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# 数学代写|期权定价理论代写Option Pricing Theory代考|ES_APPM401 INTEGRATING THE SDE FOR ST

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## 数学代写|期权定价理论代写Option Pricing Theory代考|INTEGRATING THE SDE FOR ST

From (2.1), let $\mathrm{d} S_t=\mu S_t \mathrm{~d} t+\sigma S_t \mathrm{~d} W_t$. We can write this more simply as
$$\frac{\mathrm{d} S_t}{S_t}=\mu \mathrm{d} t+\sigma \mathrm{d} W_t, \text { noting that } \frac{\mathrm{d} S_t^2}{S_t^2}=\sigma^2 \mathrm{~d} t .$$
Consider the process $X_t=f\left(S_t\right)$ defined by $f(x)=\ln (x)$. We have $f^{\prime}(x)=1 / x$ and $f^{\prime \prime}(x)=$ $-x^{-2}$. A simple application of Itô’s lemma gives
\begin{aligned} \mathrm{d} X_t &=f^{\prime}\left(S_t\right) \mathrm{d} S_t+\frac{1}{2} f^{\prime \prime}\left(S_t\right) \mathrm{d} S_t^2 \ &=\frac{\mathrm{d} S_t}{S_t}-\frac{1}{2} \frac{\mathrm{d} S_t^2}{S_t^2} \ &=\mu \mathrm{d} t+\sigma \mathrm{d} W_t-\frac{1}{2} \sigma^2 \mathrm{~d} t \end{aligned}
This can be immediately integrated to give
$$X_t=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) t+\sigma\left[W_t-W_0\right] .$$
Since $W_t$ is assumed to be a standardised Brownian motion with $W_0=0$, one obtains
$$X_T=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma W_T$$

and since $X_t=\ln \left(S_t\right) \Leftrightarrow S_t=\exp \left(X_t\right)$, one obtains the desired result
$$S_T=S_0 \exp \left(\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma W_T\right) .$$
Note that $(2.21)$ can be written as
$$X_T=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma \sqrt{T} \xi,$$
where $\xi \sim N(0,1)$.

## 数学代写|期权定价理论代写Option Pricing Theory代考|BLACK–SCHOLES PDEs EXPRESSED IN LOGSPOT

The algebra of Section $2.4$ shows that, under the assumption of geometric Brownian motion for the traded asset, it is easier to deal with the stochastic differential equation for logspot $X_t$ than the equivalent stochastic differential equation for spot $S_t$, as the drift and volatility terms for $X_t$ are homogeneous while those for $S_t$ depend on the level of the traded asset. In the same manner, the Black-Scholes PDEs (2.10) and (2.17) are simpler when expressed in terms of spatial derivatives with respect to logspot $x$, as opposed to derivatives with respect to spot $S$. We obtain, for $V=V\left(X_t, t\right)$, with a slight abuse of notation as this should be $\hat{V}=\hat{V}\left(X_t, t\right)$ :
$$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 \frac{\partial^2 V}{\partial x^2}+\left(r^d-r^f-\frac{1}{2} \sigma^2\right) \frac{\partial V}{\partial x}-r^d V=0 .$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|NTEGRATING THE SDE FOR ST

$$\frac{\mathrm{d} S_t}{S_t}=\mu \mathrm{d} t+\sigma \mathrm{d} W_t, \text { noting that } \frac{\mathrm{d} S_t^2}{S_t^2}=\sigma^2 \mathrm{~d} t .$$

$$\mathrm{d} X_t=f^{\prime}\left(S_t\right) \mathrm{d} S_t+\frac{1}{2} f^{\prime \prime}\left(S_t\right) \mathrm{d} S_t^2 \quad=\frac{\mathrm{d} S_t}{S_t}-\frac{1}{2} \frac{\mathrm{d} S_t^2}{S_t^2}=\mu \mathrm{d} t+\sigma \mathrm{d} W_t-\frac{1}{2} \sigma^2 \mathrm{~d} t$$

$$X_t=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) t+\sigma\left[W_t-W_0\right] .$$

$$X_T=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma W_T$$

$$S_T=S_0 \exp \left(\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma W_T\right) .$$

$$X_T=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma \sqrt{T} \xi,$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|BLACK-SCHOLES PDEs EXPRESSED IN LOGSPOT

$$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 \frac{\partial^2 V}{\partial x^2}+\left(r^d-r^f-\frac{1}{2} \sigma^2\right) \frac{\partial V}{\partial x}-r^d V=0$$

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