### 金融代写|金融工程作业代写Financial Engineering代考|Greeks

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## 金融代写|金融工程作业代写Financial Engineering代考|Greeks

It is often important to measure the sensitivity of the option value with respect to the variables $t, s, r$, and $\sigma$. The so-called greeks are measures of sensitivity based on partial derivatives with respect to those parameters. Explicit formulas for greeks are known only in few cases, in particular the European call option [Wilmott, 2006]. In general, since there is no explicit expression for the option value, the greeks must be approximated. This will be done in Section 1.7. Here are the main definitions and interpretations for these useful parameters.

• The sensitivity of the option value with respect to the underlying asset price, called delta, is defined by
$$\Delta=\frac{\partial C}{\partial s}$$
The delta of an option is quite useful in hedging since it corresponds to the number of shares needed to create a risk-free portfolio replicating the value of the option at maturity; see Appendix 1.A.
• The sensitivity of the option value with respect to time, called theta, is defined by
$$\Theta=\frac{\partial C}{\partial t} .$$
Note that $-\Theta$, evaluated at $\tau=T-t$, yields the sensitivity with respect to the time to maturity $\tau$.
• The sensitivity of the option value with respect to the interest rate $r$, called $r o$, is defined by
$$\rho=\frac{\partial C}{\partial r}$$

Black-Scholes Model
15

• The sensitivity of the option value with respect to the volatility, called vega, is defined by
$$\mathcal{V}=\frac{\partial C}{\partial \sigma}$$
As shown next in Section 1.6.3, the vega is also important in determining the error on the option price due to the estimation of the volatility.
• A measure of convexity, the second order derivative of the option value with respect to the underlying asset prices, called gamma, is defined by
$$\Gamma=\frac{\partial^{2} C}{\partial s^{2}}$$
$\Gamma$ is useful in some approximations.

## 金融代写|金融工程作业代写Financial Engineering代考|Greeks for a European Call Option

Using the Black-Scholes formula (1.4), it is easy to check that

• $\Delta=\frac{\partial C}{\partial s}=\mathcal{N}\left(d_{1}\right)>0 .$
• $\Theta=\frac{\partial C}{\partial t}=-\frac{\sigma s}{2 \sqrt{T-t}} \frac{e^{-d_{1}^{2} / 2}}{\sqrt{2 \pi}}-K r e^{-r(T-t)} \mathcal{N}\left(d_{2}\right)<0 .$ $\rho=\frac{\partial C}{\partial r}=K(T-t) e^{-r(T-t)} \mathcal{N}\left(d_{2}\right)>0 .$
• $\mathcal{V}=\frac{\partial C}{\partial \sigma}=s \sqrt{T-t} \frac{e^{-d_{1}^{2} / 2}}{\sqrt{2 \pi}}>0 .$
Since the vega is positive, it means that the value of the option is an increasing function of the volatility. This property is essential in determining the so-called implied volatility.
• $\Gamma=\frac{\partial^{2} C}{\partial s^{2}}=\frac{1}{s \sigma \sqrt{T-t}} \frac{e^{-d_{1}^{2} / 2}}{\sqrt{2 \pi}}>0 .$
Since the gamma is positive, it means that the value of the option is a convex function of the underlying asset value.

Remark 1.6.1 For continuously paid dividends at rate $\delta$, using formula $(1.10)$, it is easy to check that $\Delta_{\delta}(t, s)=e^{-\delta \tau} \Delta_{0}\left(t, s e^{-\delta \tau}\right) .$ Also $\Gamma_{\delta}(t, s)=$ $e^{-2 \delta \tau} \Gamma_{0}\left(t, s e^{-\delta \tau}\right)$. Next, $\Theta_{\delta}(t, s)=\Theta_{0}\left(t, s e^{-\delta \tau}\right)+s \Delta_{\delta}(t, s)$. Finally, $\rho_{\delta}(t, s)=\rho_{0}\left(t, s e^{-\delta \tau}\right)$ and $\mathcal{V}{\delta}(t, s)=\mathcal{V}{0}\left(t, s e^{-\delta \tau}\right)$.

