SoA6, Put-Call Parity

[OT]

SoA6
Put-Call Parity (Financial Economics)
Sample Problem (2002 JAM6 C-21, Altered Sample Problem #2)

I think I got this correct and thought it may be of some benefit to write it up and post it.

S=100=S_sub_0
Put Price = P = 15 (original problem has P=20),
Call Price = C = 10,
K=110=strike (for both call and put)

#1: Assuming put call parity holds what is the current risk-free rate?
Answer: C - P = S – K * [ (e) ^ (-r) ], solving for r = 0.0465



#2: Assuming r= 0.04 how could you make arbitrage-free money?
Here is where I get stuck figuring out which to sell and which to buy.
Answer: Since C + K * [ (e) ^ (-r) ] > P + S which amounts to 115.69>115, we sell high (left side) call and risk-free security and buy low (right side), put and stock.

Cash flows at time zero:
(1) Sell a call option for C = 10 with strike price, K = 110.
(2) Sell a risk-free security of 105.69 = 110 * [ (e) ^ (-0.04) ].
Gain 115.69 in cash but obligated in the call option (if exercised) and to return 110 in one year.

(3) Buy a put option for P = 15 with strike price, K = 110.
(4) Buy stock for S = 100 = S_sub_0
Loose 115 but have put option and Stock.

Net cash flow, at time zero, 115.69 – 115 = 0.69.

Cash flows at time one:
I owe 110 from risk free security I sold (2) at time zero.
To fund the 110 on the risk free security, I sell stock which I bought at time zero (4) at current price, S_sub_T. Considering the affect of the two options (call and put) I always have 110 available to fulfill the risk-free security that is now due at time one.

If S_sub_T > K, then the call option I sold at time zero (1) gets exercised against me. I thus loose S_sub_T minus K. (The call option works as follows: I am obligated to provide or sell the stock to the party I sold the call option to at strike price, K, which is below the current market value, S_sub_T. So I buy the stock at the current market price, S_sub_T, and sell it to the party who I sold the call option to for, K, taking a net loss of, S_sub_T minus K, since S_sub_T > K. In actuality I just provide, S_sub_T minus K, to the party who I sold the call option to.) With the sale of the stock, at price S_sub_T, I gain K = S_sub_T minus (S_sub_T minus K).

If S_sub_T < K, then I exercise the put option I bought at time zero (3). I thus gain, K minus S_sub_T.

(The put option works as follows: I have the right to sell the stock to the party I bought the put option from, at strike price K which is above the current market value, S_sub_T. So I buy it at the current market price, S_sub_T, and sell it to the party who I bought the put option from for, K, taking a net gain of, K minus S_sub_T, since S_sub_T < K. In actuality I just gain, K minus S_sub_T, from the party who I bought the call option from.) With the sale of the stock, at price S_sub_T, I gain K = S_sub_T plus (K minus S_sub_T).

I think I got this right but any comments or questions are welcomed and appreciated.

[OT]

Question .. not sure about this statement:
Since the risk-free rate is too low an arbitrage position exists where the long position is “in the money”. That is to say that the call option is over priced and put option is under priced. Hence I buy the long position (put option) and sell the short position (call option) while borrowing at the risk-free rate to mature at the strike price.

[OT]

Nope:

“In the money” is the term that refers to when an option is worth exercising.


Furthemore ...
In an option, one party always takes the long position and the other the short position. The short position hopes that the price will go down; the long position hopes that the price will go up.

When I sell a call option I receive a fee, and I am obligated to sell at the option strike price. (The option holder bought the option to buy at the strike price.) If the price of the security goes up, the party who bought the option from me will exercise the option and buy the security from me at a the strike price which is below market value. I hope the price stays below the strike price – so that the option never gets exercised -- hence I took the short position when I sold the call option. The party who bought the call option took the long position, hoping that market values go up.

Now if I sell a put option I receive a fee, and I am obligated to buy at the option strike price. (The option holder bought the option to sell at the strike price.) If the price of the security goes down, then the party who bought the option from me will exercise the put option and sell the security to me at the strike rate above market value. I hope the price stays above the strike price – so that the option never gets exercised -- hence I took the long position when I sold the put option. The party who bought the put option took the short position, hoping that the market values go down.