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Problem 1

1.A stock is currently priced at $55.00. The risk free rate is 4.6% per annum with continuous compounding. In 9 months, its price will be either $64.90 or $46.20.

(a) Using the binomial tree model, compute the price of an American call with strike price $52.78 expiring in 9 months.

(b) Now, compute the price of an American put option with strike price $52.78 expiring in 9 months.

2.A stock is currently priced at $51.00. Every 3 months the price will go up by 16% or down by 11%. The risk free rate is 3.4% per annum with continuous compounding.

Using the binomial tree model, compute the price of a European put option with strike $73.94 expiring in 12 months.

3.A stock is currently priced at $51.00. Every 6 months the price will go up by 13% or down by 12%. The risk free rate is 5.8% per annum with continuous compounding.

Consider a portfolio made of the following: a bond which pays $27.00 in 24 months; 3 European straddle options each with strike $55.00 expiring in 24 months.

Using the binomial tree model, compute the price of this portfolio.

4.A stock is currently priced at $46.00 and pays a dividend yield of 3.3% per annum. The risk-free rate is 5.1% per annum with continuous compounding. In 18 months, the stock price will be either $38.64 or $51.98.

Using the binomial tree model, compute the price of a 18 month European call with strike price $46.96.

5.A stock is currently priced at $52.00. The risk free rate is 6.6% per annum with continuous compounding. In 5 months, its price will be $61.36 with probability 0.5 or $44.20 with probability 0.5.

Using the binomial tree model, compute the present value of your expected profit if you buy a 5 month European call with strike price $56.00. Recall that profit can be negative.

6.A stock is currently priced at $83.00. The risk free rate is 5.3% per annum with continuous compounding.

Use a one-time step Cox-Ross-Rubenstein model for the price of the stock in 18 months assuming the stock has annual volatility of 15.3%. Compute the price of a 18 month call option on the stock with strike $85.00.

7.The current spot price of a stock is $69.00. The risk-free rate is 12.1%. You want to use a one-time step Cox-Ross-Rubinstean model for the price of the stock in 36 months. What is the range of the annual volatility that you can use that avoids arbitrage opportunities for this model?

Note: your answer should be in the form of an open interval of absolute rather than percentage values. For example, if the range of annual volatility is between 12.3% and 45.6%, your solution should be (0.123, 0.456). If an endpoint is infinity (or minus infinity), simply write "infinity" (or "-infinity")

8.Suppose the risk-free rate today is 11.3%, and that in 17 months the Federal Reserve will change the rate to either 9% or 14%. In the risk-neutral world, assume that each is likely to happen.

Compute the price of a European put option on the 8 month bond with face value $112.00, where the option expires in 17 months and the option's strike price is $102.98. Note that this means the bond matures 8 months after the option's expiration date and pays the bond holder $112.00 at the bond's maturity.

9.The current spot price of a stock is $30.00, the expected rate of return is 6%, and the volatility of the stock is 22%. The risk-free rate is 3.4%.

(a) Find the 85%-confidence interval for the stock price in 8 months.

Enter your solution as an interval of the form (123.45, 678.90). Do not include dollar signs ($) in your solution.

Hint: At some point, for the standard normal variable, you need to find the 85%-confidence interval centered at 0.

(b) Compute the expected percent change in the stock over the next 8 months.

Enter your solution as a percentage value to two decimal places. Do not include the percent (%) sign.

10.The current spot price of a stock is $34.00, the expected rate of return (per annum) of the stock is 6.1%, and the weekly volatility of the stock is 3.8%. The risk-free rate (per annum) is 5.7%.

Assume there are 52 weeks in a year. Additionally, assume the log-normal model for the stock. Let X be a random variable denoting the natural log of the price of the stock in 5 months, where X is normally distributed. Compute the mean μ and standard deviation σ of X .

Enter your solutions to two decimal places

μ=

σ=

11.The current spot price of a stock is $74.00, the expected rate of return of the stock is 9.6%, and the volatility is 25%. The risk-free rate is 5.9%.

(a) Compute the price of a European call option on the stock with strike price $78.00 expiring in 18 months.

Enter your solution as a dollar value, including dollar symbol ($), to two decimal places.

(b) Compute the price of a European put with the same strike and expiration date.

Enter your solution as a dollar value, including dollar symbol ($), to two decimal places.

12.The current spot price of a stock is $73.00, the expected rate of return is 9.7%, and the volatility of the stock is 22%. The risk-free rate is 4.6%.

Compute the price of a portfolio made of 7 stocks and 7 European straddles on the stock with strike price $80.00 expiring in 11 months.

Enter your solution as a dollar value, including dollar symbol ($), to two decimal places.

13.The current spot price of a stock is $32.00, the expected rate of return of the stock is 6.9%, and the volatility is 21%. The risk-free rate is 4.4%.

Compute the price of a derivative whose payoff in 11 months is -

ln(S)+32

Where S denotes the stock price in 11 months.

Enter your solution as a dollar value, including dollar symbol ($), to two decimal places.

14.The current spot price of a stock is $36.00, the expected rate of return of the stock is 7.6%, and the volatility is 15%. The risk-free rate is 5.8%.

Compute the price of a derivative whose payoff in 16 months is -

ln(S^5)+S^0.537

Where S denotes the stock price in 16 months.

Enter your solution as a dollar value, including dollar symbol ($), to two decimal places.

