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# 金融代写|金融工程代考Financial Engineering代写|TU-E2210 Arbitrage Pricing of Bonds

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## 金融代写|金融工程代考Financial Engineering代写|Arbitrage Pricing of Bonds

Suppose that we are given the binomial tree of bond prices $B\left(n, N ; s_n\right)$ for a bond maturing at the fixed time horizon $N$. In addition, we are given the money market process $A\left(n ; s_{n-1}\right)$. As was mentioned in the introduction to this chapter, the prices of other bonds cannot be completely arbitrary. We shall show that the prices $B\left(n, M ; s_n\right)$ for $M<N$ can be replicated by means of bonds with maturity $N$ and the money market. As a consequence of the No-Arbitrage Principle, the prices of $B\left(n, M ; s_n\right)$ will have to be equal to the values of the corresponding replicating strategies.
Example $11.5$
Consider the data in Example 11.1. At the first step the short rate is deterministic, being implied by the price $B(0,1)$. The first two values of the money market account are $A(0)=1$ and $A(1)=1.01$. As the underlying instrument we take the bond maturing at time 3 . The prices of this bond at time 0 and 1 are given in Figure 11.9, along with the prices of the bond maturing at time 2. We can find a portfolio $(x, y)$, with $x$ being the number of bonds of maturity 3 and $y$ the position in the money market, such that the value of this portfolio matches the time 1 prices of the bond maturing at time 2 . To this end we solve the following system of equations
\begin{aligned} & 0.9848 x+1.01 y=0.9948 \ & 0.9808 x+1.01 y=0.9907 \end{aligned}
obtaining $x=1$ and $y \cong 0.0098$. The value of this portfolio at time 0 is $1 \times B(0,3)+0.0098 \times A(0) \cong 0.9824$, which is not equal to $B(0,2)$. The prices in Figure $11.9$ provide an arbitrage opportunity:

• Sell a bond maturing at time 2 for $\$ 0.9828$and buy the portfolio constructed above for$\$0.9824$.

## MATLAB代写

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