Consider the following information on put and call options on an asset:
Call price: 3.1
Put price: p₀ = 9
Exercise price: X = 60
Forward price: F(0, T) = 55
Days to option expiration: 180 days
The continuously compounded risk-free rate: r(c) = 4%

To make a risk-free profit using a synthetic put, you would ______.

A. long call and bond, short forward and put
B. long call and forward, short bond and put
C. long put and forward, and short call and bond

Correct Answer: A

The present value of the bond is [X - F(0, T)] / (1 + r)^T = (60 - 55) / 1.04^180/365 = 4.9.
The price of a synthetic put would be: p₀ = short forward + c₀ + [X - F(0, T)]/(1 + r)^T = 0 + 3.1 + 4.9 = 8.
As the actual put is more expensive, we should sell the put and buy the synthetic put (long call, short forward and long bond).

The initial up-front cash is generated as -3.1 (long call) - 4.9 (long bond) + 0 (short forward) + 9 = 1.
At expiration, short forward would generate - (S_T - 55), and long bond would generate (60 - 55).

• If S_T < 60, the portfolio would generate 0 (long call) - (S_T - 55) (short forward) - (60 - S_T) (short put) + (60 - 55) (long bond) = 0.
• If S_T >= 60, the portfolio would generate (S_T - 60) (long call) - (S_T - 55) (short forward) + 0 (short put) + (60 - 55) (long bond) = 0.

The strategy would generate 1 up-front without any investment or any amount to pay back later.