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...

...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...

...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...

...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...

...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...

...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 Black...

...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...

...
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...

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