Question

class: Derivative Securities

A stock price is currently $58. You predict that at the end of 6 months stock price will increase or decrease by 10%. The risk-free interest rate is 10% per annum with continuous compounding.

1. What is the value of a 6-month European call option with a strike price of $60 using the binomial tree model?
2. If you decided to short 30 call option contracts of this stock, how would you hedge your portfolio of these 30 contracts to have a delta neutral portfolio?
3. What if stock price increased to $60, what adjustments/actions do you need to take in order to ensure that you have a delta neutral portfolio?
4. What if stock price dropped to $55, what adjustments/actions do you need to take in order to ensure that you have a delta neutral portfolio?

Q3. Consider a European call option on a stock with ex-dividends dates in one months and six months. The dividend on each ex-dividend is expected to be $0.45. The current stock price is $55, the exercise price is $40, the risk-free rate is 9% per annum, the volatility is 30% per annum, and time to maturity is six months. What is the value of the call option using the black-Scholes model?

Answer 1

A). Price at the end of 6 months will be 58*(1+10%) or 58*(1-10%) which is 63.8 or 52.2

So, payoff at expiry will be Max(Market price - Strike price, 0)

Payoff when Price = 63.8 will be 63.8 - 60 = 3.8

Payoff when Price = 52.5 will be 0.

Risk-neutral probability p = (Underlying price*exp(interest rate*time period) - DownPrice)/(UpPrice - DownPrice)

= ((58*exp(10%*0.5)) - 52.2)/(63.8-52.2) = 75.64%

1- p = 1 -75.24% = 24.36%

Average payoff = p*3.8 + (1-p)*0 = 75.64%*3.8 = 2.874

Discounting this back to the presen timem we get 2.874*exp(-interest rate*time period)

= 2.874*exp(-10%*0.5) = 2.7340

Call option price = 2.7340

B). Delta = (change in option price)/(change in underlying stock price)

= (3.8 - 0)/(63.8-52.2) = 0.3276

For 20 call options, hold 30*0.3276 = 9.83 stocks.

C). If stock price = 60, then

delta = (6-0)/(66-54) = 0.5

Then, hold 0.5*30 = 15 stocks.

D). If stock price = 55, then

delta = (0.5-0)/(60.5-49.5) = 0.0455

Then, hold 0.0455*30 = 1.36 stocks.