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() = area under the normal curve|
| d1 = [ ln(S/X) + (r + σ2/2) T ] / σ T1/2|
| d2 = d1 - σ T1/2 |

Put-call parity requires that:
P = C - S + Xe-rT
Then the price of a put option is:
P = Xe-rT N(-d2) - S N(-d1)
Let us take a simple example to understand this better:
I am interested in writing a six months call option on a particular share, which is currently selling for Rs 120. The volatility of the share returns is estimated as 67 per cent. I would like the exercise price to be Rs 120. The risk free rate is assumed to be 10 percent. How much premium should I charge for writing the call option? Sol : Let us first calculated d1 and d2 :...

...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-Scholesmodel. 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-Scholesmodel 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...

...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...

...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...

...IEOR E4707: 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-Scholesmodel, 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...

...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...

...random variable with mean (m-v) and variance v. By definition, the integral evaluates to be 1.
Proof of BlackScholes 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 BlackScholes 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 BlackScholes PDE is given by...

...Case Study: Black-Scholes Implied Volatilities in Practice
The topic for this case study is to apply the Black-Scholesmodel 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...

...
Chapter 2: Anti-Gay Stereotypes by Richard D. Mohr
Raven Tyler
Black psychology M/F 11:00-12:20
Abstract
In this article Anti-Gay stereotype gives an in-dept. look at the various issues that homosexual men and women encounter on a daily basis. It emphasizes on the ignorance of homosexual stereotypes and how these numerous misconceived notions subsidize to the violence, misunderstanding, and prejudice towards the gay community.
In relation to Richard’s Mohr perspective on the status of homosexuality in today’s society, I have to agree 100%. I feel as though Mr. Mohr drew decisions based on logical reason and reliable facts within the discriminatory history of homosexuality. Richard Mohr declares that homosexuality isn’t as unknown or rare as the society would like one to think, rather, it’s a common practice. One spiking piece of information which Richard list to support his argument was that a Gallup poll showed only 1 in 5 Americans reported having a gay acquaintance as opposed to Alfred Kinsley’s 1948 study on the sex lives of 5,000 white men, which showed that 79% of these people have had various homosexual experiences. The unit in which the 1948 and 1985 studies differ could be greatly derived from one’s incapability and fear to accept his/her sexuality because of the narrow societal standpoint on the subject matter.
Moreover Mr. Mohr gives details on America’s “profound” ignorance of the actual gay experience. With the limited discussion...