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# 金融代写|期权定价理论代写Option Pricing Theory代考|ES_APPM401 DERIVATION OF THE BLACK-SCHOLES EQUATION

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## 金融代写|期权定价理论代写Option Pricing Theory代考|DERIVATION OF THE BLACK-SCHOLES EQUATION

Suppose that the price of a contingent claim $V\left(S_t, t\right)$, which derives its value from the performance of a tradeable equity security with asset price $S_t$, is known. How can we obtain any information about this? Let $V_t=V\left(S_t, t\right)$ denote ${ }^2$ the value of the contingent claim at time $t$, conditional on the equity price being $S_t$ at that time. Applying Itô, we have
$$\mathrm{d} V_t=\frac{\partial V}{\partial t} \mathrm{~d} t+\frac{\partial V}{\partial S} \mathrm{~d} S_t+\frac{1}{2} \frac{\partial^2 V}{\partial S^2} \mathrm{~d} S_t^2 .$$
where $\mathrm{d} S_t^2$ denotes the quadratic variation of $S_t$, not to be confused with differential increments in the second asset price process $S_t^{(2)}$, such as encountered in Chapter 10. However, we know from (2.1) that $\mathrm{d} S_t^2=\sigma^2 S_t^2 \mathrm{~d} t$ and consequently at time $t$ we have
$$\mathrm{d} V_t=\left[\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right] \mathrm{d} t+\frac{\partial V}{\partial S} \mathrm{~d} S_t .$$
The term inside the square brackets above is deterministic, whereas the term appearing in front of $\mathrm{d} S_t$ is the only stochastic term. We remove the stochastic term by construction of a portfolio $\Pi_t$, which is long one unit of the contingent claim (with value $V_t$ ) and short $\partial V / \partial S$ units of the underlying asset
$$\Pi_t=V_t-\frac{\partial V}{\partial S} S_t .$$

## 金融代写|期权定价理论代写Option Pricing Theory代考|FX derivatives

The situation in $\mathrm{FX}$ is slightly more complicated in that the $\mathrm{FX}$ spot rate $S_t$ is not a natural store of wealth and cannot be regarded as a tradeable as in the analysis above. One should instead think of it as the exchange rate as a stochastic conversion rate relating two numeraires, each of which has its own natural store of wealth – the money market account in either currency.
As in Section 2.3.1 we suppose that the price of a contingent claim $V\left(S_t, t\right)$ is known, which derives its value from the performance of an FX rate $S_t$. Note that the tradeable asset in this case is not the FX rate $S_t$; it is instead the foreign bond valued in units of the domestic currency, i.e. $S_t B_t^f$.
Once again using $V_t$ to denote the price of the contingent claim at time $t$, we still have
$$\mathrm{d} V_t=\left[\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right] \mathrm{d} t+\frac{\partial V}{\partial S} \mathrm{~d} S_t,$$
but the construction of the delta-hedged portfolio is somewhat different. As remarked above, we cannot buy and sell units of the FX spot rate – the construction of the hedged portfolio $\Pi_t$ is obtained by going long one unit of the contingent claim (with value $V_t$ ) and short $\Delta_t$ units of the underlying foreign bond:
$$\Pi_t=V_t-\Delta_t S_t B_t^f .$$

## 金融代写|期权定价理论代写OPTION PRICING THEORY代 考|DERIVATION OF THE BLACK-SCHOLES EQUATION

$$\mathrm{d} V_t=\frac{\partial V}{\partial t} \mathrm{~d} t+\frac{\partial V}{\partial S} \mathrm{~d} S_t+\frac{1}{2} \frac{\partial^2 V}{\partial S^2} \mathrm{~d} S_t^2 .$$

$$\mathrm{d} V_t=\left[\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right] \mathrm{d} t+\frac{\partial V}{\partial S} \mathrm{~d} S_t$$

\Pi_t=V_t-\frac{\partial V}{\partial S} S_t .
$$## 金融代写|期权定价理论代写OPTION PRICING THEORY代考|FX DERIVATIVES 情况在 \mathrm{FX} 稍微复杂一点 \mathrm{FX} 即期汇率 S_t 不是一种天然的财富储备，不能像上面的分析那样被视为可交易的。相反，人们应该将汇率视为与两种货市相关的随机转 换率，每种货币都有自己的目然财富储备一一两种货市的货市市场账户。 正如在第 2.3.1 节中，我们假设或然债权的价格 V\left(S_t, t\right) 是已知的，它的价值来自外汇汇率的表现 S_t. 请注意，这种情况下的可交易冷产不是外汇汇率 S_t; 相反，它 是以本国货市为单位计价的外国债券，即 S_t B_t^f. 再次使用 V_t 表示或然债权的价格 t ＼mathrm{~ ， 我 们 还 有 ~}$$
\mathrm{d} V_t=\left[\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right] \mathrm{d} t+\frac{\partial V}{\partial S} \mathrm{~d} S_t,
$$但是 delta 对冲投资组合的构建有些不同。如上所述，我们不能买卖外汇即期汇率的单位一一对冲投资组合的构建 \Pi_t 通过做多或有债权的一个单位获得 withvalue \ V_t \$$ 和短 $\Delta_t$ 基础外国债券的单位:
$$\Pi_t=V_t-\Delta_t S_t B_t^f .$$

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