Derivatives

Pricing options

An option buyer has the right but not the obligation to exercise on the seller. The worst that

can happen to a buyer is the loss of the premium paid by him. His downside is limited to this

premium, but his upside is potentially unlimited. This optionality is precious and has a value,

which is expressed in terms of the option price. Just like in other free markets, it is the supply and

demand in the secondary market that drives the price of an option. On dates prior to 31 Dec 2000,

the "call option on Nifty expiring on 31 Dec 2000 with a strike of 1500" will trade at a price

that purely reflects supply and demand. There is a separate order book for each option which

generates its own price. The values shown in Table 7.1 are derived from a theoretical model,

namely the Black-Scholes option pricing model. If the secondary market prices deviate from

these values, it would imply the presence of arbitrage opportunities, which (we might expect)

would be swiftly exploited. But there is nothing innate in the market which forces the prices in

the table to come about.

There are various models which help us get close to the true price of an option. Most of these

are variants of the celebrated Black-Scholes model for pricing European options. Today most

Table 7.1 Option prices: some illustrative values

Option strike price

1400 1450 1500 1550 1600

Calls

1 mth 117 79 48 27 13

3 mth 154 119 90 67 48

Puts

1mth 8 19 38 66 102

3 mth 25 39 59 84 114

Assumptions: Nifty spot is 1500, Nifty

volatility is 25% annualized, interest rate

is 10%, Nifty dividend yield is 1.5%.

100 Pricing options

calculators and spread-sheets come with a built-in Black-Scholes options pricing formula so to

price options we don't really need to memorize the formula. What we shall do here is discuss

this model in a fairly non-technical way by focusing on the basic principles and the underlying

intuition.

7.1 Introduction to the Black-Scholes formulae

Intuition would tell us that the spot price of the underlying, exercise price, risk-free interest rate,

volatility of the underlying, time to expiration and dividends on the underlying(stock or index)

should affect the option price. Interestingly before Black and Scholes came up with their option

pricing model, there was a widespread belief that the expected growth of the underlying ought

to affect the option price. Black and Scholes demonstrate that this is not true. The beauty of

the Black and Scholes model is that like any good model, it tells us what is important and what

is not. It doesn't promise to produce the exact prices that show up in the market, but certainly

does a remarkable job of pricing options within the framework of assumptions of the model.

Virtually all option pricing models, even the most complex ones, have much in common with the

Black-Scholes model.

Black and Scholes start by specifying a simple and well-known equation that models the

way in which stock prices fluctuate. This equation called Geometric Brownian Motion, implies

that stock returns will have a lognormal distribution, meaning that the logarithm of the stock's

return will follow the normal (bell shaped) distribution. Black and Scholes then propose that

the option's price is determined by only two variables that are allowed to change: time and the

underlying stock price. The other factors - the volatility, the exercise price, and the risk-free rate

do affect the option's price but they are not allowed to change. By forming a portfolio consisting

of a long position in stock and a short position in calls, the risk of the stock is eliminated. This

hedged portfolio is obtained by setting the number of shares of stock equal to the approximate

change in the call price for a change in the stock price. This mix of stock and calls must be

revised continuously, a process known as delta hedging.

Black and Scholes then turn to a little-known result in a specialized field of probability known

as stochastic calculus. This result defines how the option price changes in terms of the change in

the stock price and time to expiration. They then reason that this hedged combination of options

and stock should grow in value at the risk-free rate. The result then is a partial differential

equation. The solution is found by forcing a condition called a boundary condition on the model

that requires the option price to converge to the exercise value at expiration. The end result is the

Black and Scholes model.

7.2 The Black-Scholes option pricing formulae

The Black-Scholes formulas for the prices of European calls and puts on a non-dividend paying

stock