Numerical analysis

Numerical analysis代写 Suppose that the short-term interest rate follows the process drt = a(b − rt)dt + σdW Q. In the pricing measure, Q.

Suppose that the short-term interest rate follows the process drt = a(b − rt)dt + σdW Q

Numerical analysis代写
Numerical analysis代写

In the pricing measure, Q Numerical analysis代写

This statement is intended to imply that bonds of all maturities T are available in the market at the risk-neutral prices given by Vasicek model. You should use this to compute the prices of bonds in the calculations below.

Parameter Value
a 0.5
b 0.05
σ 0.05
r0 0.02
λ -1
S 1
T 0.5
U 2
N 100

Table 1: Parameter values for the numerical simulation

You should discuss which P measure models are consistent with this Q- measure process for rt. In the calculations below you should use the specific P-measure model given by

 

 

 

where λ is a constant. The parameter values you should use are summarized in Table 1

 

In this model Numerical analysis代写

a trader can replicate the payoff of a T -bond (i.e. zero coupon a bond with maturity T) by a continuous time trading strategy in:

 

i) The money market account, which has instantaneous interest rate of rt;

ii)An S-bond, with maturity S > T

 

You should give the price for a bond with maturity T in this model and compute how many S-bonds should be held at time t ∈ [0, T ] to implement this replication strategy. We will call this quantity Qt.

 

In practice Numerical analysis代写

continuous time trading is not possible. Suppose that a trader has initial wealth 0 and receives the T bond price at time 0 and in exchange, promises to pay 1 unit of currency at time T. They then hedge their risk by trading at time intervals of δt = T/N for some integer N starting from time 0. At each discrete time point they ensure that they are holding Qt units of the S-bond and use the rest of their wealth to purchase zero coupon bonds with maturity t + δt.Numerical analysis代写

 

Derive equations for the traders’ wealth at each time point. Simulate the process rt. Use this to numerically compute the following statistics for the strategy in the given P-measure:

 

i)   The 1 day 95% Value at Risk of the hedging portfolio at time 0. This hedging portfolio consists of −1 unit of the T bond together with the trader’s holding in the bonds with maturity δt and S. You should use the Vasicek model to compute the market value of the bonds in your portfolio after 1

ii)   The 1 day 95% Expected Shortfall of the hedging portfolio at time0

iii)  The expected profit and loss of thestrategy

iv)  The variance of the profit and loss of thestrategy

Numerical analysis代写
Numerical analysis代写

Demonstrate numerically that as δt tends to 0 Numerical analysis代写

the variance of the strategy tends to zero, and estimate the rate of convergence. Demonstrate numerically that if the trader is also allowed to use a bond with maturity U to hedge the risk, it is possible to achieve lower Value at Risk and to achieve a faster rate of convergence. You may find that the benefits of this approach depend heavily upon the value of σ.

 

Hedging with S Bond
1 day VaR

1 day Expected Shortfall Expected Profit and Loss Variance of Profit and Loss

Hedging with S Bond U bond
1 day VaR

1 day Expected Shortfall Expected Profit and Loss Variance of Profit and Loss

Table 2: Summary of numerical results

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