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# Options

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1) A stock price is currently \$100. Over each of the next two three-month periods it is expected to increase by 12% or fall by 12%. Consider a six-month European call option with a strike price of \$105. The risk-free rate is 3%. What is the risk-neutral probability of a 12% rise in both quarters?

2) A stock price is currently \$100. Over each of the next two three-month periods it is expected to increase by 12% or fall by 12%. Consider a six-month European call option with a strike price of \$105. The risk-free rate is 3%. What is the value of the call option?

3) A stock price is \$30, the expected return is 18% per annum and the volatility is 20% per annum. What is the standard deviation of the logarithm of the stock price in two years?

4) A stock price is \$30, the expected return is 18% per annum and the volatility is 20% per annum. What is the lower 95% confidence limit for the logarithm of the stock price in two years?

5) For a call option on a non-dividend paying stock, the strike price is \$40, the stock price is \$37.65, the risk-free rate is 4% per annum, the volatility is 40% per annum and the time to maturity is 6 months. What is the price of the call option?

6) A portfolio of derivatives on a stock has a delta of -1200 and a gamma of -200. What position in the stock in the stock would create a delta-neutral portfolio

7) The delta of a European call option on a non-dividend paying stock is 0.4, its gamma is 0.08 and its vega is 0.2. What is the delta of a European put option with the same strike price and time to maturity as the call option?

#### Solution Preview

1 A stock price is currently \$100. Over each of the next two three-month periods it is expected to increase by 12% or fall by 12%. Consider a six-month European call option with a strike price of \$105. The risk-free rate is 3%. What is the risk-neutral probability of a 12% rise in both quarters?

Probability of upmoves in 2 successive quarterers = Probability of upmove in 1st quarter x Probability of upmove in 2nd quarter

u= 1.12 =1+12%
d= 0.88 =1-12%
r= 3%
t= 3 months= 0.25 years

p= {e ^(rt)-d}/(u-d) 0.53 =EXP(0.03x0.25)-0.88)/(1.12-0.88)

Probability of upmoves in 2 successive quarterers = 0.28 =0.53^2

2 A stock price is currently \$100. Over each of the next two three-month periods it is expected to increase by 12% or fall by 12%. Consider a six-month European call option with a strike price of \$105. The risk-free rate is 3%. What is the value of the call option?

Value of option= \$5.65

Option
Put or call = C =Call
American or European= E =European

Spot price=S= \$100.00
Strike price=X= \$105.00

r=risk free ...

#### Solution Summary

Calculates the value of call option, risk-neutral probability, standard deviation of the logarithm of the stock price, confidence limit for the logarithm of the stock price, position in a delta-neutral portfolio, delta of a European put option etc.

\$2.19