Part 1: Yield Curve Analysis, Nelson-Siegel-Svensson Model (NSS), and Bond Pricing

The CEO of SHB-Brooklyn Capital Management asks you 1) to analyze the term structure of
yields; 2) price a 8% semi-annual coupon bond maturing in 7 years. To price the corporate bond,
you need to implement a Nelson-Siegel-Svensson model following the below steps:

1. Open “Project Data.xlsx”
2. Using data from the worksheet titled “2017” create the yield curve
3. Using data from the worksheet titled “2018” create the yield curve
4. Analyze each shape of the yield curves from problem 1 and problem 2 and compare them. What are your findings? Provide your thoughts about market expectations as of “2017” and “2018”.
5. Using the yield rates as of 12/04/2017 (from “Project Data.xlsx”) let us implement the Nelson-Siegel-Svensson four factor model as:

a) Estimate six parameters including _β1; β 2; β 3; β 4; λ1; and λ2 solving the below equation:

b) Report 0.5 year, 1.5 year, 2.5 year, 3.5 year, …, 29.5 year, and 30 year yield rates that estimated from the model
c) Plot a combo chart displaying the original yield rates and the estimated yield rates over the entire time horizon.

1. Using the yield rates as of 12/03/2018 (from “Project Data.xlsx”) repeat steps a), b), & c) from step 5
2. Compute two 7-year semi-annual coupon bonds with a coupon rate of 8%: 1) using the spot rates from problem 4, and 2) using the spot rates from problem 5. Note that a face value is $ 1,000
3. look into two different bond prices. Do you observe a significant price difference between them? If so, what does this gap mean? Provide your thoughts.

Part 2: Quantifying Systemic Risk

Let us quantify systemic risk with VaR methods.

Steps:

1. Open “Project Data.xlsx”
2. Compute daily returns for the stock from the worksheet titled Part 2 (from “Project Data.xlsx”) by using the below equation and plot a histogram to see the distribution of the stock returns. Check the shape of this distribution and compare it with the normal distribution.
3. Using the parametric method, compute a 1 day VaR with 1% significant level ((i.e. 99% confidence interval).
4. Using the parametric method, compute a 1 day ES with 1% significant level
5. From step 2, calculate annualized returns and standard deviations, which are given by
6. Using the annualized return and standard deviation from the previous problem, estimate a 1 year VaR with 1% significant level employing the parametric method.
7. For this time, calculate a 1 day VaR and 1 day ES using the simulation method (historical simulation). (Hint: Use PERCENTILE in Excel.)
8. Compute daily returns for the stock from the worksheet titled Part 3 (from “Project Data.xlsx”) and plot a histogram. Do you think it looks like the shape of normal distribution? Why or why not?
9. Using both the 1 day VaR from the parametric method and the 1 day VaR from the historical simulation method, count how many days exceeded those VaRs from Jan. 2, 2005 to Dec.31, 2018.
10. Compare two VaRs and answer which method would be better. Provide the reason why you select either the parametric VaR or the simulation VaR.

Part 3: Black-Scholes and Binomial Option Pricing

Let us compute an option price using Black-Scholes Model and Binomial Tree model.

1. Suppose that the current price of BA is $200. The continuously compounded risk-free rate is 4% per year. The annual standard deviations are 20%. Also, assume BA pays no dividends.

a) Compute the value of a long 2 year an European put option with a strike price of $ 195 and the annualized standard deviation is 20% using a two-step binomial tree model & the Black-Sholes formula which is given by

b) Examine whether the binomial tree model and or the black-sholes model have the same answer or not

c) Use the put-call parity and compute two different 2 year call option prices with 2 different put option values

d) Calculate the Greeks that include delta, gamma, vega, rho, and theta for the put option using the below equations

where:

1. Suppose that the current price of BA stock is $200. The annual standard deviation is the same as the previous problem. The continuously compounded risk-free rate is 4% per year. Assume BA pays no dividends.

a) Compute a European call option price with the respective intrinsic values of a 2 year with a strike price of $198 using a two-step binomial model

b) Compute a European call option price with the respective intrinsic values of a 2 year with a strike price of $198 using a four-step binomial model

c) Compute a European call option price with the respective intrinsic values of a 2 year with a strike price of $198 using a eight-step binomial model

d) Compute a European call option price using the Black-Scholes model.

e) Compare four option prices and provide your findings from the results.

1. Let us compute two American option prices using the same information in problem 2.

a) Compute an American put option price with the respective intrinsic values of a 4 year with a strike price of $198 using a four-step binomial model
b) Compute an American call option price with the respective intrinsic values of a 4 year with a strike price of $198 using a four-step binomial model

c) Using the put option price from problem (a), compute an American call option.

d) Compare two call option prices from (a) and (c) and provide your findings.

Sample Solution