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Numerical analysis

数值分析作业代写 In the pricing measure, Q. This statement is intended to imply that bonds of all maturities T are available in the market

Suppose that the short-term interest rate follows the process dr_t = a(b − r_t)dt + σdW ^{Q 数值分析作业代写}

In the pricing measure, Q. 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.

Table 1: Parameter values for the numerical simulation

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

dr_t =(a(b−r_t)−λσ)dt+σdW P

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

In this model, 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: 数值分析作业代写

(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 Q_t. 数值分析作业代写

In practice, 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.

Derive equations for the traders’ wealth at each time point. Simulate the process r_t. Use this to numerically compute the following statistics for the strategy in the given P-measure: 数值分析作业代写

1. (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
2. (ii) The 1 day 95% Expected Shortfall of the hedging portfolio at time0  数值分析作业代写
3. (iii) The expected profit and loss of thestrategy
4. (iv) The variance of the profit and loss of thestrategy

Demonstrate numerically that as δt tends to 0, 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.

Table 2: Summary of numerical results

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