Question

Calculate u, d and p when a binomial tree is constructed to value an option on a foreign currency. The tree step size is one month, the domestic interest rate is 0.50% per annum, the foreign interest rate is 0.10% per annum, and the volatility is 12% per annum.
Use a three step binomial tree to value a 3m European call option on EUR/USD when spot is 1.08 $ per €, strike is 1.10 $ per €.

Answer 1

t = The tree step size = one month = 1/12 year

the domestic interest rate, rd = 0.50% per annum

the foreign interest rate, rf =  0.10% per annum

and the volatility, σ 12% per annum

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Use a three step binomial tree to value a 3m European call option on EUR/USD when spot is 1.08 $ per €, strike is 1.10 $ per €.

S0 = 1.08; K = 1.10

Please see the tree diagram of the foreign currency. The value of the call option on expiration is also mentioned. The cells highlighted in bold yellow contains the value of the call option in different possible states of S. Payoff from the call option is calculated as = max (S - K, 0). Towards the end the probability of the state is also calculated. Cells highlighted in blue color contains the probability of that state.

Hence, expected value of call option at expiration, C_3m = Sum of (probability x value of call) over all the four possible states

= 0.0944 x 0.0983 + 0.3388 x 0.0181 + 0.4053 x 0 + 0.1616 x 0

= 0.01540

Hence, price of the call option today, C₀ = C_3m x e^{-(rd - rf) x 3t} = 0.01540 x e^{-(0.05 - 0.08) x 3/12} = $ 0.0155 / €