Overview on option pricing

Overview on Option Pricing

What is an option?

In financial markets, there are a huge number of various traded assets. One can divide them into two categories: primary assets and derivatives. Assets of the former category include bonds, stocks, commodities and etc., while the latter include futures, forwards, and options. The former represents a claim on real assets. The latter represents a claim on a primary asset, which is the unerlying asset. Hence they are called derivatives.

In this step, we shall work on an example related to a class of derivatives, the pricing of options. An option gives the holder the right but not the obligation to buy or sell the underlying asset at a pre-determined price and time. Options have many variants, and the European option is probably the most widely traded one. Let us assume today is July 1st. A specific European call option can give the holder the right a buy one Apple stock at the price of 100 on August 1st. In this example, the Apple stock is an underlying, 100 is the strike price, and August 1st is the maturity. If we know the price of Apple stock today is also 100, the immediate question is how much should I pay for such an option? This is a European option pricing question that we will work on.

Pricing formula

As we said, an option is a derivative whose price depends on the underlying. Therefore one needs to model the price of the stock before calculating the option price. In the Nobel prize in Economics winning paper Black and Scholes (BS) 1973 “The Pricing of Options and Corporate Liabilities” proposed a model for a stock given by the following equation:

[mathrm{d}S_t =mu S_t mathrm{d}t + sigma S_t mathrm{d}W_t.]

Here (S_t) denotes the stock price at time (t), (mu) and (sigma) the annulaized return and volatility of the stock, and (W_t) a Brownian motion. The BS model specifies a lognormal distribution for instanteneous stock return. Based on this model, they derive the BS formula showing that the price (V) of a European option is given by

[V = S_tN(d_1) – Ke^{-rtau}N(d_2)]

where

[d_1 = frac{1}{sigma sqrt{tau}} left[ lnleft(frac{S_t}{K}right) + left(r + frac{1}{2}sigma^2right)tau right], quad text{and} quad d_2 = d_1 – sigmasqrt{tau}.]

Here (K) denotes the strike price, (tau) is the time-to-maturity, and (N) is the cumulative distribution function of a standard normal distribution.

The BS model is the most widely used model for the pricing of European options. The purpose of this assignment is to investigate the performance of a neural network model to learn the BS formula from data.

In this following exercise, for simplicity, we assume (r=0). Then the BS formula requires only 4 arguments, i.e., (V=f(S, K, sigma, tau)).

Industrial practice

In practice, the popular volatility models are usually more complex than the BS model and can capture the temporal dynamics of markets. For the moment, simply replacing the classic stochastic models with purely data-driven neural networks may be problematic due to the lack of interpretability, robustness and data availability. However, some attempts have been made with machine learning techniques. For example, calibrating model parameters with deep neural networks allows much faster calculation and opens the door to the use of more complex stochastic modelling that would not be available to use in real time.