Question

Assume that in a two period model the current stock price is $25/share. The gross rate of return on the stock over each period is either +40% or -20% while the single period simple rate of interest is 10%. Can you price a European put option on the stock with a strike of $30/share that expires at the end of the second period? Would you be able to price an American put with the same characteristics as the one above? Does it ever make sense to exercise the put at the end of the first period?

Answer 1

Up-move factor U = 1.4

Down-move factor D = 0.8

Probability of up-move =0.5

Probability of down-move = 0.5

The stock value is in yellow cell and the value of the put is below the stock value (in white cell)

Stock price in case of upmove in 1st period = 25*1.4= 35

Stock price in case of downmove in 1st period Su= 25*0.8= 20

Stock price in case of upmove in the 2nd period after upmove in 1st period Suu= 25*1.4*1.4= 49

Stock price in case of down move in 2nd period after upmove in 1st period Sud= 25*1.4*0.8= 28

Stock price in case of down move in 2nd period after downmove in 1st period Sdd= 25*0.8*0.8= 16

At t=2 the Sdd node has a positive payoff = 30-16= 14

At t=2 the Sdu node has a positive payoff = 30-28= 2

At t=2 the Sud node has a positive payoff = 30-28= 2

Value at node Sd = 0.5*2/(1.1) + 0.5*14/1.1 = 7.2727

Value at node Su = 0.5*2/(1.1) = 0.9090

Value of this payoff at node S0 = 0.5*7.2727/1.1+ 0.5*0.9090/1.1 = 3.719

Hence the price of European put option= $3.719

For an American option, we check if the value of early exercise is optimal

If option early exercised at node Su, value = 0

If option early exercised at node Sd, value = 30-20 = 10

Since this is greater than 7.2727, early exercise is optimal.

Yes, we should exercise the put at the end of the first period

Value at node Sd = 10

Value at node Su = 0.9090

Value of this payoff at node S0 = 0.5*10/1.1+ 0.5*0.9090/1.1= 4.959

Hence the price of American put option= $4.959