Statistics of Financial Markets

A derivative (derivative security or contingent claim) is a financial instrument whose value depends on the value of other, more basic underlying variables. Options, future contracts, forward contracts, and swaps are examples of derivatives. The aim of this chapter is to present and discuss various option strategies. The exercises emphasize the difference of the strategies through an intuitive approach using payoff graphs.

In this chapter we discuss the basic concepts of option management. We will consider both European and American call and put options and practice concepts of pricing, look at arbitrage opportunities and the valuation of forward contracts. Finally, we will investigate the put-call parity relationship for several cases.

This part is an introduction to standard concepts of probability theory. We discuss a variety of exercises on moment and dependence calculations with a real market example. We also study the characteristics of transformed random vectors, e.g. distributions and various statistical measures. Another feature that needs to be considered is various conditional statistical measures and their relations with corresponding marginal and joint distributions. Two more exercises are given in order to distinguish the differences between numerical statistical measures and statistical properties.

A stochastic process or random process consists of chronologically ordered random variables { X _t ; t ≥ 0 }. For simplicity we assume that the process starts at time t = 0 in X ₀ = 0: This means that even if the starting point is known, there are many possible routes the process might take, some of them with a higher probability. In this section, we exclusively consider processes in discrete time, i.e. processes which are observed at equally spaced points of time t = 0, 1, 2,.... In other words, a discrete process is considered to be an approximation of the continuous counterpart. Hence, it is important to start with discrete processes in order to understand sophisticated continuous processes. In particular, a Brownian motion is a limit of random walks and a stochastic differential equation is a limit of stochastic difference equations. A random walk is a stochastic process with independent, identically distributed binomial random variables which can serve as the basis for many stochastic processes.Typical examples are daily, monthly or yearly observed economic data as stock prices, rates of unemployment or sales figures.

In the preceding chapter we discussed stochastic processes in discrete time. This chapter is devoted to stochastic processes in continuous time. An important continuous time process is the standard Wiener process { W _t ; t ≥ 0 }.

The Black-Scholes formula is one of the most recognizable formulae in quantitative finance. The formula for the price C(S; ¿ ) of a European call option is given by: where we use y as an abbreviation for and b – r denotes the cost of carry b subtracted by the interest rate r.

A large range of options exist for which the boundary conditions of the Black-Scholes differential equation are too complex to solve analytically; an example being the American option. One therefore has to rely on numerical price computation. The best known methods for this approximate the stock price process by a discrete time stochastic process, or, as in the approach followed by Cox, Ross, Rubinstein, model the stock price process as a discrete time process from the start. The binomial model is a convenient tool for pricing European option.

Up to now we have considered mainly European options. This chapter however focuses on American Options. An American option is an option that can be exercised anytime during its life. The time at which the holder chooses to exercise the options depends on the spot price of the underlying asset St. In this sense, the exercising time is a random variable itself. It is obvious that the Black-Scholes differential equations still hold as long as the options are not exercised. However the boundary conditions are so complicated that an analytical solution is not possible.

Exotic options are financial derivatives which are more complex than normally traded options (vanilla options). They are mainly used in OTC-trading (over the counter) to meet the special needs of corporate customers. For example, a compound option allows one to acquire an ordinary option at a later date, and a chooser option is a form of the compound option where the buyer can decide at a later date which type of option he would like to have.

Pricing interest rate derivatives fundamentally depends on the underlying term structure. The often made assumptions of constant risk free interest rate and its independence of equity prices will not be reasonable when considering interest rate derivatives. Just as the dynamics of a stock price are modeled via a stochastic process, the term structure of interest rates is modeled stochastically. As interest rate derivatives have become increasingly popular, especially among institutional investors, the standard models for the term structure have become a core part of financial engineering. It is therefore important to practice the basic tools of pricing interest rate derivatives. For interest rate dynamics, there are one-factor and two-factor short rate models, the Heath Jarrow Morton framework and the LIBOR Market Model.

This section deals with financial time series analysis. The statistical properties of asset and return time series are inuenced by the media (daily news on the radio, television and newspapers) that informs us about the latest changes in stock prices, interest rates and exchange rates. This information is also available to traders who deal with immanent risk in security prices. It is therefore interesting to understand the behavior of asset prices. Economic models on the pricing of securities are mostly based on theoretical concepts which involve the formation of expectations, utility functions and risk preferences. In this section we concentrate on answering the empirical questions. Firstly, given a data set we aim to specify an appropriate model reecting the main characteristics of the empirically observable stock price process and we wish to know whether the assumptions underlying the model are fulfilled in reality or whether the model has to be modified. A new model on the stock price process could possibly effect the function of the markets. To this end we apply statistical tools to empirical data and start with considering the concepts of univariate analysis before moving on to multivariate time series.

