Computational Methods in Pricing and Model Calibration

In this week, we will study option pricing via a numerical approach. In many cases, analytical (explicit) solution of option prices is not obtainable, which requires numerical solutions. For example, if we switch the stock dynamics from geometric Brownian motion to another model, or switch the option from vanilla style to exotic style, explicit pricing formula will become unrealistic. Firstly, we start from introduction to options, where you can learn different types of options and different perspectives of option market participants. Then we will talk about option pricing via numerical integration both in general and in details. In particular, we will focus on Fourier transform and fast Fourier transform (FFT). We also provide Python codes for you to learn how to apply these techniques in practice. In the end of this week, you will be exposed to several cases studies, from time cost comparison to different models. There are lots of models which estimates the stock price evolution. Among these models, we will mainly focus on Black-Merton-Scholes (BMS), Heston, and Variance Gamma (VG) model, where you will learn the motivation and characteristic of each model. Afterwards, you will have an assignment about option pricing, where you can utilize all the theoretical knowledge and Python codes to price different options under different stock dynamics.

In this week, we will study model calibration, which follows the topics in last week. You have been exposed to many models, but you have no information about how to choose the model and parameters. Fortunately, you will learn how to solve this problem in this week from different approaches. Firstly, we start from an introduction to bid and ask prices and option surface. Then we will talk about the model calibration in regards with fitting the market option price, also with pictorial demonstration about implied volatility. Next, you will learn the calibration recipe, involving objective functions and initial parameter set. You will also learn how to do calibration in practice, which is an optimization problem. We will introduce three routines: brute-force search, Nelder-Mead algorithm, and BFGS algorithm. Except from learning these routines theoretically, you will also learn how to apply them in the optimization problem from Python codes. Followingly, you can apply what you learn about calibration in the assignment.

We will start learning interest rates and interest rate instruments from this week. Interest rates play a very important role in measuring the future and present value of financial products. People also use market interest rates to analyze the economic situation. At the very beginning, we will introduce fundamental interest rate concepts, including forward rates, spot rates, swap rates and term structures of interest rates. Then we will apply data-driven analysis to calibrate LIBOR and swap curves and cross-correlations between these rates. Using the term structure of these interest rates, we should be able to price market value of bonds, swaps and other interest rate products. We also provide you with Python codes in order to show how to obtain the LIBOR curve and how to use it to price bonds. After learning this module, you will have a brief overview of interest rates and their applications in bond and swap pricing. We will talk about complex stochastic models and calibrate interest curves with these models next week.