Page 41 - CCFA Journal - Seventh Issue

加中金融 机器学习 Machine Learning

Traditional Derivative Pricing with Monte Carlo Simulations

A traditional and original way to price derivatives such as exotic options and equity swaps is to model the possible paths of an asset
price can go within a certain period of time before their maturities. Using a European basket call option as an example which the
payoff of the option depends on the difference between the minimum price among the underlying stocks and the strike price, it will
illustrate how some derivatives would be priced with Monte Carlo Simulations. One thing worth mentioning is that it is important to
use Cholesky Decomposition on the Correlation Matrix to extract the Lower-Triangular Matrix and generate correlated random
variables for Monte Carlo Simulations.

Once inputs such as underlying stock prices, option strike price, option maturity, stock volatility and correlation data are collected,
correlated simulation paths for the underlying stocks in the basket options can be portrayed. With so many simulated paths and
combinations, millions of possible option values can be generated. The average of these possible option values will then determine
the price of this European Call Option.

传统衍生品定价方法: 蒙特卡洛模拟

一种对衍生品（如奇异期权和股权互换）定价的传统方法是对资产价格在其到期前的特定时期内的可能路径进行建模。以
欧式篮子看涨期权权为例，期权的收益取决于标的几个只股票之间的最低价格与行使价之间的差异，我们可以从中知道如
何使用蒙特卡洛模拟对一些衍生品进行定价。值得一提的是，在相关矩阵上使用科列斯基分解来提取下三角矩阵并利用蒙
特卡洛模拟生成相关随机变量是非常重要的。

一旦收集到股票价格、期权行使价、期权到期日、股票波动率和相关矩阵等数据，就可以描绘篮子期权中股票的相关模拟
路径。通过如此多的模拟路径和组合，可以生成数百万个可能的期权价值。这些期权价值的平均值将决定这个欧洲看涨期
权的价格。

Derivative Pricing with Neural Network

The major problem with the traditional derivative pricing method is that it requires huge computation power and it is time-consuming.
Pricing derivative with neural network on the other hand is more efficient as prices of derivatives can be calculated instantly once the
model is trained and it saves so much time for traders. Although periodic updates (i.e. retrain the neural network with latest data)
are required, yet they can be done overnight or on the weekend.
Replicating the research paper “Deeply Learning Derivatives” can be a good place to start. Let’s use European Call Option as an
example. First, it is required to randomly generate different inputs such as stock prices, strike prices, maturities, correlations and
variances. All these variables are randomly picked from a set of predetermined probability distributions. It is also required to get the
outputs by calculating the option prices with Monte Carlo Simulations. Then, hyperparameter tuning, model training and validation
are the next steps.

利用神经网络对衍生品进行定价

传统衍生品定价方法的主要问题是计算量大、耗时长。另一方面，使用神经网络为衍生品定价更有效，因为一旦模型经过
训练，衍生品的价格就可以立即计算出来，并且为交易者节省下大量时间。尽管模型需要定期更新（即用最新数据重新训
练神经网络），但训练一般可以在一夜间或周末完成。

复制研究论文“Deeply Learning Derivatives”可能是一个很好的起点。让我们以欧式看涨期权为例。首先，我们需要随机
生成不同的数据，例如股票价格、执行价格、期限、相关性和方差。所有这些变量数据都是从一组预先确定的概率分布中
随机挑选出来的。还需要通过使
用蒙特卡洛模拟计算期权价格来
获得期权价值。然后，接下来的
步骤便是超参数调整、模型训练
和模型验证。

CCFA JOURNAL OF FINANCE February 2022
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