A two-step binomial tree is used to value an option on the Australian dollar (AU

Question

A two-step binomial tree is used to value an option on the Australian dollar (AUD). The strike price is 1.00 USD per AUD and the expiration date is in 6 months. Each step is 3 months. The current price of one AUD is 1.04 USD. The US risk free rate is 2.0%, and the AUD risk-free rate is 2.5%. The exchange rate has a volatility of 6% per annum.

a. What is the proportional up movement, u, for the currency

b. What is the probability of an up movement, p?

c. What is the price of an American call option on the currency?

Explanation / Answer

Strike Price = K = $ 1/AUD , Current Price = $ 1.04/AUD, Implied Volatility = s = 6 % per annum, Option Maturity = 6 months and Period Size = t = 3 months, US Risk-Free Rate = 2 % and AUD Risk-Free Rate = 2.5 %

(a) u = EXP[s x t] = EXP[0.06 x 0.25] = 1.015 and d = 1/u = 1/1.015 = 0.985

(b) Let the risk neutral probability of an upward movement be denoted by p

Therefore, p = [EXP(0.02 x 0.25) - d] / [u - d] = [1.005 - 0.985] / [1.015 - 0.985] = 0.6667

Probability of downward movement = (1-p) = (1-0.6667) = 0.3333

(c)

NOTE: At each intermediate node (nodes other than the terminal ones) the value to be considered for expected payoff calculation is the one greater between IV and POV, where IV stands for Intrinsic Value and POV stands for Payoff Value.

PV of Expected Payoff at node 2 = 0.071434 x 0.6667 + 0.039766 x 0.3333 / EXP[0.02 x 0.25] = $ 0.0606

PV of Expected Payoff at node 3 = 0.039766 x 0.6667 + 0.009034 x 0.3333 / EXP[0.02 x 0.25] = $ 0.0294

PV of Expected Payoff at node 1 = American Option Value = 0.0294 x 0.6667 + 0.0606 x 0.3333 / EXP[0.02 x 0.25] = $ 0.0396

t = 0 months t = 3months t = 6 months Payoff IV = $ 0.0556, POV = $ 0.0606 $ 1.071434 (node 4) $ 0.071434 $ 1.0556 (node 2) $ 1.04 (node 1) $ 1.039766 (node 5) $ 0.039766 $ 1.0244 (node 3) IV = $ 0.0244, POV = $ 0.0294 $ 1.009034 (node 6) $ 0.009034

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