Reference no: EM132411910

FINC 3340 - Options, Futures, and Commodities Markets - Brooklyn College

Instruction

The SHB-Brooklyn Capital Management is seeking to expand to the field of high frequency tradings from low frequency trading strategies. Because this new trading area requires the company to purchase new trading platforms, super fast computing equipment, and expensive real time millisecond data, and to hire genius financial engineers from Ivy league universities and brilliant CUNY-Brooklyn alumni, it needs substantial amount of funds. To collect funds, the president and CEO agree to issue 30 year corporate bonds to invest in this new business. Since there are no 30 year corporate bonds trading in the fixed income market, they are not able to figure out what is the fair value of this bond.Thus, they ask the department of Sale and Trading Securities to report what they would like to know. Now you are retained by the department of Sale and Trading Securities in the SHB-Brooklyn Capital Management and you are asked to implement a term structure model and price this bond.

Problem 1 Yield Curve and Cubic Spline Approach

1. Find daily treasure yield curve rates on 11/15/2019 using Daily Treasury Yield Curve Rates provided by the US Department of Treasury1 and report those daily rates (i.e., 1 Month, 2 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. Using the cubic interpolation method, r^(t) = r₀ + at + bt² + ct³,

(a) Find r₀, a, b, and c. solving the below equation:

min              Σ_t=1³⁰ (r₁(t) - r^{^}₁(t))² = Σ_t=1³⁰e_i²
γ₀,a₁,b₁,c₁

(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 graphs in a figure displaying the original yield rates and the estimated yield rates over the entire time horizon.

Problem 2 Establish Vasicek term structure model

1. Based on the yield rates on Nov.15, 2019, let us develop another yield curve model applying the Vasicek model as below.

r^(t) = -1/t[ln(A(t) - B(t) x r_o)]

where:

A(t) = e(B(t) - t) (κ²µ - σ²/2)/k² - σ²B²(t)/4k

B(t) = 1-e^-κt/k

(a) Find out three parameters including κ, µ, and σ solving the below equation:

min             Σ_t=1³⁰ (r(t) - r^{^}(t))² = Σ_t=1³⁰e_i²
β₁,β₂,β₃,λ

(b) Estimate 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 graphs in a figure displaying the original yield rates and the estimated yield rates over the entire time horizon.

Problem 3 Establish CIR term structure model

1. Based on the yield rates on Nov.15, 2019, let us develop another yield curve model applying the CIR model as below.

r^(t) = -1/t [ln(A(t)) - B(t) × r₀]

where:

A(t) = ( 2γe(k+y+λ)^t/2/2κ+(κ+γ+λ)(e^γt-1))2κµ/σ²

B(t) = 2(e^γt - 1)/2κ+(κ+γ+λ)(e^γt - 1)

γ = √(κ + λ)² + 2σ²

(a) Find out three parameters including κ, µ, and σ solving the below equation:

min                Σ_t=1³⁰ (r(t) - r^{^}(t))² = Σ_t=1³⁰e_i²
β1,β2,β3,λ

(b) Estimate 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 graphs in a figure displaying the original yield rates and the estimated yield rates over the entire time horizon.

Problem 4 Establish Nelson-Siegel term structure model

1. Based on the yield rates on Nov.15, 2019, let us develop another yield curve model applying the Nelson-Siegel parsimonious approximation specified as below.

rˆ(t) = β1 + (β2 + β3)(1 e^-λt/λt) - β3e^-λt

(a) Find out four parameters including β₁, β₂, β₃, and λ solving the below equation:

min                   Σ_t=1³⁰ (r(t) - r^{^}(t))² = Σ_t=1³⁰e_i²
β1,β2,β3,λ

(b) Estimate 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 graphs in a figure displaying the original yield rates and the estimated yield rates over the entire time horizon.

(d) Report what you have observed from those four estimated models (Cublic Spline, Vasicek, CIR, and Nelson Siegel). Which model would be more accurate? Provide your evidence supporting your answers.

Problem 5 Bond Pricing

1. Using the fitted spot rates from the Nelson Siegel model and the zero rates as of Nov.15, 2019 from the US Department of Treasury, price a 6% 30 year semi-annual coupon bond. Assume that the par value of this bond is $ 100.

2. Compute a bond yield to maturity based on the bond price.

3. Calculate a duration (D) and a convexity (C) of this bond.

4. Using the obtained duration and convexity, compute a changed bond price when the yield increased by 15 basis point.2

5. Compare the approximation of the changed bond price using duration and convexity from problem 4 with the exact changed bond price for the changed yield.