Binomial Pricing for Currency Options: Practice Problems and Solutions

This is Section 21 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here.

A binomial model can be constructed to price options on currencies, using the following equations:

F_0,h = x₀e^(r-f)h

ux = xe^{(r-f)h + σ√(h)}

dx = xe^{(r-f)h - σ√(h)}

p* = (e^(r-f)h - d)/(u - d)

Δdxe^fh + Be^rh = C_d

Δuxe^fh + Be^rh = C_u

Definitions of variables:

r = annual continuously-compounded risk-free interest rate for currency 1 (the "domestic" currency or the currency in terms of which the option prices are denominated).

f = annual continuously-compounded risk-free interest rate for currency 2 (the "foreign" currency).

x₀ = spot price of "foreign" currency in terms of "domestic" currency.

u = 1 + rate of capital gain on stock if "foreign" currency price increases.

d = 1 + rate of capital loss on stock if "foreign" currency price decreases.

σ = the annualized standard deviation of the continuously compounded return on the "foreign" currency.

p* = the risk-neutral probability of an increase in the "foreign" currency's price.

h = one time period in the binomial model.

F_0,h = the time-h forward price for the currency.

∆ (delta) = the number of units of the "foreign" currency contained in the replicating portfolio for the option.

B = the number of dollars lent out in the replicating portfolio for the option.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 10, p. 332.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem BPCO1. One euro currently trades for $1.56. The dollar-denominated annual continuously-compounded risk-free interest rate is 0.02, and the euro-denominated annual continuously-compounded risk-free interest rate is 0.09. Calculate the price of a 10-year forward contract on euros, denominated in dollars.

Solution BPCO1. We use the formula F_0,h = x₀e^(r-f)h. We are given that x₀ = 1.56, h = 10, r = 0.02, and f = 0.09. Thus, F_0,10 = 1.56e^{(0.02-0.09)10} = F_0,10 = $0.7746730739

Problem BPCO2. One euro currently trades for $1.56. The dollar-denominated annual continuously-compounded risk-free interest rate is 0.02, and the euro-denominated annual continuously-compounded risk-free interest rate is 0.09. The annualized standard deviation of the continuously compounded return on the euro is 0.54. Using a one-period binomial model, calculate what the euro price in dollars will be in two years if the euro's price increases.

Solution BPCO2. We are asked to find ux = xe^{(r-f)h + σ√(h)}, given that x₀ = 1.56, h = 2, σ = 0.54 r = 0.02, and f = 0.09. Hence, ux = 1.56e^{(0.02-0.09)2 + 0.54√(2)} = ux = $2.910605529. (Can you imagine the euro costing so many dollars in two years?)

Problem BPCO3. One piece of stone from Yap (YPS) currently trades for 45 cowry shells (CS). The Yap-stone-denominated annual continuously-compounded risk-free interest rate is 0.13, while the cowry-shell-denominated annual continuously-compounded risk-free interest rate is 0.05. The annualized standard deviation of the continuously compounded return on Yap stone pieces is 0.81. Using the one-period binomial option pricing model, what is the risk-neutral probability that the price of a Yap stone will increase in two months?

Solution BPCO3. We first need to find u = e^{(r-f)h + σ√(h)} and d = e^{(r-f)h - σ√(h)}. We are given that r = 0.05, f = 0.13, σ = 0.81, and h = 1/6. So (r - f) = -0.08

Thus, u = e^{(-0.08/6) + 0.81√(1/6)} = u = 1.37348016

d = e^{(-0.08/6) - 0.81√(1/6)} = d = 0.7089186851

Now we use the formula p* = (e^(r-f)h - d)/(u - d) =

(e^(-0.08/6) - 0.7089186851)/(1.37348016 - 0.7089186851) = p* = 0.4180749069

Problem BPCO4. One piece of stone from Yap (YPS) currently trades for 45 cowry shells (CS). The Yap-stone-denominated annual continuously-compounded risk-free interest rate is 0.13, while the cowry-shell-denominated annual continuously-compounded risk-free interest rate is 0.05. The annualized standard deviation of the continuously compounded return on Yap stone pieces is 0.81. For a certain two-month European call option on one YPS, the replicating portfolio involves buying (3/4) Yap stone pieces and borrowing 23 cowrie shells. Using the one-period binomial option pricing model, what would the price of this call option be in two months if the YPS price decreased?

Solution BPCO4. We are trying to find C_d = Δdxe^fh + Be^rh. From Solution BPCO3, we are given that d = 0.7089186851. Also, B = -23, Δ = 0.75, r = 0.05, f = 0.13, h = 1/6, and x = 45.

So C_d = 0.75*0.7089186851*45e^0.13/6 -23e^0.05/6 = C_d = $1.257591655

Problem BPCO5. One piece of stone from Yap (YPS) currently trades for 45 cowry shells (CS). The Yap-stone-denominated annual continuously-compounded risk-free interest rate is 0.13, while the cowry-shell-denominated annual continuously-compounded risk-free interest rate is 0.05. The annualized standard deviation of the continuously compounded return on Yap stone pieces is 0.81. For a certain two-month European call option on one YPS, the replicating portfolio involves buying (3/4) Yap stone pieces and borrowing 23 cowrie shells. Using the one-period binomial option pricing model, what is the current price of this call option?

Solution BPCO5. In Solution BPCO3, we found that p* = 0.4180749069, and in Solution BPCO4, we found that C_d = $1.257591655. We can also find Δuxe^fh + Be^rh = C_u:

From Solution BPCO3, u = 1.37348016. Also B = -23, Δ = 0.75, r = 0.05, f = 0.13, h = 1/6, and x = 45.

Thus, Δuxe^fh + Be^rh = 0.75*1.37348016*45e^0.13/6 -23e^0.05/6 = C_u = 24.17780481

Now we use the formula C = e^-rh[p*C_u + (1 - p*)C_d] =

e^-0.05/6[0.4180749069*24.17780481 + (1 - 0.4180749069)1.257591655] = C = $10.75

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.