## 金融代写|金融工程作业代写Financial Engineering代考|Implied Distribution

One might ask why there is no sensitivity parameter corresponding to the partial derivative with respect to the strike price. In fact, there is one and it is related to the implied distribution [Breeden and Litzenberger, 1978]. Assuming that the value of a European call option is given by the expectation formula (1.8), and using the properties of expectations, namely (A.2), we obtain
$$C(t, s)=E_{Q}[\max {\tilde{S}(T)-K, 0} \mid \tilde{S}(t)=s]=\int_{K}^{\infty} Q{\tilde{S}(T)>y} d y$$
where $Q$ denotes the equivalent martingale measure. As a result,
$$\frac{\partial C}{\partial K}=-Q{\bar{S}(T)>K}=\tilde{F}(K)-1$$
where $\tilde{F}$ is the distribution function of $\bar{S}(T)$ given $\bar{S}(t)=s$, under the equivalent martingale measure $Q$. As a result $\frac{\partial C}{\partial K}$ is non-decreasing and it follows that $\frac{\partial^{2} C}{\partial K^{2}}=\tilde{f}(K) \geq 0$, where $\tilde{f}$ is the associated density, provided it exists. It also shows that the value of a call option is always a convex function of the strike. Note that in the case of the Black-Scholes model, the implied distribution is the log-normal, since $\ln {\tilde{S}(T)}$ has a Gaussian distribution with mean $\ln (s)+\left(r-\frac{\sigma^{2}}{2}\right) \tau$ and variance $\sigma^{2} \tau$, under the equivalent martingale measure. Since (1.18) is assumed to be always valid, not only for the BlackScholes model, the implied distribution function can be approximated from the market prices of the calls if there are enough strike prices available. See, e.g., Ait-Sahalia and Lo [1998].

