Derivatives and Internal Models

The explosive development of derivative financial instruments continues to provide new possibilities and increasing flexibility to manage finance and risk in a way specifically tailored to the needs of individual investors or firms. This holds in particular for banks and financial services companies who deal primarily with financial products, but is also becoming increasingly important in other sectors as well. Active financial and risk management in corporate treasury can make a significant contribution to the stability and profitability of a company. For example, the terms (price, interest rate, etc.) of transactions to be concluded at a future date can be fixed today, if desired even giving the company the option of declining to go ahead with the transaction later on. These types of transactions obviously have some very attractive uses such as arranging a long-term fixed-rate credit agreement at a specified interest rate a year in advance of the actual transaction with the option to forgo the agreement if the anticipated need for money proves to have been unwarranted (this scenario is realized using what is known as a “payer swaption”) or providing a safeguard against fluctuations in a foreign currency exchange rate by establishing a minimum rate of exchange today for changing foreign currency into euros at a future date (using a foreign currency option).

The financial services industry (banks, funds manager, insurance companies and other) faces the largest risks arising from financial instruments because trading in these instruments is their core business. Banking supervisory authorities and legislators have reacted to this situation with fundamentally new legal provisions imposing very high requirements on the risk management of banks. In 1993 the Washington-based organization Group of Thirty (G30) published a study on the risk management of derivatives entitled “Derivatives: Practices and Principles”. This study is also known as the G30 Report and puts forward recommendations which form the basis of the Minimum Requirements for Trading Activities (MaH) in Germany and other such regulations. The following list shows the most important recommendations made in the G30 Report.

The fundamental risk factors in financial markets are the market parameters which determine the price of the financial instruments being traded. They include foreign currency exchange rates and the price of commodities and stocks and, of course, interest rates. Fluctuations in these fundamental risks induce fluctuations in the prices of the financial instruments which they underlie. They constitute an inherent market risk in the financial instruments and are therefore referred to as risk factors. The risk factors of a financial instrument are the market parameters (interest rates, foreign currency exchange rates, commodity and stock prices), which, through their fluctuation, produce a change in the price of the financial instrument. The above mentioned risk factors do not exhaust the list of the possible risk factors associated with a financial instrument nor do all risk factors affect the price of each instrument; for example, the value of a 5 year coupon bond in Czech Coruna is not determined by the current market price of gold. The first step in risk management is thus to identify the relevant risk factors of a specified financial instrument.

As mentioned in the introduction, trading can be defined as an agreement between two parties in which one of the two consciously accepts a financial risk in return for the receipt of a specified payment or at least the expectation of such a payment at same future time from the counterparty. Financial instruments, also called financial products, are instruments which make such a risk mitigation or risk transfer possible. The purpose of this section is to present a classification of such instruments in a system of underlyings and derivatives, specifically for interest rate instruments. Interest rate risk is by far the most complex market risk. Correspondingly, interest rate instruments, ie. instruments having interest rates as their underlying risk factors, are among the most complex financial instruments traded on the market. Instruments on other risk factors such as stocks or foreign exchange rates can be classified analogously and will be discussed in detail in later sections of the book.

To apply the common methods for pricing and risk management we need to make assumptions which are necessary for the construction of the associated models. A complete list of all model assumptions made in this book are summarized here in order to provide an overview of the numerous conditions and assumptions arising in the various methods. For each pricing and risk management method discussed in the following chapters, we will specify which of these assumptions are needed for their application.

Present value methods determine the value of a financial instrument by discounting all future cash flows resulting from the instrument. Applying this method requires few assumptions. Only Assumptions 1, 2, 3, 4 and 5 from Section 5 are necessary.

considerations alone are sufficient for deriving relations such as the put-call parity or determining forward prices. Such arguments require only very few assumptions; we need only assumptions 1, 2, 3, 4 and 5 from Section 5 to be satisfied.

Having used arbitrage considerations to derive various properties of derivatives, in particular of option prices (upper and lower bounds, parities, etc.), we now demonstrate how such arbitrage arguments, with the help of results from stochastic analysis, namely Ito’s formula 3.19, can be used to derive the famous Black-Scholes equation. Along with the Assumptions 1, 2, 3, 4 and 5 from Section 5, the additional assumption that continuous trading is possible is essential to establishing the equation, i.e. in the following we assume that Assumption 6 from Section 5 holds. The Black-Scholes equation is a partial differential equation which must be satisfied by every price function of path-independent European derivatives on a single underlying1. Consequently, one method of pricing derivatives consists in solving this differential equation satisfying the boundary conditions corresponding to the situation being investigated. In fact, even quite a number of path-dependent options obey this differential equation. A prominent example is the barrier option. In general however, the price of path-dependent options cannot be represented as a solution to the Black-Scholes equation. It is possible to surmount these difficulties by imbedding the state space in a higher dimensional space defining one or several additional variables in an appropriate manner to represent the different paths. This method is demonstrated explicitly by Wilmott for Asian options with arithmetic means [166]. As we will see below, the valuation of American options can also be accomplished via the Black-Scholes equation (with free boundary conditions).

