Finance Project - If your field of expertise is not finance, don't waste my time.

FINC 3340 Final Project

Seungho Baek

Due: Friday, Dec. 20, 2017 11:59 P.M.

Please make a cover sheet. In a cover sheet, list your name and ID (Provide a list of names who discussed about your attempt.). To submit this electronically, use Blackboard. DO NOT send me it via email. Please submit one excel file that shows your analysis procedure through Blackboard. NO LATE SUBMISSION ALLOWED. If you cannot submit it in time, you will get zero automatically.

Part1: Estimation and Forecast

You have been retained by the Baek Derivative Analytic Group to provide a forecast about future short term interest rates, namely, the 3 month t-bill rate. You decide to use two sources of data: historical interest rate data and current forward rates. The data necessary for this forecasting exercise are contained in the Excel file Final Project2017.XLSX, which you can find on the course web site. This dataset contains daily observations of the 3 month t-bill rate until April 12, 2017, as well as the Treasury Strip Price data on April 13, 2017. You must write a report including all relevant information and computations, and provide a forecast for an horizon ranging between 6 months and 5 years. Please, follow the steps below.

1. Let us denote by rt the Bond Equivalent Yield (BEY) on day t. The data on DTB3 in Final Project2017.XLSX are quoted on a discount basis. Use the below formulas in the previous teaching note to obtain a time series of BEY (n=90). For your information, please refer to the below equations. Then you will easily compute a time series of bond equivalent yield.

P = 100[1− n 360 × d]

d = 100− P

100 × 360

n

BEY = 100− P

P × 365

90

where d is a discount yield basis, n is the number of days to maturity, and P is the price of the T-bill.

2. Estimate the AR(3) process1 for interest rates.

rt+1 = α+ β1rt + β2rt−1 + β3rt−2 + �t+1 (1)

where �t+1 ∼ N(0, σ2). 1AR represents an autoregressive linear model

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3. Let α̂, β̂1, β̂2, and β̂3be the estimated parameters from (1). Use (1) together with the most current interest rate available on Final Project2017.XLSX, call it rTODAY , to make a forecast of future interest rates rTODAY+T . Provide forecasts for horizons T = 6 month, and 1, 2, .., 5 years (a plot would suffice). Explain how you make the forecasts. (Tip: When you make the forecasts, assume there are 252 (business) days in one year).

4. Use the Treasury Strip Prices also contained on STRIPS 04132017 in Final Project2017.XLSX to compute both the current yield curve and forward rates. Compare the forecasts of future interest rates that are implicit in the forward rates to those obtained in step 3 above. Plot the forecasts and the corresponding forward rates. Discuss your findings.

Part2: Piecewise Cubic Spline and Nelson-Siegel-Svensson Model

The CEO of SHB-Brooklyn Capital Management asks you to implement two different term struc- ture models: a piece-wise cubic spline and a Nelson-Siegel-Svensson model, and to examine their accuracy of the fitting the yields curve. So now you are planning to develop two models following the below steps.

1. Find daily treasure yield curve rates on 12/04/2017 using Daily Treasury Yield Curve Rates provided by the US Department of Treasury2 and report those daily rates (i.e., 1 Month, 3 Month, 6 Month, 1 Year, 2 Year, 3 Year, 5 Year, 7 Year, 10 Year, 20 Year, 30 Year) and plot the yield curve.

2. Let us use three piecewise cubic interpolations:

• r̂1(t) = γ0 + a1t+ b1t2 + c1t3 for 1 month to 1 year • r̂2(t) = δ0 + a2t+ b2t2 + c2t3 for 1.5 years to 10 years • r̂3(t) = ω0 + a3t+ b3t2 + c3t3 for 10.5 years to 30 years

denote r1(t) as the original yield rates for 1 month to 1 year; denote r2(t) as the original yield rates for 1.5 years to 10 years; denote r3(t) as the original yield rates for 10.5 years to 30 years

(a) Find out all the parameters (γ0, δ0, ω0, a1, a2, a3, b1, b2, b3, c1, c2, and c3) solving three equations:

i. min γ0,a1,b1,c1

∑1 t=0.08(r1(t)− r̂1(t))2

ii. min δ0,a2,b2,c2

∑10 t=1.5(r2(t)− r̂2(t))2

iii. min ω0,a3,b3,c3

∑30 t=10.5(r3(t)− r̂3(t))2

(b) Given the estimated model, predict 0.5 year, 1.5 year, 2.5 year, 3.5 year, ..., 29.5 year, and 30 year yield rates.

(c) Plot two series of charts in a figure displaying the original yield rates and the estimated yield rates over the entire time horizon.

2http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?

data=yield

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3. Using the Nelson-Siegel-Svensson four factor model as:

r̂(t) = β1 + β2( 1− e−λ1t

λ1t ) + β3(

1− e−λ1t

λ1t − e−λ1t) + β4(

1− eλ2t

λ2t − e−λ2t)

where λ1, λ2 > 0.

