Question

A stock index is currently 810 and has a volatility of 20% and a dividend yield of 2%. The risk-free rate is 5%. Value a European six-month put option with a strike price of 800 using a two-step tree.

Answer 1

Theorem

Assume that the stock price S0 goes either up or down by a factor u > 1 and d < 1 resp. in the time steps δt. Let fuu and fud and fdd the payoffs of the option at maturity time T = 2δt in the different cases of stock movements. Let r be the riskless interest rate.Then the price f of the european option is

f = e^-2rt [ p² fuu + 2p(1 − p)fud + (1 − p)² fdd ]

where p = (e rδt − d )/ ( u − d ).

We have 2 steps each of 3 months in which stock goes up or down by 20%

So u =1.2 d= 0.8 given r = 0.05 or 5%

We get p =( e^rt – d )/ u-d

                = (e^0.05*3/12 -0.8)/(1.2-0.8)

                =0.532

And f_uu = 0   since 1166 > srike price 800

       f_ud = 22 since 800-778 =22

      f_dd = 282 since 800-518 =282

f = e ^-2rt [p² f_uu +2p(1-p)f_ud + (1-P)² f_dd ]

   = e ^{-2 x 0.05 x 3/12} [ (0.532)²(0) + 2 (0.532 ) (1-0.532)(22) +(1-0.532)²282]

= 70.902