In addition to Assumptions 1, 2, 3, 4, 5 and 6 from Section 5 required to set up the differential equation in Section 8, we will now further simplify our model by assuming that the parameters involved (interest rates, dividend yields, volatility) are constant (Assumptions 9, 11 and thus 7 from Section 5) despite the fact that these assumptions are quite unrealistic. These were the assumptions for which Fischer Black and Myron Scholes first found an analytic expression for the price of a plain vanilla option, the famous Black-Scholes option pricing formula. For this reason, we often speak of the Black-Scholes world when working with these assumptions. In the Black-Scholes world, solutions of the Black-Scholes differential equation (i.e. option prices) for some payoff profiles (for example for plain vanilla calls and puts) can be given in closed form. We will now present two elegant methods to derive such closed form solutions.

Among the many numerical procedures available to solve partial differential equations the finite difference method stands out as probably the most widely used method in mathematical finance. Finite difference methods are very powerful and flexible as well. They can be applied to a wide variety of different derivatives with either European or American payoff modes. In this section, we will provide a very detailed discussion of these important methods in a generality far exceeding that which is usually presented in comparable books.

Binomial and trinomial trees are very popular tools commonly used in practice to calculate prices and sensitivity parameters of derivatives while avoiding direct reference to the fundamental differential equations governing the price of the instrument. These methods provide a useful alternative to those (numerical or analytical) methods presented in the previous sections for solving differential equations. In principle (ignoring for the moment the potential computing-time problems), binomial and trinomial trees can also be used in pricing path-dependent derivatives.

Having recognized the fact that prices of financial instruments can be calculated as discounted future expectations (with respect to a risk-neutral probability measure), the idea of calculating such expectations by simulating the (stochastic) evolution of the underlyings several times and subsequently averaging the results somehow is not far removed. In fact, this relatively simple idea is widely used and is successful even in the valuation of very exotic options for which other methods are either too complicated or completely unsuitable, the only requirement being the availability of sufficient computation time. Before proceeding with financial applications of Monte Carlo techniques, we begin with a presentation of the technique itself.

The replication of derivatives with a portfolio consisting of underlyings and a bank account as, for example, in Equation 11.3, can of course also be used to hedge the derivative’s risk resulting from the stochastic movement of its underlying (or conversely a derivative could be used to hedge such a portfolio). This is accomplished by going short in the portfolio and long in the derivative or vice versa. This idea can be extended to hedging against influences other than the underlying price, for example, changes in the volatility, interest rate, etc. Such concepts of safeguarding against a risk factor have already made their appearance in arbitrage arguments in previous chapters and will be presented in their general form in this chapter. In addition to the fundamental Assumptions 1, 2, 3, 4 and 5 from Section 5, continuous trading will also be assumed below, i.e. Assumption 6. We will allow the underlying to perform a general Ito process1 of the Form 3.16 and assume that it pays a dividend yield q.

The most important and profound concept that the reader may have gained from the material presented in this book so far is that of risk neutrality, which can be summarized as follows

Equation 9.10, that interest rates are non-stochastic directly contradicts the very existence of interest rate options. If interest rates were deterministic and hence predictable with certainty for all future times, we would know at time t which options will be in or out of the money upon maturity T. The options which are out of the money at maturity would be worthless at all earlier times t < T as well. The options which are in the money at maturity would be nothing other than forward transactions. Thus, the assumptions made in pricing interest rate options using the Black-76 model imply that these very options should not even exist! In spite of this fact, the option prices obtained by applying the Black-76 model are surprisingly good. The results of recent research [94][126] have shown that the effects of several “false” assumptions (in particular, the assumed equality of forward and futures prices) tend to cancel out each other.

In Part II, a whole array of very workable methods for the valuation and hedging of financial instruments was introduced. We now continue in Part III with the explicit valuation of the most important and common financial instruments. We restrict our considerations to simple (for the most part, plain vanilla) instruments which still represent the largest proportion of all trades in financial markets today. The methods presented in Part II do in fact allow much more complicated instruments than those introduced here to be effectively priced. Seen from this point of view, the application of the material introduced in Part II is much more extensive than its restriction to the instruments defined in Part III might suggest. More involved applications (such as term structure models for Bermudan swaptions or multi-dimensional finite difference schemes for convertible bonds, etc.) often contain so many details specific to the individual implementation, that it is easy to become distracted from the essential ideas. Therefore such complicated examples are not particularly appropriate for discussion in an introductory text. However, Part II enables the reader to develop pricing techniques for quite complex products even if they don’t receive specific treatment in Part III.

