Question

An American put option to sell a Swiss franc for dollars has a strike price of $0.75 and a time to maturity of one year. The volatility of the Swiss franc is 20%, the dollar interest rate is 4%, the Swiss franc interest rate is 2%, and the current exchange rate is 0.78. Use a three-time-step tree to value the option.

1. Please fill in the value for the blank cells in the tree diagram below. Yellow (top cells) for stock price, green (middle cells) for European put option premium , and blue (bottom cells) for American put option premium .

						1.10291
						0
				0.98263
		0.87547		0		0.87547
		0.0134				0
0.78000		0.0134		0.78		0
				0.02716
		0.69494				0.69494
				0.61915		0.05506
				0.13085		0.55163
						0.19837

Answer 1

For applying the Binomial model, the Swiss franc may be considered as the asset here and Swiss Franc interest rate of 2% may be considered as the dividend rate.

time of one period  t= 1/3 years (for a three step model)

Volatility s= 0.20  

So, here, u = exp(s*t^0.5) =  exp (0.2 * (1/3)^0.5) = 1.1224

d = 1/u = 0.8909473

So, the Swiss franc binomial lattice can be constructed as shown in the image of question

Now, value of option at expiration (t=3) = max(K-St,0)

So, if Swiss franc = 1.10291 , value of both options (American and European) = max(0.75-1.10291,0) = 0

if Swiss franc = 0.87547 , value of both options (American and European) = max(0.75-0.87547,0) = 0

if Swiss franc = 0.69494 , value of both options (American and European) = max(0.75-0.69494,0) = 0.05506

& if Swiss franc = 0.55163 , value of both options (American and European) = max(0.75-0.55163,0) = 0.19837

For calculating value of options backward , we need option lattice parameters

a = (1+0.04/3)/(1+0.02/3) = 1.0066225

risk neutral probability p = (a-d)/(u-d) = (1.0066225- 0.8909473)/(1.1224009-0.8909473) = 0.4997772

Value of European put option when Swiss franc = 0.98263 (t=2)

= present value of the expected value of European options one period ahead

= (p*0+ (1-p)*0) /(1+0.04/3)

=0

Value of American put option when Swiss franc = 0.98263 (t=2)

= max (present value of the expected value of American options one period ahead, K-St)

= max ((p*0+ (1-p)*0) /(1+0.04/3), 0.75-0.98263)

=0

Similarly , we can calculate the values of options at other points at t=2 and then at t=1 and finally at t=0

and the completed tree diagram is as given below

			1.10291
			0.00000
		0.98263	0.00000
		0.00000
	0.87547	0.00000	0.87547
	0.01342		0.00000
0.78000	0.01342	0.78000	0.00000
0.04371		0.02718
0.04512	0.69494	0.02718	0.69494
	0.07515		0.05506
	0.07800	0.61915	0.05506
		0.12508
		0.13085	0.55163
			0.19837
			0.19837
t=0	t=1	t=2	t=3

So, value of European Put option today = $0.04371

and value of American Put option today = $0.04512