# 金融代写|期权定价理论代写Option Pricing Theory代考|MATH451

## 金融代写|期权定价理论代写Option Pricing Theory代考|Numerical experiments with vanilla options

In this section, we price vanilla options with maturity $T=1$ under a local volatility model. The time-homogeneous local volatility function is inferred from the one-year implied volatility given in Table 2.1.
Interest rates and repos are zero, and the underlying delivers no dividend. We plot the estimator of the price and its standard deviation as functions of the time step $(1 / 2,1 / 4,1 / 8$, etc.), respectively, for $R-1$ (classical Euler scheme), $R=2$ (classical Romberg extrapolation), $R=3$ (iterated Romberg extrapolation with time steps $T / n, T /(2 n)$ and $T /(4 n)$ ), and $R=4$ (iterated Romberg extrapolation with time steps $T / n, T /(2 n), T /(4 n)$, and $T /(8 n))$. We use $# p=2^{18}$ paths in the Monte Carlo sampling.
Put
We first price a put option with strike $80 \%$; see Figures $2.1$ and $2.2$. The estimators of the price for $R=3$ and $R=4$ have very small bias, even for large time steps. However, the standard deviation of the price estimator for

$R=4$ is much larger than that for $R=1,2$, or 3 : the variance of the estimator is not yet controlled by the asymptotic variance. The waves on the price curve for $R=4$ are evidence of a large variance. Even for desirable time steps, such as $1 / 8$ or $1 / 16$, the standard deviation for $R=4$ is more than $1.5$ times those for $R=1$, 2, or 3 . For those time steps, the basic estimator $(R=1)$ shows a clear bias ( 17 and $11 \mathrm{bps}$, resp. 4 and $2.5$ standard deviations), whereas the estimators for $R=2$ ( 4 and $1 \mathrm{bps}$ ) and $R=3$ ( 2 and $5 \mathrm{bps}$ ) lie within the Monte Carlo error zone.
In Figure 2.3, in order to check how the bias of the price estimator depends on $R$, we plot
$$\log {10}(1 / n) \mapsto \log {10} \mid \text { Estimator } r_R(1 / n) \text { – Exact Price } \mid$$
for various values of $R$. For $R-1$, we distinguish three different regimes:

• for large time steps, the asymptotic regime $(n \rightarrow \infty)$ is not reached;
• for time steps smaller than $1 / 8$, we enter into the asymptotic regime: as expected, the graph is affine with a slope close to 1 ;
• for time steps smaller than $1 / 128$, the asymptotic regime is hidden by the Monte Carlo sampling error.

## 金融代写|期权定价理论代写Option Pricing Theory代考|Numerical experiments with path-dependent options

Using iterated conditional expectations, one can easily see that expansion (2.14) remains valid for the Euler scheme for discretely monitored pathdependent options. One just needs to slightly adapt the scheme by adding the discrete observation dates involved in the payout of the option. However, if there are a lot of observation dates, the asymptotic regime may be visible only for very small time steps, that is, time steps smaller than the typical length of time between two observation dates. Here we consider a path-dependent option with a one-year maturity, and with 4 uniformly spread constatation dates $t_i$, i.e., one every 3 months. As in the previous section, we price under a local volatility modell, with a time-homogeneous local volatility function inferred from the one year implied volatility given in Table 2.1.

This option delivers $\left(K-\min {i \in{1,2,3,4}} X{t_i} / X_{t_0}\right)_{+}$. The estimated price and standard deviation are shown in Fignres $2.6$ and $2.7$. I ike for vanilla options, for $R=1$, we observe a linear behavior of the bias of the estimator as a function of the time step. We also distinguish the quadratic behavior of the estimator for $R=2$. This estimator has the most desirable properties: no visible bias for $\Delta t=1 / 8$ and $# p=2^{18}$, and almost no increase in variance, compared to $R=1$.
Of course, computational time matters a lot. For instance, in Figure 2.8, we plot the computational time (in arbitrary units) as a function of the time step for $R=1,2,3,4$, in the case of the lookback put option. Similar graphs were obtained for the vanilla options tested in the previous section. It takes less time to get an unbiased result (i.e., the bias is hidden by the sampling error) with $R=2$ and $\Delta t=1 / 8$ than it takes to get a biased result with $R=1$ and $\Delta t=1 / 64$. Hence, we definitely advise using the Romberg extrapolation with $R=2$ and a reasonably large time step.

# 期权理论代考

## 金融代写|期权定价理论代写Option Pricing Theory代考|Numerical experiments with vanilla options

$R=4$ 远大于 $R=1,2$, 或 3 : 估计量的方差尚末由渐近方差控制。价格曲线上的波浪 $R=4$ 是大 方差的证据。即使对于理想的时间步长，例如 $1 / 8$ 或者 $1 / 16$ ，标准差为 $R=4$ 超过 $1.5$ 乘以那些 $R=1$ 、 2 或 3 。对于这些时间步长，基本估计器 $(R=1$ )显示出明显的偏差 ( 17 和11bps，分 别 4和 $2.5$ 标准差)，而估计量 $R=2$ ( 4 和 $1 \mathrm{bps})$ 和 $R=3$ (2和5bps) 位于蒙特卡洛淏差区 内。

• 对于大时间步长，渐近状态 $(n \rightarrow \infty)$ 末达到;
• 对于小于的时间步长 $1 / 8$ ，我们进入渐近状态: 正如预期的那样，该图是仿射的，斜率接近 1
• 对于小于的时间步长 $1 / 128$ ，渐近状态被蒙特卡洛抽样误差隐藏。

## 金融代写|期权定价理论代写Option Pricing Theory代考|Numerical experiments with path-dependent options

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