A forward rate agreement, abbreviated as FRA, is a contract in which both contract parties agree to a fixed rate K on a principal N to be paid for some future interest period between T and T_. An FRA can be interpreted as an agreement loan to be made in the future with an interest rate already fixed today. The party receiving the loan makes the fixed interest payments. In contrast to bonds, we will refer to this party’s position as a long position in the FRA, whereas the counterparty receiving the interest payments is short in the FRA. A (long) FRA can thus be interpreted as an agreement on two future cash flows: a receipt of the principal N at time T (the loan is made) and a payment at maturity T_ of the FRA in the amount of the principal N compounded at the agreed rate K over the period T_−T (the loan plus interest is paid back).

In this chapter, the pricing of options on different underlying types using the Black- Scholes model and the Black-76 model is presented. The values listed in Figure 18.1 will repeatedly be used in the following examples.

An option gives its purchaser the right to buy (call ) or sell (put ) a specified underlying at a fixed price (strike ) at (European ) or up to (American ) a fixed date (maturity ). In the age of exotic options, this traditional definition is no longer sufficient. In order to include the “exotics”, this definition requires generalization: an option on one or several underlyings with prices given by S 1, S 2, ...Sm is characterized by its payoff profile. The payoff of the option is a function F (S 1, S 2, ...Sm ) of the underlying prices and indicates the cash flows arising for the option’s holder upon exercise1. American options can be exercised at any time during the lifetime of the option in contrast to the European option, which can only be exercised at maturity. There are also options which can be exercised at several specific times during their lifetime. These are called Bermuda options or Atlantic options since, in a sense, they are an intermediate form between American (exercise is possible at any time during the option’s lifetime) and European (exercise is possible at a single time, namely at maturity) options.

It is often the case that financial instruments are not traded in the form described in the previous chapters, but that combinations of these instruments are collected and traded as new financial instruments. Such constructions are called structured products or structured instruments. In order to understand the behavior, the valuation and the risk of such structured products, they must first be decomposed into their elementary instruments in a process called stripping. The following elementary instruments play a particularly important role in this process

The term risk signifies the uncertainty of the future developments in risk factors (for example, interest rate curves, stock prices, foreign exchange rates, volatilities, etc.) resulting in a negative deviation of the quantity of interest (e.g. the value of a portfolio) from a certain reference value. Or expressed in terms of a bank balance sheet, risk is the possibility that the value of assets decreases or that the value of liabilities increases. Among all the conceivable types of risk, for example, market risk, credit risk, operational risk, legal risk, etc., market risk and credit risk are those which are most commonly traded via various financial instruments.

The variance-covariance method makes use of covariances (volatilities and correlations) of the risk factors and the sensitivities of the portfolio values with respect to these risk factors with the goal of approximating the value at risk. This method leads directly to the final result, i.e. the portfolio’s value at risk; no information regarding market scenarios arises. The variance-covariance method utilizes linear approximations of the risk factors themselves throughout the entire calculation, often neglecting the drift as well.

In the calculation of the value at risk by means of Monte Carlo simulations, all of the risk factors influencing a portfolio are simulated over the liquidation period δt as stochastic processes satisfying, for example, Equation 3.13 or even more general processes of the form 3.16. The value at risk of the risk factors themselves are taken into complete consideration using Equation 21.15 sometimes neglecting the drift in the simulation if the liquidation period is short

A consistent relationship between interest rate risk factors and the changes in the value of a financial instrument induced by them is established through the decomposition of the financial instrument into cash flows insofar as such a decomposition is possible. The philosophy behind this approach is that the value of any financial instrument is equal to the sum of all discounted, expected future cash flows. The decomposition into expected future cash flows can, in general, be accomplished to a point where the size of the cash flow no longer depends explicitly on the interest rate (in the case of interest rate options, this only holds when the option price is approximated linearly as a function of the interest rate). The interest rate risk of an instrument then consists solely in the fact that the discount factors used in computing the instrument’s present value at a specified value date change with changes in the interest rates. However, the size of the cash flows can by all means be affected by other risk factors; these could be foreign exchange risk or other price risks influencing the value of the instrument.

In the Excel workbook ValueAtRisk.xls to be found on the CD accompanying this text, several of the value at risk concepts already introduced are applied to explicitly compute the VaR of a concrete portfolio within the delta-normal method. The example is quite dense in the sense that many of the concepts introduced above (as well as several concepts to be presented in later chapters, in particular in Sections 28.3.3 and 28.3.4) are collected in one calculation. However, it is by all means reasonable to present such a summary at this point as it will provide the reader with a complete reference containing all the essential steps for computing a value at risk (at least, the “simple” delta-normal version). We will proceed step by step through the example.

