# Black Scholes and Binomial Option Valuation Models

Question 1) Panera's (PNR) stock price is $45. Its standard deviation is 35%. Interest rates are 3%. You are interested in an 8-month call. u and d are computed based on the formulas:

1. Draw the two-period binomial tree for PNR call options with exercise price = $40. . 2. Compute the hedge ratios at time 0 and after 4 months.

3. Discuss the observed changes in the hedge ratios.

Question 2) Based on the information in number 1 compute the value of a K=$40 European call using the two-period binomial model.

Question 3) Based on your answer to Question number 2:

1. Compute the maximum price an 8-month PNR call with K = $60 could sell for.

2. Describe an arbitrage opportunity if the actual price of the $60 call was less than this amount.

3. Describe how you could create a synthetic share of stock using options with K=$40. What price would the synthetic stock sell at? Based on this what is the must the K=40 put sell for?

Question 4) Bank of America stock sells for $3.19. Its implied volatility is 125%. Interest rates are 1%.

1. K=$5 options are among the most actively traded due to the uncertainty around BAC's future. Using the BSM formula compute N(d1) and N(d2). Briefly, discuss what each represents.

2. Based on the BSM compute the price of a $5 call that expires in 18 months.

3. Do you feel that the BSM or the binomial model is best equipped to price options on distressed, highly volatile stocks like BAC? Briefly explain.

Question 5) Below is information regarding 10.5 month BAC options:

BSM Call Value

1,08

BSM PUT VALUE

2,85

Delta

0.5970 (CALL)

-0.4030 (PUT)

Vega (per %)

0.0115 (CALL)

0.0115 (PUT)

Theta (per day)

-0.0024(CALL)

-0.0022(PUT)

Rho (per %)

0.0072(CALL)

-0.0362(PUT)

1. BAC has agreements to give its employees shares of stock worth $3.19 million in 10.5 months. It has 1 million shares of stock set aside to meet this obligation and does not want to issue more shares to fund the liability. What option position could BAC enter into to delta hedge its exposure?

2. What are estimated profits/losses after one month if the price of BAC has increased to $3.65? (Estimate price change based only on delta and theta)

3. How will gamma impact your answer to B? Briefly explain.

See attached file for questions.

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#### Solution Summary

Answers questions on valuation of options and delta hedging.