Question

A stock with current price of $30 can only go up or down by $2 each month. The probability that the stock goes up in price is 60% each month. The continuously compounding interest rate is 3% p.a. Use a binomial tree with 2 time steps to price an American put option with a strike of $32. Compute the value by setting up delta-hedged portfolios along the way and working out the cost.

Answer 1

Let f be the price of put option.

At node B, we compute the price using p= 0.6 (since probability of going up is 60%)

Therefore, fu = e^(-r)(t^1/2) * [p.fuu + (1-p).fud]

= e^(-0.03*1) * [0.6*0 + (1-0.6)*2]

= 0.9704 * (0.8)

= 0.7763

At node C, we compute the price using p= 0.6 (since probability of going up is 60%)

therefore, fd = e^(-r)(t^1/2) * [p.fud + (1-p).fdd]

= e^(-0.03) * [0.6*2 + (1-0.6) *4]

= 0.9704 * (1.2+1.6)

= 2.7171

We now have the right to sell of the option at node C for strike price of $ 32, so the payoff is 32 -28 = 4$ which is higher than value of the put option at this time which is $2.7171.So the value of American Put option at node C is now 4 and not 2.7171

At node A, f = e^(-r)(t^1/2) * [p.fu + (1-p).fd]

= e^(-0.03) * [ 0.6 * 0.7763 + 0.4 (4)]

= 0.9705 * [0.4658 + 1.6]

= 2.0048

Pay off at this node is 2 which is less than F.

And the value of Put option is 2.0048$