University of California, Los AngelesDepartment of Statistics

Statistics C183/C283 Instructor: Nicolas Christou

Practice final examSome of the problems are from Options Futures and Other Derivatives by John Hull, Prentice Hall 6th

Edition, 2006.

Problem 1Answer the following questions:

a. Suppose you want to value a European call option with the following data: S0 = $100, E = $95, r = 0.10, σ =0.50, T = 3 months. Find the price of this call using the Black-Scholes-Merton model.

b. Refer to question (a). Find the price of a European put written on the same stock with identical exercise priceand same time to expiration.

c. Refer to question (a). Ignoring the time value of money, by how much the stock price has to rise for thepurchaser of the call to break even? By how much has to fall for the purchaser of the put?

d. Refer to question (a). Suppose that the call option actually is selling for $16. Is its implied volatility more orless than 0.50? Explain.

Problem 2Answer the following questions:

a. A stock price is currently $40. Over each of the next two 3-month periods it is expected to go up by 10% ordown by 10% according to a binomial model. The risk-free rate is 12% per year with continuous compounding.You are considering buying an “as-you-like it option” (also known as “chooser option”), under which you willhave the right to choose whether you want a European call or a European put at expiration (i.e. in 6 months,same strike E = $42). What is the value of this option?

b. The process followed by a stock price is dS = µSdt+ σSdz. What is the coefficient of dt followed by Q = S3.What is the coefficient of dz followed by Q = S3. Express both answers in terms of Q.

c. Assume the stock price at the end of each of 5 consecutive weeks is 31.2, 32.5, 33.7, 35.5, 36.5. (Most recentprice is 31.2.) Estimate the annual volatility from these data. Assume lognormal distribution of stock prices.

d. Any function of f(S, t) that is a solution of the Black-Scholes-Merton differential equation is the theoreticalprice of a derivative that could be traded. Consider the function es. Does it satisfy the Black-Scholes-Mertondifferential equation?

Problem 3Answer the following questions:

a. Suppose that a financial institution offers a security that will pay off a dollar amount equal to S2T at time T .

Assume this security follows the lognormal distribution. What is the expected value of the payoff?

b. Refer to question (a). Use risk-neutral valuation to calculate the price of this security at time t, where (t < T ).

c. Verify that the price you found in (b) satisfies the Black-Scholes-Merton differential equation. Show all yourwork.

d. Assume the lognormal property of stock prices holds. Consider a stock with a price today of $38, expectedreturn of 18% per year, and a volatility of 20% per year. Find a and b such that P (a < ST < b) = 80% in 3months from now.

Problem 4Answer the following questions:

a. Calculate the delta of an at-the-money 6-month European call option on a non-dividend paying stock whenrf = 0.10 per year, and the stock price volatility is 25% per year.

b. Find the hedge ratio (∆) for a European put. Please show the entire derivation.

c. Consider the following scenario: The price of a stock is $100 and in one month its price can go up to $120, stayat $100 or go down to $90. A European call has exercise price $105 and price c = $5 and another Europeancall has exercise price $95 and price c = $10. How many stocks and how many calls you need to buy (or sell)so that the payoff in one month will be $1 regardless of the price of the stock? Note: You must use all threesecurities.

d. Refer to question (c). What would be the risk-free interest rate during this one month period?

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Problem 5Answer the following questions:

a. The current price of a stock is S0 = $50 and suppose we know that at the end of 6 months will be either $60or $42. The risk-free rate with continuous compounding is 12% per year. Calculate the value of a 6-monthEuropean call option with exercise price $48 using no-arbitrage arguments and using risk-neutral valuation andverify that they give the same answer.

b. Consider the dynamic Delta hedging handout (table 2). Complete the second row of the table. Please show allyour work.

c. Suppose Y1 and Y2 are the prices of two stocks and let Y ∼ N2(µ,Σ), where Y =

(Y1

Y2

), µ =

(µ1

µ2

),

and Σ =

(σ21 σ12

σ12 σ22

). Find the mean vector and variance covariance matrix of the random vector Z =(

Z1

Z2

)=

(eY1

eY2

).

Problem 6Answer the following questions:

a. A power option pays off [max(ST −E), 0]2 at time T where ST is the stock price at time T and E is the exerciseprice. Consider a situation where E = 26 and T is one year. The stock price is currently $24 and at the endof one year it will be either $30 or $18. The risk-free annual continuously compounded interest is 5%. Whatis the risk-neutral probability of the stock rising to $30?

b. Refer to question (b). What position (and how much) in the stock is necessary to hedge a short position inone such power option?

d. Refer to question (b). What is the value of this power option?

e. Show that vega of a European call is given by V = S0Φ′(d1)√t, where t is the time to expiration. (Vega is the

rate of change of the value of the option with respect to the volatility σ of the underlying asset). Is vega of aEuropean put the same as the vega of European call?

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The standard normal distribution table. Note: P (Z ≤ 1.13) = 0.8708.

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