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Finance Problems and Solutions

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?

## Problem 2

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.

## Problem 3

(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.

## Problem 4

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.

## Problem 5

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

(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