### 金融代写|金融工程作业代写Financial Engineering代考|Estimation of Greeks using the Broadie-Glasserman

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

While it is generally impossible to find explicit expressions for the option value, we can however fairly easily estimate them with a Monte Carlo approximation of the expected value in (1.8). On a similar note, expressions for the greeks are often not available. An easy way to circumvent this problem, which is often used in practice, is to estimate them with a finite difference approximation. For example, the delta could be approximated as
$$\Delta \approx \frac{C(t, s+\epsilon)-C(t, s)}{\epsilon}$$
where $\epsilon$ is a small positive scalar. However, such procedures are plagued by an inevitable tradeoff; a large $\epsilon$ will produce biased estimations of the greeks, while small $\epsilon$ values will results in high estimation variance.

Fortunately, Broadie and Glasserman [1996] proposed methods to estimate an option’s value, together with unbiased estimations of the greeks. They considered several models, including the Black-Scholes model.

According to formula (1.8), the value of a European option with payoff $\Phi$ at maturity is
\begin{aligned} C(t, s) &=e^{-r \tau} E\left[\Phi\left{s e^{\left(r-\frac{\alpha^{2}}{2}\right) \tau+\sigma \sqrt{\tau} Z}\right}\right] \ &=e^{-r \tau} \int_{-\infty}^{+\infty} \Phi\left{s e^{\left(r-\frac{\sigma^{2}}{2}\right) \tau+\sigma \sqrt{\tau} z}\right} \frac{e^{-z^{2} / 2}}{\sqrt{2 \pi}} d z \ &=e^{-r \tau} \int_{0}^{+\infty} \Phi(x) \frac{e^{-\frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{2}}{2}\right) \tau\right}^{2}}}{x \sigma \sqrt{2 \pi \tau}} d x \end{aligned}
where $Z \sim N(0,1)$ and $\tau=T-t$ is the time to maturity.
Suppose that one generates $Z_{1}, \ldots, Z_{N} \sim N(0,1)$, with $N$ large enough. Further set $\bar{S}{i}=s e^{\left(r-\frac{a^{2}}{2}\right) \tau+\sigma \sqrt{\tau} Z{i}}, i \in{1 \ldots, N}$.
Then, an unbiased and consistent estimation of $C(t, s)$ is given by
$$\hat{C}=\frac{1}{N} \sum_{i=1}^{N} e^{-r \tau} \Phi\left(\bar{S}_{i}\right)$$
This Monte Carlo approach was proposed a long time ago by Boyle [1977]. However, no unbiased estimation of the greeks was proposed until Broadie and Glasserman [1996]. In their article, the authors proposed in fact two methodologies to estimate greeks, based respectively on representations (1.19) and $(1.20)$. These methodologies have the advantage of being computed in parallel with the option price, not sequentially.

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

The first methodology, called pathwise method, is based on representation (1.19). To be applicable, one has to assume that the payoff function $\Phi$ is differentiable “almost everywhere,” i.e., everywhere but possibly at a countable set of points ${ }^{11}$. However, note that the partial derivatives of any order for $\tilde{S}_{i}$ exist for any possible parameter $\theta \in{s, r, t, \sigma}$.

Proposition 1.7.1 Suppose that $\Phi$ is differentiable almost everywhere. Then simultaneous unbiased estimations of the option value and its first order derivatives are given respectively by
$$\hat{C}=\frac{1}{N} \sum_{i=1}^{N} e^{-r \tau} \Phi\left(\tilde{S}_{i}\right)$$

Remark 1.7.1 Since these estimations are averages of independent and identically distributed random vectors $X_{1}, \ldots, X_{N}$ with mean $\mathcal{G} \in \mathbb{R}^{p}$, one can determine the asymptotic behavior of the estimation errors. In fact, the central limit theorem (Theorem B.4.1) applies to yield
$$\sqrt{N}(\bar{X}-\mathcal{G}) \leftrightarrow N_{p}(0, V),$$
where $V$ is estimated by
$$\frac{1}{N-1} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)\left(X_{i}-\bar{X}\right)^{\top}$$
Example 1.7.1 For a European call option, $\Phi(s)=\max (s-K, 0)$, so $\Phi^{\prime}(s)=$ $\mathbb{I}(s>K)$ almost everywhere. As a result,
$$\hat{\Delta}=e^{-r \tau} \frac{1}{N} \sum_{i=1}^{N} \frac{\tilde{S}{i}}{s} \mathbb{I}\left(\bar{S}{i}>K\right)$$
and
$$\hat{\mathcal{V}}=e^{-r \tau} \frac{1}{N} \sum_{i=1}^{N}\left(Z_{i} \sqrt{\tau}-\sigma \tau\right) \tilde{S}{i} \mathbb{I}\left(\tilde{S}{i}>K\right) .$$
Therefore $\mathcal{G}=(C, \Delta, \mathcal{V})^{\top}$ can be estimated as the mean of the 3dimensional random vectors
$$X_{i}=e^{-r \tau} \mathbb{I}\left(\bar{S}{i}>K\right)\left(\bar{S}{i}-K, \frac{\bar{S}{i}}{s}, \bar{S}{i}\left(Z_{i} \sqrt{\tau}-\sigma \tau\right)\right)^{\top}$$
$i \in{1, \ldots, N} .$

## 金融代写|金融工程作业代写Financial Engineering代考|Likelihood Ratio Method

The second method proposed by Broadie and Glasserman $[1996]$ is based on representation $(1.20)$. For $x>0$, set
$$f(x)=\frac{e^{-\frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{2}}{2}\right) \tau\right}^{2}}}{x \sigma \sqrt{2 \pi \tau}} .$$
Then $f$ is the density of $\tilde{S}(T)$ given $\tilde{S}(t)=s$. Note that $f$ is differentiable with respect to every parameter $\theta \in{s, r, \sigma, t}$.

