Question

A stock's current price is $72. A call option with 3-month maturity and strike price of $ 68 is trading for 6, while a put with the same strike and expiration is trading for $20. The risk free rate is 2%. How much arbitrage profit can you make by selling the put and purchasing a synthetic put? (Provide your answer rounded to two decimals.)

You have purchased a put option for $ 11 three months ago. The option's strike price is $ 340. How much is your profit, if the stock price at expiration is $ 364?

A call option with X=$63 expires in half a year. The underlying stock's current price is $70 and its standard deviation is 40%. What is the Black-Scholes price of the call if the risk free rate is 2%? (Enter your answer rounded to two digits.)

Answer 1

Part A. As per the put call parity, we have C + PV(Strike Price) = P + S0 ; where C is the price of call, P is the price of put and S0 is current stock price.

PV (strike Price ) = 68 * e^-rt = 68 * e^-2%*3/12 = 67.66

Hence we have LHS if equation = 6+ 67.66 = 73.66

RHS of equation = 72 + 20 = 92. Hence there is arbitrage as below:

• Sell 1 Put, 1 stock and buy 1 Call . The net cash inflow from this will be 86 which should be deposited at risk free rate for 3 months. After 3 three months we get 86.43
• Expiry Pay off off - if expiry price (EP) is equal to or above 68:
  • Call pay off = EP - 68 ; Put will expire worthless ; Stock pay off = -EP and deposit = 86.43
  • Net Pay off = 18.34 (arbitrage profit)
• Expiry Pay off off - if expiry price (EP) is less than 68:
  • Call pay off : expires worthless ; Put pay off = EP - 68 (loss) ; stock pay off = -EP and deposit = 86.43
  • Net pay off = 18.34 (arbitrage profit)

Part B : Since the expiry price is above the put strike price, there will be loss equal to the premium paid which is $11

Part C : B.S formula for call option:

C = N(d1) S0 - Xe^-rtN(d2)

d1 = [(ln(S0/X) + t*(r + ((annualised volatility)²)/2)]/(annual volatility * (t)^1/2)

d2 = d1 - (annual volatility * (t)^1/2)

Plugging in the values and solving we get :

S0 = 70 ; X = 63 ; annualised vol = 40% ; r = 2% t = 0.5 years

d1 = 0.3753 & d2 = 0.0925

N(d1) = 0.6443 (used normal distribution table which is for 2 digits only)

N(d2) = 0.5359

C = 0.6443 * 70 - 63*e^-2%*0.5*0.5359 = 11.68 which is the price of call option.