# Minicourses

T. Björk: An introduction to interest rate theory (slides 1, slides 2)

• Basic interest rate theory
• Bond markets and interest rates.
• Short rate models. The market price of risk.
• Martingale modelling.
• Affine term structures.
• Inverting the yield curve.
• Forward rate models
• Bond prices and forward rates. A toolbox.
• The HJM approach to forward rate modelling.
• Examples.
• Change of numeraire
• The normalized price system.
• Pricing.
• Forward measures.
• A general option pricing formula.
• New developments
• Credit risk.
• ``Market models''.
D. Heath: Risk Measures

• Overview of Financial Risk Management
• Risks to be managed
• How these are managed
• Measurement of risk
• Structure of the firm
• VaR
• Definitions of VaR
• Models to compute VaR
• Properties of VaR
• Consequences of requirements based on VaR
• Coherent Measures of Risk
Properties of a good risk measure
• Structure of good regulatory risk measures
• Structure of good management risk measures
• Examples of risk measures
• Managing Risk using Coherent Measures
• Allocation of risk limits
• Charging for use of risk limits
• Implications of Risk Management on Market Prices
• Valuation and stress measures
• "No good deal" pricing
• Dependence on the firm's position
• How risk limits affect prices
• # Special invited lectures

S. Hodges: No Good Deal Bounds

The principle of "no-good-deal" pricing can provide much tighter bounds than no-arbitrage pricing in an incomplete market without the need for the much stronger assumptions of equilibrium pricing. We will consider:
Bounds based on the Sharpe Ratio, on the Generalized Sharpe Ratio, the more general formulation of Cerny and Hodges, and also the relationship with coherent risk measures. (slides, paper Cerny, Hodges)

A. Pelsser: Mathematical Foundation of Convexity Correction

A broad class of exotic interest interest rate derivatives can be valued simply by adjusting the forward interest rate. This adjustment is known in the market as convexity correction. Various ad hoc rules are used to calculate the convexity correction for different products, many of them mutually inconsistent. In this paper we put convexity correction on a firm mathematical basis by showing that it can be interpreted as the side-effect of a change of probability measure. This provides us with a theoretically consistent framework to calculate convexity corrections. Using this framework we provide exact expressions for libor in arrears, and diff swaps. Furthermore, we propose a simple method to calculate analytical approximations for general instances of convexity correction.

L.C.G. Rogers: Monte Carlo Valuation of American Options

This paper introduces a `dual' way to price American options, based on simulating the path of the option payoff, and of a judiciously-chosen Lagrangian martingale. Taking the pathwise maximum of the payoff less the martingale provides an upper bound for the price of the option, and this bound is sharp for the optimal choice of Lagrangian martingale. As a first exploration of this method, three examples are investigated numerically; the accuracy achieved with even very simple-minded choices of Lagrangian martingale is surprising. The method also leads naturally to candidate hedging policies for the option, and estimates of the risk involved in using them.

# Short lectures

A.J.G. Driessen: The Cross-Firm Behaviour of Credit Spread Term Structures

We apply a term structure approach to analyze the relation between term structures of credit spreads on corporate bonds of many different firms. We use the intensity-based framework of Duffie and Singleton (1999) and model the instantaneous credit spread of each firm as a function of common factors and a firm-specific factor, thereby generalizing the purely firm-specific model of Duffee (1999). Using data on US corporate bond prices of 104 firms, we estimate the model for the credit spread term structures of all firms with Quasi Maximum Likelihood based on the Kalman filter. The results provide strong evidence for the presence of common factors in credit spreads across firms. These common factors influence credit spreads of all firms in the same direction. We find that the risk associated with the common factors is priced, while the firm-specific factor risk is not. Changes in the common factors and firm-specific factors are negatively correlated with stock returns and positively correlated with changes in stock return volatility.

J.K. Hoogland: Symmetries in Jump-Diffusion Models with Applications in Option Pricing and Credit Risk

It is a well known fact that local scale invariance plays a fundamental role in the theory of derivative pricing. Specific applications of this principle have been used quite often under the name of `change of numeraire', but in recent work it was shown that when invoked as a fundamental first principle, it provides a powerful alternative method for the derivation of prices and hedges of derivative securities, when prices of the underlying tradables are driven by Wiener processes. In this article we extend this work to the pricing problem in markets driven not only by Wiener processes but also by Poisson processes, i.e. jump-diffusion models. It is shown that in this case too, the focus on symmetry aspects of the problem leads to important simplifications of, and a deeper insight into the problem. Among the applications of the theory we consider the pricing of stock options in the presence of jumps, and Levy-processes. Next we show how the same theory, by restricting the number of jumps, can be used to model credit risk, leading to a `market model' of credit risk. Both the traditional Duffie-Singleton and Jarrow-Turnbull models can be described within this framework, but also more general models, which incorporate default correlation in a consistent way. As an application of this theory we look at the pricing of a credit default swap (CDS) and a first-to-default basket option.

J. Kerkhof: A Quantitative Assessment of Model Risk

In this paper we propose a general approach to quantify model risk for market risk measures. We present a theoretical framework to compute model reserves for trading desks based on coherent market risk measures. We decompose total model risk in a model risk premium due to estimation risk and a model risk premium due to misspecification risk. We compute these components using Monte-Carlo simulation and the bootstrap. The presented approach is general and can be applied to portfolios of basis assets and all types of derivatives. We apply the procedure to the equity, FX, and interest rate market for the heavily used Black-Scholes type models.

P. Klaassen: Loan maturity, economic cycles and bank insolvency risk

Credit risk portfolio models usually assume a one-year time horizon when evaluating the credit risk of loan portfolios. The average maturity of a bank's loan portfolio is typically significantly longer than one year, however. We develop a stylized model to investigate how a bank's insolvency risk changes when different holding periods are considered for the loan portfolio. To this end, the development of the bank's balance sheet is modelled over time. We furthermore include economic cycles in our multi-period model, and study how this influences a bank's required capital for a chosen insolvency probability. (joint work with Andre Lucas)

M.R. Pistorius: Option pricing under a phase type model

One of the functions of derivatives, such as the American Put option, is to provide the buyer an insurance against certain risks. We will focus on the problem of pricing such derivatives under the "phase type" model. As is well known, the classical Black-Scholes model is often uncapable of capturing certain features seen in empirical data. As an alternative, we propose an exponential Levy model to model the risk (price of the underlying) where the Levy process is given by a Brownian motion with added a compound Poisson process, where the jumps are of both signs and of "phase type", that is, the Laplace transform of the jump sizes is the quotient of two polynomials. In this talk, we address the question of analytical tractability of this model, exemplified by pricing the Russian option - a perpetual expiration version of a lookback option. This talk is based on joint work with Soren Asmussen and Florin Avram.

M.H. Vellekoop: Pricing Methods for Defaultable Assets

We discuss the pricing and hedging of contingent claims on assets which may be subject to default: a sudden unpredictable loss in value. We show that a unique price can be found if we assume that the stochastic processes which cause default are idiosyncratic, and we introduce a nice interpretation of this result in terms of insurance contracts which complete the market. Trinomial tree models which can be used to find prices and hedges are developed, and we study in particular the convergence properties of such methods in detail.

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