A strangle is to hold a long European put at strike K1 and a long European call

at strike K2, where S is the underlying and K1 < S < K2. Assume that the underlying

stock has a price of 100, an expected yearly return of µ = 7% and a volatility of

35%. The (continuously compounded) risk-free rate is given by r = 1%. The time to

maturity is one year.

a) How can you replicate the payoff using only in-the-money options, the stock

and the money market account?

[10 marks]

b) What is the price of the strangle in the Black-Scholes-Merton model if K1 = 80

and K2 = 120?

[8 marks]

c) How should you chose the strikes K1 and K2 if you would like the price of the

strangle to be $9.25 and still require that both K1 and K2 have the same

distance to S (the distance of K1 and K2 to S in (b) was 20)?

[8 marks]

d) Note: If you need values for any other parameters to answer the questions

below, make reasonable assumptions and justify these. Simulate the return

distribution of the strangle if it is held until maturity. Use 1,000 simulation runs.

What is the probability that you loose all your investment with this strategy?

[7 marks]

e) Simulate the return distribution of the strangle if it is held for only 6 months.

Again, use 1,000 simulation runs. What is the probability of loosing more than

50% after 6 months?

[7 marks]


Discuss the empirical evidence that suggests that the assumption that the S&P 500

index follows a Geometric Brownian Motion is rejected. Provide at least two stylised

Financial Derivatives


empirical facts using daily returns of the index. For each of the stylised facts,

discuss whether you expect out-of-the money put option prices to increase or

decrease (compared with the Geometric Brownian Motion assumption) and why.

Note: You can refer to discussions from class and lecture notes, but you

need to provide references for any other sources you use. You do not have to

use your own data.

[30 marks]


Consider the strategy of buying two call options and one put option with the same

strike and time to maturity. The underlying stock price S is $50, the maturity T is in

two months, the volatility of the underlying is 40 percent per year, and the risk-free

interest rate (continuously compounded) is 1.1 percent per year. The stock pays no


a) Explain how the Black-Scholes-Merton delta of this strategy depends on the

chosen strike level. Would the delta be positive or negative if you chose (i) a

very low strike, (ii) an at-the money strike or (iii) a very high strike value?

[10 marks]

b) In a Black-Scholes-Merton model, what would be the strike for which the

strategy has a zero delta?

[5 marks]

c) Now assume that one month later the underlying stock price S is $57, has the

delta of the strategy from (b) changed? If so, what is the new delta?

[5 marks]

d) Explain how you could delta hedge the strategy and what the advantages and

disadvantages are of doing so.