Financial Applications

1. Relation between GARCH and Continuous-Time Model
1. Some Properties of Stochastic Differential Equations
2. Convergence of Markov Chains to Diffusions
2. Option Pricing
1. Derivatives, Options
2. The Black-Scholes Approach
3. Historic Volatility and Implied Volatilities
4. Option Pricing when the Underlying Process Is a GARCH
3. Value at Risk (VaR) and Other Risks Measures
1. VaR
2. Other Risk Measures
3. Estimation Methods
4. Bibliographical Notes
5. Exercises

Keywords: Black-Scholes's formula, Diffusions, GARCH-M, Historical Volatility, Implied Volatility, Itô's lemma, Markov Chain, Options, Risk Measures, Riskmetrics, Stochastic Discount Factor, Stochastic Volatility Model, VaR (Value at Risk), Volatility.

Description: In this chapter we discuss several financial applications of GARCH models. Connecting these models with those frequently used in mathematical finance requires elaboration, because the latter are generally written in continuous time. We start by studying the relation between GARCH and continuous-time processes. We present sufficient conditions for a sequence of stochastic difference equations to converge in distribution to a stochastic differential equation as the length of the discrete time intervals between observations goes to zero. We then apply these results to GARCH(1,1)-type models. The second part of this chapter is devoted to the pricing of derivatives. We introduce the notion of stochastic discount factor and show how it can be used in the GARCH framework. The last part of the chapter is devoted to risk measurement.