Question

1. Consider contracts on natural gas maturing in 2 months with strike price $3/MMbtu. The call price is 0.3245 and the put price is 0.2625. The current futures price of a contract maturing in 2 months is 3.062.

a) What is the implied risk-free interest rate to avoid arbitrage? Hint: c0 - p0 = PV[ST] - PV[X] = (PV[ST] - PV[F0]) + (PV[F0] - PV[X]).

b) Suppose the risk-free rate is 3% per year. Describe an arbitrage opportunity.

Answer 1

a) Let us assume that S is the present stock price and r is the risk free rate per annum.

Future price, F = S (1+r)^(1/6) ......................(1)

c = 0.3245, p=0.2625, X= 3

From Put-call parity:

S+ p = C + PV(X),

S = c-p + X*(1+r)^(-1/6)

S = 0.3245-0.2625+ 3/(1+r)^(1/6) = 0.062+ 3/(1+r)^(1/6)

Substituting the value of S in eqn 1, we get

3.062 = 0.062*(1+r)^(1/6) + 3

0.062 = 0.062*(1+r)^(1/6)

(1+r)^(1/6) = 1 or r = 0%

b) If r = 3%

from eqn 1:

S = F/[(1+r)^(1/6)] = 3.062/[1.03^(1/6)] = 3.047

This will disturb the put-call parity

S+P = 3.047 + 0.2625 = 3.3095

C+ PV(X) = 0.3245 + PV(3) = 3.3097

Since, C+ PV(X) > S+P an arbitrage opportunity

to make profit from this opportunity:

Borrow PV(X) at 3% for 2 months and short call option.

use the proceeds of this to buy the underlying and the put option.