BLACK-SCHOLES-MERTON MODEL

Course Project IME 625A

By

Mohit Shukla 11435

Varun Tomar 11792

USEFUL TERMS

Option

A financial derivative that represents a contract sold by one party (option writer) to another party (option holder). The contract offers the buyer the right, but not the obligation, to buy (call) or sell (put) a security or other financial asset at an agreed-upon price (the strike price) during a certain period of time or on a specific date (exercise date).

Premium of an option

The income received by an investor who sells or "writes" an option contract to another party. The current price of any specific option contract that has yet to expire. For stock options, the premium is quoted as a dollar amount per share.

INTRODUCTION

It provides a theoretical estimation of the premium of an European-style options. 

Assumption: price of heavily traded assets follow a geometric Brownian motion with constant drift and volatility.

Many empirical tests have shown that the Black–Scholes price is "fairly close" to the observed prices.

 Option traders generally go for the buy options which are priced less than and sell options which are priced higher than the calculated value.

MOTIVATION

dSt = μStdt + S𝜎 tdWt

Black and Scholes derived option price formula under this assumption on stock market dynamics

Where,

St is the current stock price.

μ is drift

𝜎 is volatility/ diffusion

The process Wt generates a random variable that is Normally distributed with mean 0 and variance t, φ(0,t). (Also referred to as Gaussian.)

Main driver of this process is Wt, a Brownian Process

THE BLACK–SCHOLES EQUATION AND FORMULA

C = SN(d1) – Ke(-rt) N(d2)

C = Theoretical call premium

S = Current Stock Price

t = time until option expires

K = option striking price

r = risk free interest rate

N = cumulative standard normal distribution

d1 = {ln(S/K) + (r + (s2 / 2))t}/s

d2 = d 1 - s

s = standard deviation of stock returns

ASSUMPTION

The options are European and can only be exercised at the expiration.

No dividends are paid.

Efficient Market.

No commissions.

Risk free rate and volatility is known.

returns on the underlying stock are normally distributed.

INTERPRETATION

Delta = N(d1)measure sensitivity that calculated option value has on changes in share price

Gamma = measures calculated delta’s sensitivity to small change in share price

Theta = measures calculated option value’s sensitivity to small change in time till maturity

EXAMPLE

Consider an item option with 20 days to expiration. The strike price is 105 and the price of the stock is 100 and the stock has an daily volatility of 0.02. Assume an interest rate of 0.01 (1% annual).S= 105, K = 100 , = 0.02 and r = 1%𝜎d1 = {ln(S/K) + (r + (s2 / 2))t}/s = -0.494d2 = d 1 - s = -0.584

C = SN(d1) – Ke(-rt) N(d2) = 1.7

LIMITATIONS

Understanding of extreme moves like market crashAssumption of instant, cost-less tradingAssumption of stationary processesAssumption of continuous time trading In reality security prices do not follow normal distribution

This is a simplified model which doesn’t take into account some real life factors

MAIN REASONS FOR USING

Easy to calculateA useful approximation to determine the direction of price movement

A robust basis for more refined models

THANK YOU!