(a) Estimate six parameters including β1, β2, β3, β4, λ1, and λ2 solving the below equation:

min β1,β2,β3,β4,λ1,λ2

30∑ t=1

(r(t)− r̂(t))2

(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 two series of charts in a figure displaying the original yield rates and the estimated yield rates over the entire time horizon.

4. Suggest your opinion about the accuracy of the models and provide your evidence of your suggestion.

Part3: Quantifying Systemic Risk

Let us quantify systemic risk with VaR methods.

1. Download daily BAC stock price from either Yahoo Finance or Bloomberg. The data period is from December 31, 2002 and December 31, 2007. Compute daily returns for the stock 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.

RT = PT − PT−1 PT−1

, t = T, T − 1, . . . , 1

where RT is a stock return for BAC at time T (day) and PT is an adjusted close price at time T .

2. Calculate both arithmetic average and standard deviation of daily stock returns.

R̄ =

T∑ i=1

Ri T

σ =

√√√√ T∑ i=1

Ri − R̄ T − 1

3. Using the parametric method, compute a 1 day VaR with 1% significant level and a 1 year VaR.

4. Using the parametric method, compute a 1 day ES with 1% significant level.

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5. Calculate both geometric average and standard deviation of daily stock returns.

R̄g = e ∑T

i=1 ln(Ri)

T

σg = e

√∑T i=1

(ln(Ri)−ln(R̄g))2 T−1

6. From problem 2 and 4, calculate annualized returns and standard deviations, which are given by

Annualized R̄ = R̄× √

252

Annualized R̄g = R̄g × √

252

Annualizedσ = σ × √

252

Annualizedσg = σg × √

252

7. Using the annualized arithmetic return and standard deviation from the previous example, estimate a 1 year VaR with 1% significant level employing the parametric method.

8. For this time, calculate a 1 day VaR and 1 day ES using the simulation method (historical simulation).

9. Download daily BAC stock price from either Yahoo Finance or Bloomberg. The data period is from December 31, 2007 and December 31, 2009. Compute daily returns and plot a histogram. Do you think it looks like the shape of normal distribution? Why or why not?

10. Using both the 1 day VaR from the parametric method and the 1 day VaR from the simulation method, count how many days exceeded those VaRs from Jan 3, 2008 to Dec. 31, 2009.

11. Compare two VaRs and answer which method would be better. Provide the reason why you select either the parametric VaR or the simulation VaR with evidence.

Part4: 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 Bank of America(BAC) is $17. The continuously com- pounded risk-free rate is 4% per year. The annual standard deviations are from the previous problem. Also, assume BAC pays no dividends.

(a) Compute the value of a long 2 year an European put option with a strike price of $ 17 and the annualized standard deviation from problem2 in part3 using 1) a two-period binomial tree (BT) model and 2) the BLACK-SCHOLES (BS) formula,which is given by

• call option : c = S0N(d1)−Xe−rTN(d2)

• put option : p = S0N(−d2)−Xe−rTN(−d1)

d1 = ln(S0/X) + (r + σ

2/2)T

σ √ T

d2 = ln(S0/X) + (r − σ2/2)T

σ √ T

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That is, d2 = d1 − σ

√ T

where:

T : Time to maturity

S0 : Asset price

X : Exercise price

r: Continuously compounded risk free rate

σ : Volatility of continuously compounded return on the stock

N(∗) : Cumulative Normal Probability

(b) Compute the value of a 2 year an European put option with a strike price of $ 17 and the annualized geometric standard deviation from problem 4 in part3 using 1) a two-period binomial model and 2) the BLACK-SCHOLES formula.

(c) Examine whether BT and BS models have the same answer or not.

(d) Examine whether there is any difference between option values from arithmetic standard deviation and those from geometric standard deviation.

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

• Delta: ∆ = ∂V∂S = −N(−d1)

• Gamma: Γ = ∂∆∂S = N ′(d1)

S0·σ √ T

• Vega: ν = ∂V∂σ = S √ TN ′(d1) 100

• Rho: ρ = ∂V∂r = −KTe−rTN(−d2)

100

• Theta: Θ = ∂V∂T = −SσN

′(d1) 2 √ T

+Kre−rTN(−d2) 365

where:

V : Value of a put option, which is the same as p

N ′(d1) = e − d

2 1 2 · 1√

2π

2. Suppose that the current price of BAC stock is $20. The annual standard deviation is the same as the previous problem. The continuously compounded risk-free rate is 4% per year. Assume BAC pays no dividends.

(a) Compute the respective intrinsic values of a 5 year American call option with a strike price of $22 using a five-step binomial model.

(b) Compute the value of a 5 year American call option and put option with a strike price of $22 using a five-step binomial tree.

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