This is from Q17, page 278 on reading 63 in the CFAI text. Burke uses the one-period binomial model to calculate the price of the call option on a stock to be $7.44 when the market price for the call option = $7.75 The non-dividend paying stock could either rise by 12% or fall by 15% over one year annual risk free rate = 4.00% exercise price = $45 current stock price = $50 An appropriate set of arbitrage transactions is to A. sell 1,000 calls and buy 815 shares B. sell 815 calls and buy 1,000 shares C. buy 1,000 calls and sell 815 shares I was able to figure out the answer by elimination but I don’t understand the explanation about how to calculate the number of shares.
Up price on stock = 50 * 1.12 = 56 down price = 50*.85=42.5 On Up price call price = Max(0, S-X) = 11 On down price Call price = 0 Call delta = (11-0)/(56-42.5)=0.815 if you sold 1000 calls - you must buy 815 shares. Ans A. 1000 calls - sell 1000 calls = 1000 * 7.44 = 74400 buy 815 shares @ 50$ cost = (40750) =========================== total: 33650 =========================== at expiration: positions of the stock + call are identical: when S=56: +815 * 56 - 1000 ( 56 - 45 ) = 34640 when S=42.5: 815 * 42.5 - 1000 * 0 = 34637.5 end position due to the delta hedging is identical whether stock price went up or down.
It’s the hedge ratio: For calls: n = (c+ - c-)/(S+ - S-)
Thanks guys, for some reason I kept thinking I needed the probabilities of an up or down move to solve it. I’m glad I’m reviewing this now.
Question, On page 167 it says that d < 1 + r < u “This statement says that if the price of the underlying goes up, it must do so at a rate better than the risk-free rate. If it goes down, it must do so at a rate lower than the risk-free rate. If the underlying always does better than the risk-free rate, it would be possible to buy the underlying, financing it by borrowing at the risk-free rate, and be assured of earnings a greater return from the underlying than the cost of borrowing. This would make it possible to generate an unlimited amount of money. If the underlying always does worse than the risk-free rate, one can buy the risk-free asset and finance it by shorting the underlying. This would make it possible to earn an unlimited amount of money. Thus, the risky underlying asset cannot dominate or by dominated by the risk-free rate.” Given that statement, wouldn’t the cost of borrowing have to be higher than the appreciation of the underlying asset to prevent unlimited arbitrage? Wouldn’t u < 1 + r ? Obviously, my thought process is wrong…
cpk123 Wrote: ------------------------------------------------------- > > > 1000 calls - > sell 1000 calls = 1000 * 7.44 = 74400 > buy 815 shares @ 50$ cost = (40750) > =========================== > total: 33650 > =========================== > CP, When we sell 1000 calls, we receive a premium which is the market price of call, Shouldn’t we use the 7.75, instead of 7.44?? 1000*7.75 = 7750 I dont know if i am correct? Lemme know Regards
right… it should be 7.75 market price - but that number is irrelevant for this question. Obviously there is an arbitrage opportunity that can be made use of here.
This may be trifling, but on the test, will they ask you to use the number of “calls” or the number of “contracts,” meaning #calls/100?
^ contracts and it’s assumed you know that each contract has 100 calls
Hello, If someone can explain this: If we have sold 1000 calls option, and we are covering with 815 shares only, so when person executes, call option, we will be just delivering 815 shares ? -Thanks,
Yes, whatever isn’t “covered” by the options contract will be purchased at market price (determined by the binomial tree) and covered at the current price
Bradleyz, there are two ways of finding the call option value. One is using probabilities and the other way uses the hedge ratio as detailed in your question.