Question

Assume that put options on a particular stock are very thinly traded and have very high transaction costs with a large bid-ask spread.

You desire to sell a 3-month out option on this particular stock (sell to open) but want to avoid the high transactions cost. you understand put-call parity. Assume that we are referring to European style options

A) using the principles of put call parity, how could you create a "short synthetic put" by transacting the stock, the comparable European-style call option, and risk free bond? make sure to state your answers in words in terms of what your transactions would be and be as specific as possible.

B) assume you set up the "synthetic put" Above. Then assume that the 3 month pass and the option reaches expiration. what is the payoff at option expiration of this "synthetic put"if the stock price ends less than the exercise price of the put? how does this compare to the future payoff at option expiration of an actual short put position under the same stock price outcome?

Answer 1

Part (A)

Recall call put parity equation:

C - P = S - PV (K) where K is the strike price

Hence, short put position = - P = S - C - PV (K)

Hence, synthetic short put position can be created as:

• Buy (Long) the underlying stock
• Short (Sell) the call option of 3 months matuirity and strike price of K on the same underlying stock
• Borrow present value of strike price today (at risk free rate over three months maturity)

Part (B)

S < K on expiration

Hence, the payoff from the synthetic position on maturity = S - C - K = S - max (S - K, 0) - K = S - K

Payoff from an actual short put position = - max (K - S, 0) = - (K - S) = S - K

Hence, the two payoffs are identical.