Question

5. (Risk and return of option; Sharpe ratio) James would like to speculate on a possible rise in the stock price of QQQ (an exchange traded fund launched and managed by PowerShares Capital Management LLC). The current stock price of QQQ is $82. James expects that in one year the stock price of QQQ will be either $110 (up move; 60% chance) or $60 (down move; 40% chance). The exercise price of one-year European call option of QQQ=$80 and risk-free rate r=3% per annum. James would like to construct a portfolio with the stock (n shares) and cash from borrowing ($B) to replicate the payoff of 500 units of European call options of QQQ.

(a) Calculate the option’s leverage ratio [nS/(nS+B)] (up to 4 decimal places).

(b) Calculate the expected return for the call option.

(c) Calculate the standard deviation of returns for the European call option.

(d) Calculate the Sharpe ratio for the European call option.

Answer 1

S = 82; Su = 110; Sd = 60; K = 80; r = 3% = 0.03; u = Su / S = 110 / 82 =  1.3415 ; Sd = Sd / S = 60 / 82 =  0.7317

Cd = max (Sd - K, 0) = max (60 - 110, 0) = 0

Cu = max (Su - K, 0) = max (110 - 80, 0) = 30

Hedge ratio, n = (Cu - Cd) / [S (u - d)] = (30 - 0) / [82 x (1.3415 -  0.7317)] =  0.6000 shares per option

Borrowing per option = B = (dCu - uCd) / [(u - d)(1 + r)] = (0.7317 x 30 - 1.3415 x 0) / [(1.3415 - 0.7317) x (1 + 3%)] =  34.9515

Part (a) the option’s leverage ratio [nS/(nS+B)] = 0.6 x 82 / (0.6 x 82 + 34.9515) =  0.5847

Part (b) Price of the call option, C = Cost of the replicating portfolio = nS - B = 0.6 x 82 - 34.9515 = $  14.25

Return in up state = Cu / C -1 = 30 / 14.25 - 1 = 110.55%; probability = 60%

Return in down state = Cd / C - 1 = -100%; probability = 40%

Hence, expected return = 110.55% x 60% - 100% x 40% = 26.33%

Part (c) Variance = 60% x (110.55% - 26.33%)² + 40% x (-100% - 26.33%)² =  1.0639

Hence, standard deviation = Variance^1/2 =  1.0639^1/2 = 103.15%

Part (d) Sharpe ratio = (Return in excess of risk free rate) / Standard deviation = (26.33% - 3%) / 103.15% =  0.2262