Question

A put and a call option have the same maturity and strike price. If they also have the same price, which one is in the money? Mathematically show how you reached your conclusion.

Answer 1

Let p = Price of put option = premium of put option

c = Price of call option = Premium of call option

T = time to maturity

r = risk free rate

X = Exercise price = Strike price

S = Spot price of asset

We know that according to Put call parity

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

0 = S - X / (1+r)^T                  [ As both call and put option have same price therefore c - p = 0]

S = X / (1+r)^T

or we can write X = S (1+r)^T

It is a well known fact that Strike price(X), Risk free rate(r) and Time(T) to maturity are positive.

Also S is equal to discounted value of X at discount rate of R for time period T. In other words X can be obtained by compounding S over time period T using rate R.

By this relation we can interpret that S must be smaller than X or X > S .

It is known that if Spot price is less than strike price, then Put option is in the money.

Answer: Put option is in the money.