Question

As a financial analyst at Citibank Derivative Trading desk, you have collected data for a power option. A power call option pays off (max(S_T-X, 0))² at time T, where S_T is the stock price at time T and X is the exercise price. A stock price is currently $60. It is known that at the end of one year it will be either $66 or $54. The risk-free rate of interest with continuous compounding is 5% per annum. Calculate the value of a one year power call option with an exercise price of $60.

What is the delta of the option ?
What is the risk neutral probability of up move ?
What is the value of the option ?

Answer 1

Stock Price today = $60

Price in case of Upmove = $66

Price in case of Downmove = $54

Interest Rate = 5% p.a. Continuous Compounding

Strike Price of Call Option = $60

Time to maturity (t) = 1 Year

Type of Option = Power Option

Payoff of Option = Max(S_T - X, 0)²

Using no - arbitrage condition:

Investor can construct the risk free portfolio by combining assets and call option on that asset. They can sell the call option and buy the stock to create the risk free portfolio. This can be illustrated as follows:

Payoff in Case of Upmove: * $66 - Max($66 - $60, 0)² = 66 - 36

Payoff in Case of Downmove: * $54 - Max($54 - $60, 0)² = 54

Equating both equations, we can find the value of

66 - 36 = 54

66- 54 = 36

12 = 36

= 36 / 12 = 3

Delta of the Option is 3. It means that if stock price changes by $1, option value would change by $3.

Payoff in Case of Upmove = 3 * 66 - 36 = 198 - 36 = $162

Similarly in case of Downmove = 3 * 54 = $162.

Thus, risk less portfolio would give us payoff of $162 in either case. Thus, it should only earn risk free rate of return i.e. 5%. In case of excess or small return than this, there is arbitrage condition. So, assuming no arbitrage opportunity exists,

Cost of Risk Free Portfolio = Present Value of future payoff

Cost of Risk Free Portfolio = $162 * e ^{-r * T}

Cost of Risk Free Portfolio = $162 * e ^{-5% * 1}

Cost of Risk Free Portfolio = $162 * 0.95

Cost of Risk Free Portfolio = $154.1

Cost of Risk Free Portfolio = 3 * Stock - Call Option Premium

154.1 = 3 * 60 - Call Option Premium

Call Option Premium = 180 - 154.1

Call Option Premium = $25.9

Thus, Call Option Premium should be $25.9 as per no arbitrage condition.

Upmove (u) = Price at Upmove / Stock Price today

u = $66 / $60 = 1.1

Downmove (d) = Price at Downmove / Stock Price today

d = $54 / $60 = 0.9

Risk Netrual Probability of Upmove =  

Risk Netrual Probability of Upmove = (e^0.05 - 0.9) / (1.1 - 0.9)

Risk Netrual Probability of Upmove = (1.05 - 0.9) / (1.1 - 0.9)

Risk Netrual Probability of Upmove = 0.7564 i.e. 75.64%

Thus, risk neutral probability of Upmove is 75.64%.