The autoregressive moving average (ARMA) model defined as \( X_t = \nu + \alpha _1 X_{t - 1} + \ldots + \alpha _p X_{t - p} + \beta _1 \varepsilon _{t - 1} + \ldots + \beta _q \varepsilon _{t - q} + \varepsilon _t \) deals with linear time series. That means, the time series should be covariance stationary processes, see Franke et al. (2008). The model consists of two parts, an autoregressive (AR) part of order p and a moving average (MA) part of order q. When an ARMA model is not stationary, the methods of analyzing stationary time series cannot be used directly. In order to handle those processes within the framework of the classical time series analysis, we must first form the differences to get a stationary process. The autoregressive integrated moving average (ARIMA) models are an extention of ARMA processes by the integrated (I) part. Sometimes ARIMA models are refered to as ARIMA(p; d; q) whereas p and q denote the order of an autoregressive (AR) respective a moving average (MA) part and d describes the integrated (I) part.

In the previous chapters, we have already discussed that volatility plays an important role in modeling financial systems and time series. Unlike the term structure, volatility is unobservable and thus must be estimated from the data.

Value-at-Risk (VaR) is probably the most commonly known measure for quantifying and controlling the risk of a portfolio. Establishing VaR is of central importance to a credit institute. The description of risk is attained with the help of an “internal model”, whose job is to reect the market risk of portfolios and similar uncertain investments over time. The objective parameter in the model is the probability forecast of portfolio changes over a given period. Whether the model and its technical application correctly identify the essential aspects of the risk, remains to be seen and verified. The backtesting procedure serves to evaluate the quality of the forecast of a risk model by comparing the actual results to those generated with the VaR model. For this the daily VaR estimates are compared to the results from hypothetical trading that are held from the end-of-day position to the end of the next day, the so-called “clean backtesting”. The concept of clean backtesting is differentiated from that of “mark-to-market” profit and loss (“dirty P&L“) analyses in which intra-day changes are also observed. In judging the quality of the forecast of a risk model it is advisable to concentrate on the clean backtesting.

In order to investigate the risk of a portfolio, the assets subjected to risk (risk factors) should be identified and the changes in the portfolio value caused by the risk factors evaluated. Especially relevant for risk management purposes are negative changes - the portfolio losses. The Value-at-Risk (VaR) is a measure that quantifies the riskiness of a portfolio. This measure and its accuracy are of crucial importance in determining the capital requirement for financial institutions. That is one of the reasons why increasing attention has been paid to VaR computing methods.

When we model returns using a GARCH process with normally distributed innovations, we have already taken into account the second stylised fact. The the random returns automatically have a leptokurtic distribution and larger losses occur more frequently than under the assumption that the returns are normally distributed. If one is interested in the 95%-VaR of liquid assets, this approach produces the most useful results. For extreme risk quantiles such as the 99%-VaR and for riskier types of investments, the risk is often underestimated when the innovations are assumed to be normally distributed, since a higher probability of particularly be extreme losses than a GARCH process εt with normally distributed Z _t can be produced.

There is a close connection between the value of an option and the volatility process of the financial underlying. Assuming that the price process follows a geometric Brownian motion, we have derived the Black-Scholes formula (BS) for pricing European options. With this formula the option price is, at a given time point, a function of the volatility parameters when the following values are given: τ (time to maturity in years), K (strike price), r (risk free, long-run interest rate) and S (the spot price of the financial underlying).

Financial institutions are interested in loss protection and loan insurance. Thus determining the loss reserves needed to cover the risk stemming from credit portfolios is a major issue in banking. By charging risk premiums a bank can create a loss reserve account which it can exploit to be shielded against losses from defaulted debt. However, it is imperative that these premiums are appropriate to the issued loans and to the credit portfolio risk inherent to the bank. To determine the current risk exposure it is necessary that financial institutions can model the default probabilities for their portfolios of credit instruments appropriately. To begin with, these probabilities can be viewed as independent but it is apparent that it is plausible to drop this assumption and to model possible defaults as correlated events.