# Put-Call-parity

An investor is following the real-time changes in the price of options on a particular asset. She notices that both a European call and a European put on the same underlying asset each have an exercise price of\$45. The two options have six months to expiration and are both selling for \$4. She also observes that the underlying asset is selling for \$43 and that the rate of return on a I-year Treasury bill is 6%. According to put-call parity, what series of transactions would be necessary to take advantage of any mispricing in this case? A. Sell the call, sell a T-bill equal to the present value of \$45, buy the put, and buy the underlying asset. B. Buy the call, buy a T-bill equal to the present value of \$45, sell the put, and sell the underlying asset. Ans: A

Construct 2 portfolios: 1 (covered call) - buy security at P and sell (short) a call option on the security for premium c. 2 (protected put) - buy t-bill with value P and buy a put on the security. These portfolios have the same value for all times (verify by checking the values when in and out of the money). Thus, if one is cheaper we would buy the cheaper one and sell the expensive one, realizing a risk free profit. Cost of covered call = 43 + 4 = 47. Cost of protected put = 45/(1.06)^0.5 + 4 = 47.71. The covered call is cheaper, so we buy that and sell the protect put for a profit of 0.71.

Alright so we know that “Sexy Pamela is an X-rated Cougar” (this is how I remember it) or S + P = X/(1+r)^t + Co this is the put-call parity relationship Let’s calculate S + P S is the underlying asset price = 43, P is the put price = 4 So S + P = 43 + 4 = 47 Now let’s calculate the other side, Co is the call price, 4, and calculate the risk free bond, X/(1+r)^t (where X is the strike, r is the risk free rate, and t is the time to expiration), 45/(1+0.06)^0.5 = 43.71 Add the two, Co + X/(1+r)^t = 47,7 So the long call and risk free rate is worth more than the underlying asset and the put as 47.7 > 43. So short the overpriced call and bond, and long the put and underlying.

2 (protected put) - buy t-bill with value P and buy a put on the security. Isura, I’m sure you meant protective put is long the underlying (S) and long the put. You buy the stock and a put at the same time to protect you from loss on your stock.