Question

Assuming that

a. you can buy a euro call with strike price of $1.50 for 3 cents

b. you can sell a euro put at the same strike price for 4 cents

c. the prevailing forward rate is $1.54 per euro

d. the annual risk-free rate in the US is 6%.

Show how arbitrageurs can generate riskless profit. Explain how the different prices will adjust

Answer 1

As a first step, let's check if Call Put parity is satisfied.

Call Put Parity equation is:

C + PV (K) = P + S

Or, C + PV (K) - P - S = 0

Where C = Call premium = 3 cents = $ 0.03

P = Put premium = 4 cents = $ 0.04

K = strike price = $ 1.50

S = Current forward rate = $ 1.54

Risk free rate, r = 6%

Time to maturity, t = = not specified, let's assume 1 year

PV (K) = Present value of strike = K x (1 + r)^-t = 1.50 x (1 + 6%)^-1 =  1.42

Hence, LHS of the call put parity = C + PV (K) - P - S = 0.03 + 1.42 - 0.04 - 1.54 = - 0.13 which is not equal to zero

Hence, call put parity is not satisfied and there exists an arbitrage opportunity. The opportunity can be exhausted by creating a position C + PV (K) - P - S in the following manner:

Sl. No.	Action	Cash flows at t = 0	Cash flows on expiry / maturity at t = 1
1	Short (Sell) a Put	+ P = + 0.04	- max (K - S1, 0)
2.	Short (Sell) the current forward	+ S = + 1.54	- S1
3.	Buy (Long) a call	- C = - 0.03	max (S1 - K, 0)
4.	Lend PV (K)	- 1.42	+ 1.42 x (1 + 6%) = 1.50
	Total	0.13	1.50 + max (S1 - K, 0) - max (K - S1, 0) - S1

Cash flows at t = 1: 1.50 + max (S₁ - K, 0) - max (K - S₁, 0) - S₁

Scenario 1: S₁ > K; hence max (S₁ - K, 0) = S₁ - K; max (K - S₁, 0) = 0

Hence, cash flows at t = 1 will be = 1.50 + S₁ - K - 0 - S₁ = 1.50 - K = 1.50 - 1.50 = 0

Scenario 1: S₁ < K; hence max (S₁ - K, 0) = 0; max (K - S₁, 0) = K - S₁

Hence, cash flows at t = 1 will be = 1.50 + 0 - (K - S₁) - S₁ = 1.50 - K = 1.50 - 1.50 = 0

Hence, cash flows at t = 1 will always be 0.

And cash flows at t= 0 will be $ 0.13.

Hence, we are getting a positive cash flow to begin with without any risk of payout in future. Thus we are making return without any investment or risk. This is the arbitrage where we make a riskless profit of $ 0.13