A comparison of the value at risk figures delivered by a risk management system with the actual value changes of a portfolio allows an estimation of the qualitative and quantitative “goodness” of the risk model. Comparisons of realized values with previously calculated values are called backtesting procedures.

Interest rate curves are constructed from the prices of bonds traded in the market. In order to construct the spot rate curve (also called spot rate term structure for term structure, for short), for example, the yields of zero bonds for all possible maturities are required. Observing an entire array of conditions in order to be consistent with the assumption of an arbitrage-free market, such an interest rate curve can be determined from market data characterizing traded interest rate instruments. Such market data are for instance prices of traded bonds (not necessarily zero bonds), spot rates, par rates, swap rates, etc. All of these variables can be traced back to a single common nucleus. If the market is arbitrage-free (and assuming the same credit-worthiness for all cash flows involved) then for every value date t and maturity date T there exists a unique discount factor BR(t, T). If all possible discount factors are known, then the present value of every instrument or portfolio consisting of cash flows can be determined. Thus, the spot rate curve R(t, T) (which is nothing other than the yields of the discount factors BR(t, T)) has to be constructed in such a way that the observed market prices of interest rate instruments can be reproduced by discounting the future cash flows of the instruments using BR(t, T).

The most up-to-date estimation for the volatility of a risk factor can be obtained when the risk factor is the underlying of a liquid option. Since the option is liquid, it is traded with a volume which ensures that the bid-ask spreads are sufficiently small to define a precise option price. The volatility can then be determined from the pricing model valid for the valuation of the option in the market if all other factors influencing the price of the option (interest rates, price of the underlying, etc.) are known. The parameter volatility is then varied in the model until the it yields the market price of the option. In this way, the implied volatility is determined. If there is agreement in the market as to which model should be used for the determination of the option price, then there is a one-to-one relation between option price and this implied volatility; if one of these values is known, the other can be uniquely determined. This method is so common in the market that many options are quoted directly in terms of their implied volatility instead of their price.

Having shown in the previous sections how statistical parameters such as the volatility can be obtained implicitly from the prices of derivatives traded in the market, we now proceed with what is perhaps the more natural approach, which builds directly on the definition of the statistical values, namely the analysis of historical time series. Time series analysis is a broad topic in the field of statistics whose application here will be limited to those areas which serve the purposes of this book. A much more general and wide-reaching presentation can be found in [72]

Time series analysis goes a significant step further than merely determining statistical parameters from observed time series data (such as the variance, correlation, etc.) as described above. Indeed, it is primarily used as a tool for deriving models describing the time series concerned. Estimators such as those appearing in Equation 29.5 are examples of how parameters can be estimated which are subsequently used to model the stochastic process governing the time series (for example, a random walk with drift μ and volatility σ). Building a model which “explains” and “describes” the time series data is the principle goal of time series analysis. The object is thus to interpret a series of observed data points {X _t }, for example a historical price or volatility evolution (in this way acquiring a fundamental understanding of the process) and to model the processes underlying the observed historical evolution. In this sense, the historical sequence of data points is interpreted as just one realization of the time series process. The parameters of the process are then estimated from the available data and can subsequently be used in making forecasts

Having selected a model and fitted its parameters to a given times series, the model can then be used to estimate new data of the time series. If such data are estimated for a time period following the final data value XT of the given time series, we speak of a prediction or forecast. The estimation of data lying between given data points is called interpolation. The question now arises as to how a model such as those given in Equations 30.6 or 30.13 could be used to obtain an “optimal” estimate.

In addition to the autoregressive models described above, which are used for instance in the form of GARCH models when modeling volatility, a further technique of time series analysis, called principle component analysis (abbreviated as PCA), is widely applied in the financial world. This technique is employed in the analysis of term structure evolutions, for instance. As mentioned in the introduction of Chapter 15 on term structure models, the approach described in Section 27 in which the term structure was constructed by interpolating between vertices is usually not used to model the (stochastic) dynamics of the term structure. Rather than modeling the interest rate at the vertices as risk factors, the stochastic evolution of the term structure is reduced to a small number of stochastic variables (one, two and sometimes three) which act as the driving factors of the entire term structure. This approach has its justification in principle component analysis. Principle component analysis is a statistical technique which extracts the statistical components from the time series which are most relevant for the dynamics of the process in order of their importance. By means of this method applied to interest rates, it can be shown that usually more than 90% of the term structure’s dynamics can be ascribed to the one or two most important components.

The pre-treatment for the transformation of a given data set into a stationary time series has been mentioned several times in the preceding sections and will receive detailed treatment in this section. The basis for pre-treating a time series is its decomposition into a trend component gt, a seasonal component st, and a random component Z _t