How to determine the current price of derivative

Question

For a non-dividend paying share of a company whose price at time t is denoted by St, the current price of the share is S0=£100. In any year the price of the share can either increase by 20% or decrease by 20%. The continuously compounded constant annual risk-free interest rate is r, such that e^r = 1.1 The maturity payoff for a 2 year derivative contract is

K × I(S2>K);

(the option striking price is K=£90 and I(S2>90) is the indicator function, i.e. I(S2>90)=1 if S2>K, 0 if S2≤90)

I've calculated the current price of the derivative with maturity payoff S2 × I (S2 > K) as 27.46669982, using the Black-Scholes formula

How do I determine the current price of the derivative with maturity payoff K × I(S2>K)?

Answer 1

You cannot use B-S as the stock price distribution is binomial, not log-normal.

I am going to assume that:

1. The probability of stock going up/dn 20% is 50%
2. It is a European option that can only be exercise at the end of 2 year contract. (Not a big deal - as we know it never makes sense to exercise an American call early)

You need to build a binomial tree for two years. You have 2 first-year branches (up and dn) and 3 second-year branches (up-up, up-dn = dn-up, dn-dn). The price of the stock in two years will be:

Branch up-up (25% probability) price = 100*1.2*1.2 = 144

Branch up-dn/dn-up (50% probability) price = 100 * 1.2*0.8 = 96

Branch dn-dn (25% probability) = 100 * 0.8*0.8 = 64

Now let's find the option price in each branch:

Branch up-up (25% probability) price = min(144-90, 0) = 54

Branch up-dn/dn-up (50% probability) price = min(96-90, 0) = 6

Branch dn-dn (25% probability) = 100 * 0.8*0.8 = min(64-90, 0) = 0

The future value (FV) of the option is then FV = 54*.25 + 6 * .5 = 16.5

Lastly, discount the price to its present value (PV) given the interest rate e^2r = 1.1^2 is

PV = 16.5/1.1^2 = 13.6