Question

1 Write short answers to the following questions. Use your own words rather than quoting verbatim from the notes.

a) The normal distribution is used extensively in financial mathematics. What properties of the normal distribution make it useful in general, and why nonetheless is its appearance in the Black-Scholes option pricing formula not ideal?

b) What does it mean to to say there is a possibility of arbitrage in an investment decision, and how is the concept of an investment situation being arbitrage free related to option pricing? Give an example of the use of the concept of an investment situation being arbitrage free to prove a result in option pricing.

c) Give a definition of the law of one price and explain how it is used to price options.

d) In modelling the behaviour of financial returns the geometric random walk model is used. Explain why this model is preferred to the random walk model.

e) Outline some ways in which put options differ from call options.

Answer 1

A) and B)

Exampel -

if we have to create a portfolio comprising of these two assets (call option and underlying stock) such that irrespective of where the underlying price goes ($110 or $90), the net return on portfolio always remains the same. Suppose we buy ‘d’ shares of underlying and short one call option to create this portfolio.

If the price goes to $110, our shares will be worth $110*d and we’ll lose $10 on short call payoff. The net value of our portfolio will be (110d – 10).

If the price goes down to $90, our shares will be worth $90*d, and option will expire worthless. The net value of our portfolio will be (90d).

If we want the value of our portfolio to remain the same, irrespective of wherever the underlying stock price goes, then our portfolio value should remain the same in either cases, i.e.:

=> (110d – 10) = 90d

=> d = ½

This portfolio value, indicated by (90d) or (110d -10) = 45, is one year down the line. To calculate its present value, it can be discounted by risk free rate of return (assuming 5%).

=> 90d * exp(-5%*1 year) = 45* 0.9523 = 42.85 => Present value of the portfolio

Since at present, the portfolio comprises of ½ share of underlying stock (with market price $100) and 1 short call, it should be equal to the present value calculated above i.e.

=> 1/2*100 – 1*call price = 42.85

=> Call price = $7.14 i.e. the call price as of today.

Since this is based on the above assumption that portfolio value remains the same irrespective of which way the underlying price goes, the probability of up move or down move does not play any role here. The portfolio remains risk-free, irrespective of the underlying price moves.

In both cases (assumed to be up move to $110 and down move to $90), our portfolio is neutral to the risk and earns the risk free rate of return.

C) The law of one price is the economic theory that states the price of an identical security, commodity or asset traded anywhere should have the same price regardless of location when currency exchange rates are taken into consideration, if it is traded in a free market with no trade restrictions.

It is used to price options with arbitrage opprtunity becuase,

The law of one price exists because differences between asset prices in different locations should eventually be eliminated due to the arbitrage opportunity.

D) The random walk theory suggests that changes in stock prices have the same distribution and are independent of each other, therefore, the past movement or trend of a stock price or market cannot be used to predict its future movement.

And The essential idea underlying the random walk for real processes is the assumption of mutually independent increments of the order of magnitude for each point of time. However, economic time series in particular do not satisfy the latter assumption.

Hence we are using Geometric Random Walk mdoel.

E)put options differ from call options.

Call Option -

1)Call option grants right to the buyer, not the obligation, to buy the underlying asset by a particular date for the strike price.

2) Potential gain is unlimited

3) It allows buying a stock

PUT Option -

1)Put option grants the right to the buyer, not the obligation, to sell the underlying asset by a particular date at the strike price.

2) Potential gain is limited

3) It allows Selling a stock