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# 金融代写|金融微积分代写Financial Calculus代考|MATH205 Bond-only strategy

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## 金融代写|金融微积分代写Financial Calculus代考|Bond-only strategy

All is not lost, though. Consider a portfolio of just the cash bond. The cash bond will grow by a factor of $\exp (r \delta t)$ across the period, thus buying discount bonds to the value of $\exp (-r \delta t)[(1-p) f(2)+p f(3)]$ at the start of the period will provide a value equal to $(1-p) f(2)+p f(3)$ at the end. Why would we choose this value as the target to aim for? Because it is the expected value of the derivative at the end of the period – formally:
Expectation for a branch
Let $S$ be a binomial branch process with base value $s_1$ at time zero, downvalue $s_2$ and up-value $s_3$. Then the expectation of $S$ at tick-time 1 under the probability of an up-move $p$ is:
$$\mathbb{E}_p\left(S_1\right)=(1-p) s_2+p s_3$$
Our claim $f$ on $S$ is just as much a random variable as $S_1$ is – we can meaningfully talk of its expectation. And thus we can meaningfully aim for the expectation of the claim, via the cash bonds. This strategy of construction would at the very least be expected to break even. And the value of the starting portfolio of cash bonds might be claimed to be a good predictor of the value of the derivative at the start of the period. The price we would predict for the derivative would be the discounted expectation of its value at the end.
But of course this is just the strong law of chapter one all over again just thinly disguised as construction. And exactly as before we are missing an element of coercion. We haven’t explicitly constructed the two possible values the derivative can take: $f(2)$ and $f(3)$; we have simply aimed between them in a probabilistic sense and hoped for the best.

And we already know that this best isn’t good enough for forwards. For a stock that obeys a binomial branch process, its forward price is not suggested by the possible stock values $s_2$ and $s_3$, but enforced by the interest rate $r$ implied by the cash bond $B$ : namely $s_1 \exp (r \delta t)$. The discounted expectation of the claim doesn’t work as a pricing tool.

## 金融代写|金融微积分代写Financial Calculus代考|Stocks and bonds together

But can we do any better? Another strategy might occur to us, we have after all two instruments which we can build into a portfolio to hold for the tick-period. We tried using the guaranteed growth of the cash bond as a device for producing a particular desired value, and we chose the expected value of the derivative as our target point. But we have another instrument tied more strongly to the behaviour of both the stock and the derivative than just the cash bond. Namely the stock itself. Suppose we attempted to guarantee not an amount known in advance which we hope will stand as a reasonable predictor for the value of the derivative, but the value of the derivative itself, whatever it might be.

Consider a general portfolio $(\phi, \psi)$, namely $\phi$ of the stock $S$ (worth $\phi s_1$ ) and $\psi$ of the cash bond $B$ (worth $\psi B_0$ ). If we were to buy this portfolio at time zero, it would cost $\phi s_1+\psi B_0$.
One tick later, though, it would be worth one of two possible values:
$\phi s_3+\psi B_0 \exp (r \delta t) \quad$ after an ‘up’ move,
and $\phi s_2+\psi B_0 \exp (r \delta t) \quad$ after a ‘down’ move.
This pair of equations should intrigue us – we have two equations, two possible claim values and two free variables $\phi$ and $\psi$. We have two values $f(3)$ and $f(2)$ which we want to duplicate under the appropriate move of the stock, and we have two variables $\phi$ and $\psi$ which we can adjust. Thus the strategy can reduce to solving the following two simultaneous equations for $(\phi, \psi)$
\begin{aligned} \phi s_3+\psi B_0 \exp (r \delta t) &=f(3) \ \phi s_2+\psi B_0 \exp (r \delta t) &=f(2) \end{aligned}
Except if perversely $s_2$ and $s_3$ are identical – in which case $S$ is a bond not a stock – we have the solutions:
\begin{aligned} \phi &=\frac{f(3)-f(2)}{s_3-s_2} \ \psi &=B_0^{-1} \exp (-r \delta t)\left(f(3)-\frac{(f(3)-f(2)) s_3}{s_3-s_2}\right) \end{aligned}
What can we do with this algebraic result? If we bought this $(\phi, \psi)$ portfolio and held it, the equations guarantee that we achieve our goal – if the stock moves up, then the portfolio becomes worth $f(3)$; and if the stock moves down, the portfolio becomes worth $f(2)$. We have synthesized the derivative.

## 金融代写|金融微积分代写Financial Calculus代考|Bond-only strategy

$$\mathbb{E}_p\left(S_1\right)=(1-p) s_2+p s_3$$

## 金融代写|金融微积分代写Financial Calculus代考|Stocks and bonds together

$\phi s_3+\psi B_0 \exp (r \delta t)$ 在“向上”移动之后，

$$\phi s_3+\psi B_0 \exp (r \delta t)=f(3) \phi s_2+\psi B_0 \exp (r \delta t) \quad=f(2)$$

$$\phi=\frac{f(3)-f(2)}{s_3-s_2} \psi \quad=B_0^{-1} \exp (-r \delta t)\left(f(3)-\frac{(f(3)-f(2)) s_3}{s_3-s_2}\right)$$

## MATLAB代写

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