## 金融代写|期权理论代写Mathematical Introduction to Options代考|MATH4380

2022年10月14日

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## 金融代写|期权理论代写Mathematical Introduction to Options代考|SIMPLE EXAMPLE

(i) Suppose a company is awaiting a crucially important yes/no decision from a government regulator, to be announced in one month. The outcome will radically alter the company’s future in a way which is predictable, once we know which way the decision goes. If the decision is “yes”, the stock price will rise to $S_{\text {high }}$ but for a “no” the price will fall to $S_{\text {low. }}$. Obviously, $S_{\text {high }}$ and $S_{\text {low }}$ must be above and below the present stock price $S_0$ (if they were both above, $S_0$ would rise immediately). Let us further assume that everyone knows that given the political climate, the yes probability is $70 \%$ and the no probability $30 \%$.

We are equity derivatives investors and are holding an unquoted option on this company’s stock which matures immediately after the announcement. The payoff of the option is $f_{1 \text { month }}$, which takes values $f_{\text {high }}$ or $f_{\text {low }}$ depending on whether the stock price becomes $f_{\text {high }}$ or $S_{\text {low }}$. How would we go about working out today’s value for this option?
(ii) Considering first the stock price itself, the expected value in one month and the expected growth rate over that month $\mu$ are defined by
$$\mathrm{E}\left[S_{1 \text { month }}\right]=0.7 S_{\text {high }}+0.3 S_{\text {low }}=(1+\mu) S_0$$
At the risk of emphasizing the obvious, let us be clear on this point: $\mu$ is definitely not the rate by which $S_0$ will grow, since the final stock price will be either $S_{\text {high }}$ or $S_{\text {low. }}$. It is the mathematical expectation of the stock price growth. In this example we can work out $\mu$ from our knowledge of the probabilities of yes and no; alternatively, if we knew $\mu$ at the beginning, we could work out the probabilities.
The expected value for $f_{1 \text { month }}$ is similarly given by
$$\mathrm{E}\left[f_{1 \text { month }}\right]=0.7 f_{\text {high }}+0.3 f_{\text {low }}$$
which we can evaluate since we know the payoff values. It should not be too hard to calculate the present value, but how? The simplest way might be just to discount back by the interest rate, but remember that this is only valid for finding the present value of some certain future amount; for a risky asset, we must discount back by the rate of return (growth rate) of the particular asset. This is clear from the slightly rewritten equation (4.1):
$$S_0=\frac{\mathrm{E}\left[S_{1 \text { month }}\right]}{(1+\mu)}$$
Maybe the answer is to use $(1+\mu)$ as the discount factor; but $\mu$ is the growth rate of the underlying equity stock, not the option. There is nothing to suggest that the expected growth rate of the stock $\mu$ should equal the expected growth rate of the option $\lambda$. Nor is there any simple general way of deriving $\lambda$ from $\mu$. This was the point at which option theory remained stuck for many years. At this point, we enter the world of modern option theory.

## 金融代写|期权理论代写Mathematical Introduction to Options代考|CONTINUOUS TIME ANALYSIS

(i) The simple “high-low” example of the last section has wider applicability than a reader might expect at this point. However this remains to be developed in Chapter 7, and for the moment we will extend the theory in a way that describes real financial markets in a more credible way. Following the reasoning of the last section, we assume that we can construct a little portfolio in such a way that a derivative and $-\Delta$ units of stock hedge each other in the short term. Only short-term moves are considered since it is reasonable to assume that the $\Delta$ units of short stock position needed to hedge one derivative will vary with the stock price and the time to maturity. Therefore the hedge will only work over small ranges before $\Delta$ needs to be changed in order to maintain the perfect hedge.

The value of the portfolio at time $t$ may be written $f_t-S_t \Delta$. The increase in value of this portfolio over a small time interval $\delta t$, during which $S_t$ changes by $\delta S_t$, may be written
$$\delta f_t-S_t \Delta-S_t q \Delta \delta t$$
The first two terms are obvious while the last term is just the amount of dividend which we must pay to the stock lender from whom we have borrowed stock in the time interval $\delta t$, assuming a continuous dividend proportional to the stock price.

The quantity $\Delta$ is chosen so that the short stock position exactly hedges the derivative over a small time interval $\delta t$; this is the same as saying that the outcome of the portfolio is certain. The arbitrage arguments again lead us to the conclusion that the return of this portfolio must equal the interest rate:
$$\frac{\delta f_t-\delta S_t \Delta-S_t q \Delta \delta t}{f_t-S_t \Delta}=r \delta t$$
or
$$\delta f_t-\delta S_t \Delta+(r-q) S_t \Delta \delta t=r f_t \delta t$$
These equations are the exact analogue of equations (4.2) for the simple high-low model of the last section.

# 期权理论代写

## 金融代写|期权理论代写期权数学介绍代考|简单示例

.

(ii)首先考虑股价本身，一个月后的期望值和当月的预期增长率$\mu$由
$$\mathrm{E}\left[S_{1 \text { month }}\right]=0.7 S_{\text {high }}+0.3 S_{\text {low }}=(1+\mu) S_0$$

$$\mathrm{E}\left[f_{1 \text { month }}\right]=0.7 f_{\text {high }}+0.3 f_{\text {low }}$$

$$S_0=\frac{\mathrm{E}\left[S_{1 \text { month }}\right]}{(1+\mu)}$$

## 金融代写|期权理论代写期权数学介绍代考|CONTINUOUS TIME ANALYSIS

.

$$\delta f_t-S_t \Delta-S_t q \Delta \delta t$$

$$\frac{\delta f_t-\delta S_t \Delta-S_t q \Delta \delta t}{f_t-S_t \Delta}=r \delta t$$

$$\delta f_t-\delta S_t \Delta+(r-q) S_t \Delta \delta t=r f_t \delta t$$

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