191010K02: Risky Asset Models with Dependence

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191010K02: Risky Asset Models with Dependence

Overview

It is well known that log-returns from financial time-series often exhibit a significant dependence structure. In the classical geometric Brownian motion model, the log returns are increments of Brownian motion with drift and hence due to independent increments of Brownian motion, the log-returns are independent. To overcome these limitations of geometric Brownian motion and related exponential Levy models, Fractal activity time Geometric Brownian motion models were introduced.

In this course, we will cover different models for risky assets and discuss their limitations and will introduce more general models which have long-range dependence properties.

Further, we will discuss the option pricing under these risky asset models.

Objectives

i. Exposing participants to fundamental of risky asset pricing

ii. To make them aware about different models that are used in risky asset pricing iii. To discuss the limitations of these models

iv. To discuss the theoretical aspects of the risky asset models

v. Introducing the risky asset models with long-range dependence which is often found in log-returns from financial time-series

Lecture Schedule (February 27 to March 03, 2023)

Day 1 (February 27, 2023) Lecture 1: 90 Minutes

Geometric Brownian motion, Black-Scholes model for risky asset and its significant shortcomings (log returns are uncorrelated but long range dependence is present in the absolute and squared returns, returns have leptokurtic empirical distributions, etc).

Lecture 2: 90 Minutes

Stochastic models with long-range dependence: log-linear transformation of Gaussian processes, Gamma-correlated processes

Tutorial 1: 90 Minutes

Exercises related to Brownian motion, geometric Brownian motion, long-range dependence, simulation of Brownian motion and Geometric Brownian motion

Day 2 (February 28, 2023) Lecture 3: 90 Minutes

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Superposition of Levy-driven Ornstein-Uhlenbeck (OU) type and mean-reversing diffusion processes with given marginals. Student processes.

Limit theorems for stochastic processes with long-range dependence Tutorial 2: 90 Minutes

Exercises related to OU type process, mean reversion, self-similarity.

Day 3 (March 01, 2023) Lecture 5: 90 Minutes

Rosenblatt processes and Hermite processes and their self-similar properties. Fractal activity time Geometric Brownian motion. Risky asset models with long-range dependence.

Gamma, Inverse Gaussian and tempered stable log-returns.

Practice problems on Rosenblatt processes, Hermite processes and their self-similarity properties, fractal activity time processes.

Mean-reverting and skew-reverting martingales. Pricing formulae in risky asset modes with dependence.

Variance Gamma process and applications Tutorial 4: 90 Minutes

Exercises realted to Mean-reverting and skew-reverting martingales. Pricing formulae in risky asset modes with dependence

Calibration of risky asset models with dependence.

Risky asses model with real life applications on financial data.

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