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# 数学代写|金融衍生品代写Financial Derivatives代考|NBA6730 STOCHASTIC VOLATILITY MODELS

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## 数学代写|金融衍生品代写Financial Derivatives代考|STOCHASTIC VOLATILITY MODELS

When the instantaneous volatility is a known function of time, the risk-neutral process followed by the stock price is
$$d S=(r-q) S d t+a(t) S d z$$
The Black-Scholes formulas are then correct providing the variance rate is set equal to the average variance rate during the life of the option. The variance rate is the square of the volatility. Suppose that during a one-year period the volatility of a stock will be $20 \%$ during the first six months and $30 \%$ during the second six months. The average variance rate is
$$0.5 \times 0.20^2+0.5 \times 0.30^2=0.065$$
It is correct to use Black-Scholes with a variance rate of $0.065$. This corresponds to a volatility of $\mathrm{V} 0.065=0.255$, or $25.5 \%$. model can be used to calculate volatility term structures.Equation (1) assumes that the instantaneous volatility of an asset is perfectly predictable. Inpractice volatility varies stochastically. This has led some researchers to develop more complex models where there are two stochastic variables: the stock price and its volatility.

## 数学代写|金融衍生品代写Financial Derivatives代考|GAMMA MODEL

The gamma of a portfolio of options on an underlying asset, $F$, is the rate of change of theportfolio’s delta with respect to the price of the underlying asset. It is the second partialderivative of the portfolio with respect to asset price:
$$r-\frac{\partial^2 n}{d s^2}$$
If gamma is small, delta changes slowly, and adjustments to keep a portfolio delta neutral need tobe made only relatively infrequently. However, if gamma is large in absolute terms, delta is highlysensitive to the price of the underlying asset. It is then quite risky to leave a delta-neutral portfoliounchanged for any length of time. Figure illustrates this point. When the stock price movesfrom $S$ to $S^{\prime}$, delta hedging assumes that the option price moves from $C$ to $C^{\prime}$, when in fact itmoves from $C$ to $C^{\prime \prime}$. The difference between $C$ and $C^{\prime \prime}$ leads to a hedging error. This errordepends on the curvature of the relationship between the option price and the stock price. Gammameasures this curvature.
Suppose that $S S$ is the price change of an underlying asset during a small interval of time, $S t$, and $<5 n$ is the corresponding price change in the portfolio. If terms of higher order than $S t$ are ignored, a delta-neutral portfolio,

$$s n=\delta t+\frac{1}{2} \Gamma \delta S^2$$
where 0 is the theta of the portfolio. Figure $14.8$ shows the nature of this relationship between $S U$ and $S S$. When gamma is positive, theta tends to be negative. The portfolio declines in value if there is no change in $S$, but increases in value if there is a large positive or negative change in $S$.

## 数学代写|金融衍生品代写金融衍生品代考|随机波动模型

$$d S=(r-q) S d t+a(t) S d z$$

$$0.5 \times 0.20^2+0.5 \times 0.30^2=0.065$$

## 数学代写|金融衍生品代写金融衍生工具代考|GAMMA MODEL

. GAMMA MODEL . GAMMA MODEL . GAMMA MODEL

$$r-\frac{\partial^2 n}{d s^2}$$

$$s n=\delta t+\frac{1}{2} \Gamma \delta S^2$$

## MATLAB代写

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