15.The current spot price of a stock is $33.00, the expected rate of return of the stock is 6%, and the volatility is 15%. The risk-free rate is 4.2%.

Compute the price of a derivative whose payoff in 15 months is

- $13.00 if the stock price in 15 months is below $35.00,

- Nothing if the stock price in 15 months is above $35.00

Enter your solution as a dollar value, including dollar symbol ($), to two decimal places.

16.The current spot price of a stock is $73.00, the expected rate of return of the stock is 9.8%, and the volatility is 19%. The risk-free rate is 4.7%.

Compute the price of a derivative whose payoff in 18 months is

- $13.00 if the stock price in 18 months is below $78.00,

- $6.00 if the stock price in 18 months is between $78.00 and $87.00,

- Nothing if the stock price in 18 months is greater than $87.00

Enter your solution as a dollar value, including dollar symbol ($), to two decimal places.

17.The current spot price of a stock is $83.93, the expected rate of return of the stock is 6.3%, and the volatility is 17%. The risk-free rate is 5.3%.

Compute the price of a derivative whose payoff in 14 months is

- S^0.537+100 if S< $50.00, where S denotes the stock price in 14 months

- S^0.537if S≥ $50.00

Enter your solution as a dollar value, including dollar symbol ($), to two decimal places.

18.The current spot price of a stock is $77.00, the expected rate of return is 9.7%, and the volatility of the stock is 24%. The risk-free rate is 3.1%. Assume the log-normal model.

(a) Calculate the Delta of a portfolio containing 3 European calls and 3 European puts, each with the same strike $85.00 and expiring in 10 months.

Enter your solution to three decimal places.

(b) If the stock price drops suddenly by 12%, use part (a) to estimate the change in price.

Enter your solution as a dollar value, including dollar symbol ($), to two decimal places. Note that a loss in value corresponds to a negative change in price.

19.The current spot price of a stock is $78.00, the expected rate of return is 9.1%, and the volatility of the stock is 17%. The risk-free rate is 3.2%. Assume the log-normal model.

(a) Calculate the Delta Δc and Vega νcνc of a European call with strike $86.00 expiring in 9 months.

Enter your solution for Δc to three decimal places. Enter your solution for νc as a dollar value, including dollar symbol ($), to two decimal places.

Δc=

νc=

(b) Calculate the Delta Δs and Vega νs of a European straddle with strike $86.00 expiring in 9 months.

Enter your solution for Δs to three decimal places. Enter your solution for νs as a dollar value, including dollar symbol ($), to two decimal places.

Δs=

νs=

20.The current spot price of a stock is $74.00, the expected rate of return is 6.3%, and the volatility of the stock is 20%. The risk-free rate is 4.9%. Assume the log-normal model.

A derivative pays you S^2 in 9 months, where S denotes the price of the stock at that time. Calculate its Delta Δ and Gamma Γ.

Enter your solutions to three decimal places.

Δ=

Γ=

21.Your portfolio Δ is 1.047 and Γ is -0.936.

(a) Suppose your portfolio suddenly increases in value by $17.00. Estimate the implied change in value of the portfolio's underlying asset (stock) using Δ.

Enter your solution as a dollar value, including dollar sign ($), to two decimal places.

22.A stock is currently priced at $39.00. The risk free rate is 4.9% per annum with continuous compounding. Every 6 months, its price will either go up by 17% or down by 19%. Consider a European put with strike $42.00 expiring in 12 months.

(a) Using the binomial tree model, compute the price of a European put option at the initial node, the two intermediate nodes, and the three terminal nodes.

Enter the following solutions as dollar values, including dollar symbols ($), to two decimal places.

Note: The solutions to later parts of this problem may require more precision than solutions to earlier parts. Therefore, you need to record your solutions to a higher precision than required for use in subsequent computations in order to avoid rounding errors later on.

Top terminal node:

Middle terminal node:

Lower terminal node:

Upper intermediate node:

Lower intermediate node:

Initial node:

(b) Estimate the Δ of the put at the two intermediate nodes and the initial node using the solutions to part (a)

Enter the following solutions to three decimal places.

Upper intermediate node:

Lower intermediate node:

Initial node:

(c) Estimate the Γ of the put at the initial node using the solutions to part (b)

Enter your solution to three decimal places.

(d) You own 1 put today. Use part (b) to determine how much stock you should buy to be Δ-neutral (that is, to ensure that the portfolio value does not change regardless of where the stock goes to at the intermediate time step 6 months from today).

Enter your solution to three decimal places. Note that a negative answer means selling stocks

Buy shares

(e) Suppose the stock goes up at the intermediate time step. How should you change your position in order to remain Δ-neutral? In other words, how many stocks should you add to your holding from part (d)? This is an example of dynamic hedging.

Enter your solution to three decimal places. Note that a negative answer means selling stocks

Buy shares

23.Derivative A has the following Greeks today: ΔA=0, νA= 0.52, and ΓA=1.42. Derivative B has the following Greeks today: ΔB=0, νB= 0.57, and ΓB= 2.11. You write a derivative X with the following Greeks: ΔX=332, νX= 124, and ΓX=58. All three derivatives are for the same underlying asset.

How many shares of the underlying asset, derivative AA, and derivative B should you add to your portfolio to make it simultaneously Δ, ν, and Γ−neutral?

Enter the following solutions to three decimal places.

Amount of stock: shares

Amount of derivative AA: shares

Amount of derivative BB: shares

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