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Text Box: How to Compute the Price of a Put from the Price of a Call (and Vice Versa) Using Put-Call Parity Note from Joe: This tip is covers a range of problems the test designers may ask such as given the price of a call with strike $40 is $5, the stock is presently priced at $30, the risk-free interest rate is 5%, and it is two years until exercise time, what would be the price of the corresponding put with strike $40? Or “If the Put at strike 35 = the call at strike 35, what is the forward price of the underlying stock? See whether you can answer those after reading this. Let me start this tip by giving you the punch line: If ‘S0’ is the present stock price, ‘Put(K)’ is the price of a put with a strike price of K, due t years in the future; and ‘Call(K)’ is the price of a call with a strike price of K, due t years in the future, and ‘r’is the risk-free interest rate; then Call(K)-Put(K) = S0 – K(1+r)-t. {or, if we call Fwd(S) the forward price of S at time t, so Fwd(S) = S0(1+r)t, we get (1+r) t[Call(K)-Put(K)] = Fwd(S) - K.} Further, since S0(1+r)t = the forward price of the stock (see our tip on risk-adjusted pricing underlying forward pricing here for the details), if K = the forward price of the stock Call(K) = Put(K) {ie. Call(S0(1+r)t) = Put(S0(1+r)t),or Call(Fwd(S)) = Put(Fwd(S)) } Why is this true? First, let me introduce one piece of notation. We called ‘S0’ the price of the stock now. So, let’s call the price of the stock when option expires (at time t) ‘St’. So, then the value of the call t months hence is St – K, but only when St > K (specifically when St – K >0). Otherwise the option is worthless and the value is zero. So, we call the value of the option at expiry (St – K)+, and under the risk-adjusted pricing the cost of the option now = Call(K) = (1+r)-tE[(St – K)+], where the expectation is again using the risk-adjusted probabilities, and the (1+r)-t = vt is the discounting induced by paying now for an option payoff t years in the future. . By a similar argument, Put(K) = (1+r)-tE [(K - St)+]. But, taking negatives –Put(K) = (1+r)-tE[-(K - St)+] = (1+r)-tE[(St – K)-], where (St – K)- is St – K when it is negative and 0 otherwise. So, combining those two, we get Call(K) - Put(K) = (1+r)-tE[(St – K)+] + (1+r)-tE[(St – K)-] = (1+r)-tE[(St – K)+ + E[(St – K)-] = (1+r)-tE[St – K] = (1+r)-t {E[St]– E[K]}= (1+r)-tE[St] – K(1+r)-t Now E[St] is just the forward price of the stock, which we know to be S0(1+r)t by virtue of the risk-adjusted probabilities and the risk-adjusted parity with bond investments (see our previous tip for the details). Further, K is a constant, so E[K] = K under any set of probabilities. So Call(K)-Put(K) = S0 – K(1+r)-t. What can you use this for? If you know the strike price, and the interest rate you can compute a put price from the call with an equivalent strike. Put(K) = K(1+r)-t - S0 + Call(K). Or you can compute a call price from the corresponding put price Call(K) = S0– K(1+r)-t + Put(K). You can even compute the interest rate from the put and call prices, r = , although that is likely to be a less common question. Also, it is worth remembering that when the strike price = the forward price, the put and call prices are equal (parity). There are a number of avenues they can branch into in that case, but if you remember the equality and that that it applies when the strike price = the forward price occurs, you can likely work your way through whatever the exam throws you. So, what’s the skinny? Iif you leave this with one principle, it should be either the general formula valued at option expiry (1+r) t [Call(K)-Put(K)] = (1+r)t S0– K = Fwd(S) – K, or the general formula when the option is sold Call(K)-Put(K) = S0– (1+r)-tK. If you remember one of those and formula for the forward price, you can work most of the problems using basic algebra. We hope this helps.