past history is fully reected in the present price, however this does not hold any further information. (Past performance is not indicative of future returns) Markets respond immediately to any new information about an asset.
Understanding the Stock Pricing Model
22M:303:002
Wiener Process Ito's Lemma Derivation of Black-Scholes
Markov Process
Solving...

...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
mathematicians and computer scientists.
In the early 1970’s, Myron Scholes, Robert Merton, and Fisher Black made
an important breakthrough in the pricing of complex ﬁnancial instruments by
developing what has become known as the Black-Scholes model. In 1997, the
importance of their model was recognized world wide when Myron Scholes
and Robert Merton received the Nobel Prize for Economics. Unfortunately,
Fisher Black died in 1995, or he would have also received the award [Hull,
2000]. The Black-Scholes model displayed the importance that mathematics
plays in the ﬁeld of ﬁnance. It also led to the growth and success of the new
ﬁeld of mathematical ﬁnance or ﬁnancial engineering.
In this paper, we will derive the Black-Scholes partial diﬀerential equation
and ultimately solve the equation for a European call option. First, we
will discuss basic...

...Financial Engineering: 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 the Black-Scholes partial diﬀerential equation. We will use this PDE and the Feynman-Kac equation to
demonstrate that the price we obtain from the replicating strategy argument is consistent with martingale
pricing.
We will also discuss the weaknesses of the Black-Scholes model, i.e. geometric Brownian motion, and this leads
us naturally to the concept of the volatility surface which we will describe in some detail. We will also derive and
study the Black-Scholes Greeks and discuss how they are used in practice to hedge option portfolios. We will
also derive Black’s formula which emphasizes the role of the forward when pricing European options. Finally, we
will discuss the pricing of other derivative securities and which securities can be priced uniquely given the
volatility surface. Change of numeraire / measure methods will also be demonstrated to price exchange options....

...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 Option Price o formula. Then an analytical solution for the equation will be provided by using mathematical tools such as Fourier Transformation and properties of the heat equation. Finally, I will implement a finite difference numerical scheme in MATLAB to simulate the original Black-Scholes equation for both European call and put options and compare to analytic solutions.
1
Contents
1 Introduction and Background 1.1 1.2 1.3 1.4 What is financial mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction of option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some economic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 4 5
2 Brownian Motion
7
3 Itˆ’s Lemma o
12
4 Black-Schloes Partial Differential Equation
15
5 Analytical Solution of the Black-Scholes Equation 5.1 5.2 The...

...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 and consequently "eliminate risk" (Ray, 2012). The derivative asset we will be most interested in is a European call option. A call option gives the holder of the option the right to buy the underlying asset by a certain date for a certain price, but a put option gives the holder the right to sell the underlying asset by a certain date for a certain price. The date in the contract is known as the expiration date or maturity date; the price in the contract is known as the exercise price or strike price. The market price of the underlying asset on the valuation date is spot price or stock price. Intrinsic value is the difference between the current stock market price and the exercise price or simply higher of zero. American options can be exercised at any time up to the expiration date. European options can be exercised only on the expiration date itself. (Hull, 2012).
For example, consider a July European call option contract on XYZ with strike price $70. When the contract...

...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 option price in the exchange against the theoretical value derived by the Black-Scholes Model in order to determine if a particular option contract is over or under valued, hence assisting them in their options trading decision.
This model is based on following Assumptions:
1. The rates of return on a share are log normally distributed.
2. The value of the underlying share and the risk free rate are constant during the life of the option.
3. The market is efficient and there are no transaction costs and taxes.
4. There is no dividend to be paid on the share during the life of the option.
The Black-Scholes formula calculates the price of a call option to be:
C = S N(d1) - X e-rT N(d2)
where
| C = price of the call option |
| S = price of the underlying stock |
| X = option exercise price |
| r = risk-free interest rate |
| T = current time until expiration |
| N() = ...

...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:
Valuation of currency Europearn call option | Valuation of currency Europearn put option |
C= S0*e^(-Rf*T)*N(d1) - Ke^(-R*T)*N(d2) | P=Ke^(-R*T)*N(-d2) - S0*e^(-Rf*T)*N(-d1) |
d1 = (ln(S/K)+(R - Rf+ σ^2/2)*T)/(σ*sqrt(T)) | d1 = (ln(S/K)+(R - Rf+ σ^2/2)*T)/(σ*sqrt(T)) |
d2 = d1 - σ*sqrt(T) | d2 = d1 - σ*sqrt(T) |
Δ= e^(−Rf *T)*N(d1) | Δ = e^(−Rf *T)*[N(d1) − 1] |
Q1. Complete the following table, by entering the strikes of the 50-delta options:
Answer:
Date | Option Strikes (measured in one GBP in terms of USD) |
| 1 week | 1 month | 3 months | 6 months | 1 year | 2 years |
14-Jan | USD 1.9578 | USD 1.9556 | USD 1.9496 | USD 1.9397 | USD 1.9185 | USD 1.8717 |
Detailed explanations:
Step 1: The below information is given in the questions as below:
14-Jan | 1 wk | 1 mth | 3 mths | 6 month | 1 yr | 2 yrs |
Delta | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 |
S0 | 1.9584 | 1.9584 | 1.9584 | 1.9584 | 1.9584 | 1.9584 |
σ | 0.0890 | 0.0918 |...

...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 can affect business operations. Do you think it is possible to use a typical US management style in India or should you adjust to the local Indian managerial style and employment practices? Explain.
Forum Discussion Week 3 – Question #2
Businesses create strategies for setting up relationships with other countries based upon the type of product involved and whether their entry is in the product market or the resource market.
“Strategy” within this global framework most closely resembles a ‘marketing strategy’ that addresses target markets, segmentation, positioning and allocation of resources.
Discuss the various strategies outlined within the text reading and which one or two seem to fit your final project country and product best.
Provide a glimpse of your final paper by discussing the product and country you have selected and post and respond to students related to what product you have chosen and whether you are entering in the product or resource markets.
Written...