Proposition 1.7.2 Simultaneous unbiased estimations of the value of the option and derivatives of order 1 are given by
$$\hat{C}=\frac{1}{N} \sum_{i=1}^{N} e^{-r \tau} \Phi\left(\tilde{S}{i}\right)$$ and $$\widetilde{\partial{\theta} C}=-\hat{C} \partial_{\theta}(r \tau)+\left.e^{-r \tau} \frac{1}{N} \sum_{i=1}^{N} \Phi\left(\tilde{S}{i}\right) \partial{\theta}[\ln {f(x)}]\right|{x=\tilde{S}{i}}$$
In particular
$$\hat{\Delta}=e^{-r \tau} \frac{1}{N} \sum_{i=1}^{N} \frac{Z_{i}}{s \sigma \sqrt{\tau}} \Phi\left(\tilde{S}{i}\right)$$ and $$\hat{\mathcal{V}}=e^{-r \tau} \frac{1}{N} \sum{i=1}^{N} \frac{\left(Z_{i}^{2}-1-Z_{i} \sigma \sqrt{\tau}\right)}{\sigma} \Phi\left(\tilde{S}{i}\right) .$$ Moreover, an unbiased estimation of the gamma is given by $$\hat{\Gamma}=e^{-r \tau} \frac{1}{N} \sum{i=1}^{N} \frac{\left(Z_{i}^{2}-1-Z_{i} \sigma \sqrt{\tau}\right)}{s^{2} \sigma^{2} \tau} \Phi\left(\bar{S}_{i}\right)=\frac{\hat{\mathcal{V}}}{s^{2} \sigma \tau} .$$

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

Δ≈C(吨,s+ε)−C(吨,s)ε

\begin{aligned} C(t, s) &=e^{-r \tau} E\left[\Phi\left{s e^{\left(r-\frac{\alpha^{2}}{2 }\right) \tau+\sigma \sqrt{\tau} Z}\right}\right] \ &=e^{-r \tau} \int_{-\infty}^{+\infty} \Phi\left {s e^{\left(r-\frac{\sigma^{2}}{2}\right) \tau+\sigma \sqrt{\tau} z}\right} \frac{e^{-z^{ 2} / 2}}{\sqrt{2 \pi}} d z \ &=e^{-r \tau} \int_{0}^{+\infty} \Phi(x) \frac{e^{- \frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{2}}{2}\right) \tau\对}^{2}}}{x \sigma \sqrt{2 \pi \tau}} d x \end{对齐}\begin{aligned} C(t, s) &=e^{-r \tau} E\left[\Phi\left{s e^{\left(r-\frac{\alpha^{2}}{2 }\right) \tau+\sigma \sqrt{\tau} Z}\right}\right] \ &=e^{-r \tau} \int_{-\infty}^{+\infty} \Phi\left {s e^{\left(r-\frac{\sigma^{2}}{2}\right) \tau+\sigma \sqrt{\tau} z}\right} \frac{e^{-z^{ 2} / 2}}{\sqrt{2 \pi}} d z \ &=e^{-r \tau} \int_{0}^{+\infty} \Phi(x) \frac{e^{- \frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{2}}{2}\right) \tau\对}^{2}}}{x \sigma \sqrt{2 \pi \tau}} d x \end{对齐}

C^=1ñ∑一世=1ñ和−rτ披(小号¯一世)

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

C^=1ñ∑一世=1ñ和−rτ披(小号~一世)

ñ(X¯−G)↔ñp(0,在),

1ñ−1∑一世=1ñ(X一世−X¯)(X一世−X¯)⊤

Δ^=和−rτ1ñ∑一世=1ñ小号~一世s一世(小号¯一世>ķ)

X一世=和−rτ一世(小号¯一世>ķ)(小号¯一世−ķ,小号¯一世s,小号¯一世(从一世τ−στ))⊤

## 金融代写|金融工程作业代写Financial Engineering代考|Likelihood Ratio Method

Broadie 和 Glasserman 提出的第二种方法[1996]是基于表示(1.20). 为了X>0， 放
f(x)=\frac{e^{-\frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{ 2}}{2}\right) \tau\right}^{2}}}{x \sigma \sqrt{2 \pi \tau}} 。f(x)=\frac{e^{-\frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{ 2}}{2}\right) \tau\right}^{2}}}{x \sigma \sqrt{2 \pi \tau}} 。

C^=1ñ∑一世=1ñ和−rτ披(小号~一世)和∂θC~=−C^∂θ(rτ)+和−rτ1ñ∑一世=1ñ披(小号~一世)∂θ[ln⁡F(X)]|X=小号~一世

Δ^=和−rτ1ñ∑一世=1ñ从一世sστ披(小号~一世)和在^=和−rτ1ñ∑一世=1ñ(从一世2−1−从一世στ)σ披(小号~一世).此外，伽马的无偏估计由下式给出Γ^=和−rτ1ñ∑一世=1ñ(从一世2−1−从一世στ)s2σ2τ披(小号¯一世)=在^s2στ.

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