Abstract
The financial markets use stochastic models to represent the seemingly random behavior of assets such as stocks, commodities, relative currency prices such as the price of one currency compared to that of another, such as the price of US Dollar compared to that of the Euro, and interest rates. These models are then used by quantitative analysts to value options on stock prices, bond prices, and on interest rates. This chapter gives an overview of the stochastic models and methods used in financial risk management. Given the random nature of future events on financial markets, the field of stochastic processes obviously plays an important role in quantitative risk management. Random walk, Brownian motion and geometric Brownian motion processes in risk management are explained. Simulations of these processes are provided with some software codes.
TopBackground
Definitions of risk vary based on context. The International Organization for Standardization (ISO) in the ISO Guide 73 defines risk as the “effect of uncertainty on objectives” (ISO, 2009). The reason for a risk is uncertainty. Uncertainty is “the state of being uncertain.” Uncertain means “not able to be relied on; not known or definite.” Imagine that you would like to purchase a property in San Francisco, which is known for its susceptibility to earthquakes. We know that there is a risk of an earthquake occurring at any time, but we cannot say if there will be an earthquake during the next three years or not. It is uncertain.
Key Terms in this Chapter
Black Scholes Model: A model of price variation over time of financial instruments such as stocks that can, among other things, be used to determine the price of a European call option. The model assumes that the price of heavily traded assets follow a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option's strike price and the time to the option's expiry. Also known as the Black-Scholes-Merton Model.
Stochastic Differential Equation: It is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is itself a stochastic process. They are used to model diverse phenomena such as fluctuating stock prices or physical systems subject to thermal fluctuations.
Wiener Process: In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance and physics.
Assets: An item of current or future economic benefit to an organization. Examples include: cash, short-term investments, accounts receivable, grants receivable, inventories, prepaid expenses, buildings, furniture, equipment, vehicles, and long-term investments.
Stock Price: The cost of purchasing a security on an exchange. Stock prices can be affected by a number of things including volatility in the market, current economic conditions, and popularity of the company.
Markov Property: The term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state, not on the sequence of events that preceded it.
Volatility: It is a measure for variation of price of a financial instrument over time in finance.
Option Pricing Theory: Any model- or theory-based approach for calculating the fair value of an option. The most commonly used models today are the Black-Scholes model and the binomial model. Both theories on options pricing have wide margins for error because their values are derived from other assets, usually the price of a company's common stock. Time also plays a large role in option pricing theory, because calculations involve time periods of several years and more. Marketable options require different valuation methods than non-marketable ones, such as those given to company employees.