Question

5. Use the Black-Scholes methodology to find, by direct calculation, an explicit formula for the fair price (at time t) of the following contingent claims (European type options). The price of the underlying (stock) at time t is denoted by S(0); the time of maturity by T; the risk-free interest rate by r; the volatility of the underlying by o (a) The stock or nothing call option: This is a claim that will pay exactly the price of the underlying in all cases where the price of the underlying at time T is larger or equal to K and 0 in all other cases).

Answer 1

Formula for a regular call option = c= S(t) N(d₁) - e^-rT K N(d₂)

N(d₁) & N(d₂) are the standard normal cumulative distributive functions wherein N (d1) is the probability linked to the stock price and N(d₂) is the probability that the call will end in money.

Black scholes Merton Model of a regular call option can viewed as Leveraged stock investment wherein

We buy N(d1) units of the stock with borrrowed funds e^-rT K N(d2).

The pay off therefore is the value of the underlying less exercise price and the premium paid for the option.

(a)

Stock or nothing option can be better understood with understanding the payoff received on this option.

Here the payoff is the underlying price once the exercise price exceeds the underlying price by paying a premium for the option. If the underlying price does not exceed the exercise price then payoff is zero. So, any given point time it is the underlying price or zero.

So the fair price of such an option is simply S(t) N(d₁) if the underlying price exceeds the exercise price else zero.

The only payoff is the stock price.