SABR模型数值模拟方法比较

SABR 模型在金融领域被广泛使用，但由于该模型结构复杂，我们无法获得解析表达式解决该模型下金融衍生品的定价问题。因此大多数对SABR模型的研究通过探究数值模拟方法以解决相关问题。对于SABR 模型的数值模拟，可以依据弹性参数β的取值分类讨论。当 β = 1 或者 β < 1 并且标的物远期价格过程和波动率过程瞬时不相关（即 ρ = 0）时， Cai，Song和Chen提出了精确模拟方法。因此本文关注于 β < 1，ρ=0 且标的物远期价格 F_t 在 0 处为吸收边界的情况进行模拟并对模拟结果进行比较，通过大量模拟路径数多组模拟参数的数值实验，从各种模拟方法结果的准确性、均方根误差的收敛性，模拟的时间等角度上比较各种模拟方法。

模拟的方法上我们使用了 Euler 方法、精确模拟方法、以及本文在精确模拟的基础上修改过后的无偏模拟方法。数值实验部分我们分别对欧式期权和远期起点期权进行模拟定价。在对欧式期权进行定价的部分，我们加入了计算隐含波动率得到 B-S-M 价格的结果和条件模拟方法的结果，与三种模拟方法的结果进行比对。

数值实验的结果表明，当到期日T或者波动率过程的波动率 ν 较大时，隐含波动率渐进展开式的结果就会产生偏差。Euler 方法容易实现，但想要得到较为精确的结果，需要增加模拟路径数和模拟步数，需要花费大量的计算时间。条件模拟方法结果准确，并且能显著降低模拟结果的方差，但该方法只能用于欧式期权定价。精确模拟方法和无偏模拟方法能够模拟得到标的物远期价格，适用于各种衍生品的定价问题，相比 Euler 方法仅需较少的模拟路径数就能得到准确的结果。相较于精确模拟方法，无偏模拟方法在模拟时间上更有优势。

The SABR model is widely used in the financial field. However, due to its complex structure, we cannot obtain analytical expressions to solve the pricing problem of financial derivatives under the SABR model. Most of the research on SABR model has been focused on exploring numerical simulation methods to solve related problems. The numerical simulation of the SABR model can be discussed based on the classification of the values of the elasticity parameter β. When β = 1 or β < 1 and the forward price of the underlying is instantaneously uncorrelated with the volatility process(i.e., ρ = 0), Cai, Song, and Chen proposed an exact simulation method. Therefore, we focuses on the situation where β < 1, ρ = 0 and the forward price of the underlying F_t at 0 with an absorbing boundary condition. For numerical experiments, we simulate multiple sets of parameters with a lot of paths. We compare various simulation methods from the perspectives of the accuracy, convergence of the results' root mean square errors, and the simulation time.

We use the Euler method, the exact simulation method, and the unbiased simulation method modified from the exact simulation method. For the numerical experiments, we simulate the prices of European options and forward starting options. In the section of pricing European options, we add the results of the implied volatility calculation to obtain the B-S-M price and the conditional simulation method to compare with the results of the three simulation methods.

The results of the numerical experiments show that the implied volatility asymptotic expansion formula is biased when the maturity T or the volatility ν of the volatility process is large. The Euler method is easy to implement, but to obtain more accurate results, we need to increase the number of simulation paths and the simulation steps, which takes a lot of computational time. The conditional simulation method gives accurate results and significantly reduces the variance of the simulation results, but it can only be used for pricing European options. The exact simulation method and the unbiased simulation method are able to simulate forward prices of the underlying, thus they can be used for pricing variety of derivatives. The unbiased simulation method is more efficient than the exact simulation method.

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