As an example, consider the values of call options on Apple, on January $14^{\text {th }}, 2011$, with a 24-day maturity. The first data are shown in Table $1.2$; the complete data set is in the MATLAB structure AppleCalls containing the strikes and call market values for four different maturities. The graph is displayed in Figure 1.1. One can notice that the value of the call for a strike $K=\$ 210$seems too low, while the call values for strikes$K=\$160$ and $K=\$ 170$are too close, destroying the (theoretical) convexity of the curve. TABLE 1.2: Some market values of call options on Apple with a 24-day maturity, on January$14^{\text {th }}, 2011$. \begin{tabular}{|c|c|c|c|c|c|c|} \hline Strike &$160.00$&$170.00$&$200.00$&$210.00$&$220.00$&$240.00$\ \hline Call &$169.62$&$169.60$&$147.85$&$112.25$&$125.70$&$108.00$\ \hline \end{tabular} ## 金融工程代写 ## 金融代写|金融工程作业代写Financial Engineering代考|Greeks 衡量期权价值对变量的敏感性通常很重要吨,s,r， 和σ. 所谓的 greeks 是基于对这些参数的偏导数的灵敏度度量。希腊人的显式公式仅在少数情况下为人所知，尤其是欧洲看涨期权 [Wilmott, 2006]。一般来说，由于选项值没有明确的表达，希腊语必须是近似的。这将在第 1.7 节中完成。以下是这些有用参数的主要定义和解释。 • 期权价值相对于标的资产价格的敏感性，称为 delta，定义为 Δ=∂C∂s 期权的 delta 在对冲中非常有用，因为它对应于创建无风险投资组合所需的股票数量，该投资组合在到期时复制期权的价值；见附录 1.A。 • 期权价值对时间的敏感性，称为 theta，定义为 θ=∂C∂吨. 注意−θ, 评价为τ=吨−吨, 产生关于到期时间的敏感性τ. • 期权价值对利率的敏感性r, 称为r这, 定义为 ρ=∂C∂r 布莱克-斯科尔斯 15型 • 期权价值对波动率的敏感性称为 vega，定义为 在=∂C∂σ 如下 1.6.3 节所示，由于波动率的估计，vega 在确定期权价格的误差方面也很重要。 • 凸度的度量，即期权价值相对于标的资产价格的二阶导数，称为 gamma，定义为 Γ=∂2C∂s2 Γ在某些近似值中很有用。 ## 金融代写|金融工程作业代写Financial Engineering代考|Greeks for a European Call Option 使用 Black-Scholes 公式 (1.4)，很容易检查 • Δ=∂C∂s=ñ(d1)>0. • θ=∂C∂吨=−σs2吨−吨和−d12/22圆周率−ķr和−r(吨−吨)ñ(d2)<0. ρ=∂C∂r=ķ(吨−吨)和−r(吨−吨)ñ(d2)>0. • 在=∂C∂σ=s吨−吨和−d12/22圆周率>0. 由于 vega 是正数，这意味着期权的价值是波动率的增函数。该属性对于确定所谓的隐含波动率至关重要。 • Γ=∂2C∂s2=1sσ吨−吨和−d12/22圆周率>0. 由于 gamma 为正，意味着期权的价值是标的资产价值的凸函数。 备注 1.6.1 按利率连续派发股息d, 使用公式(1.10), 很容易检查Δd(吨,s)=和−dτΔ0(吨,s和−dτ).还Γd(吨,s)= 和−2dτΓ0(吨,s和−dτ). 下一个，θd(吨,s)=θ0(吨,s和−dτ)+sΔd(吨,s). 最后，ρd(吨,s)=ρ0(吨,s和−dτ)和在d(吨,s)=在0(吨,s和−dτ). ## 金融代写|金融工程作业代写Financial Engineering代考|Implied Distribution 有人可能会问，为什么没有对应于执行价格的偏导数的敏感性参数。事实上，有一个与隐含分布有关[Breeden and Litzenberger, 1978]。假设欧式看涨期权的价值由期望公式（1.8）给出，并利用期望的性质，即（A.2），我们得到 C(吨,s)=和问[最大限度小号~(吨)−ķ,0∣小号~(吨)=s]=∫ķ∞问小号~(吨)>是d是 在哪里问表示等价鞅测度。因此， ∂C∂ķ=−问小号¯(吨)>ķ=F~(ķ)−1 在哪里F~是分布函数小号¯(吨)给定小号¯(吨)=s, 在等价鞅测度下问. 因此∂C∂ķ是非减少的，因此∂2C∂ķ2=F~(ķ)≥0， 在哪里F~是相关密度，前提是它存在。它还表明，看涨期权的价值始终是行使价的凸函数。请注意，在 Black-Scholes 模型的情况下，隐含分布是对数正态分布，因为ln⁡小号~(吨)具有均值的高斯分布ln⁡(s)+(r−σ22)τ和方差σ2τ，在等价鞅测度下。由于假设 (1.18) 始终有效，不仅对于 BlackScholes 模型，如果有足够的执行价格可用，隐含分布函数可以从看涨期权的市场价格近似。例如，参见 Ait-Sahalia 和 Lo [1998]。 例如，考虑 Apple 的看涨期权价值，1 月14th ,2011，到期日为 24 天。第一个数据如表所示1.2; 完整的数据集位于 MATLAB 结构 AppleCalls 中，其中包含四种不同期限的行使价和看涨期权的市场价值。图表如图 1.1 所示。可以注意到呼吁罢工的价值ķ=$210似乎太低，而罢工的呼吁价值ķ=$160和ķ=$170太接近了，破坏了曲线的（理论）凸度。

\begin{tabular}{|c|c|c|c|c|c|c|} \hline 罢工 & $160.00$ & $170.00$ & $200.00$ & \$210

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