Minicourses

Hans Föllmer: Probabilistic Aspects of Financial Uncertainty

The price fluctuation of liquid financial assets is usually modeled as a stochastic process which satisfies some form of the "efficient markets hypothesis". Such assumptions can be made precise in terms of martingale measures. In these lectures we will discuss the role of these martingale measures in analyzing financial options and in solving problems of optimal investment. The emphasis will be on "incomplete" financial market models where the equivalent martingale measure is not unique. We will also focus on robust approaches where model uncertainty is taken into account explicitely. This will involve recent developments in the theory of convex risk measures and of robust projection and optimization problems. In particular we will discuss dynamic risk measures and their connections to the pricing problem for American options and to the theory of backward stochastic differential equations.
References
[1] Hans Föllmer and Alexander Schied (2004), Stochastic Finance, An Introduction in Discrete Time (Chapter IV)
[2] A. Gundel (2005), Robust utility maximization for complete and incomplete market models, Finance and Stochastics 9 (2), 151-177
[3] H. Föllmer and A. Gundel (2006), Robust projections on the class of martingale measures. To appear in: Illinois Journal of Mathematics (special volume in honour of J. L. Doob)

Special invited lectures

Monique Jeanblanc: Pricing And Trading Credit Default Swaps

The goal of this note is a detailed study of stylized credit default swaps within the framework of a generic reduced-form credit risk model. We provide results concerning the valuation and trading of credit default swaps under the assumption that the default intensity is deterministic and the interest rate is zero. We derive a closed-form solution for replicating strategy for an arbitrary non-dividend paying defaultable claim in a market in which a bond and a credit default swap are traded, and we examine the market completeness. Finally, we extend some of previously established results to the case of stochastic default intensity. (Joint work with T. Bielecki and M. Rutkowski)
References
[1] Tomasz R. Bielecki, Monique Jeanblanc, Marek Rutkowski (2005), Pricing and trading credit default swaps (preprint)
[2] Monique Jeanblanc et al. (2006), Pricing and trading credit default swaps (slides of the lecture)
[3] Tomasz R. Bielecki, Monique Jeanblanc, Marek Rutkowski (2006), Credit risk (lecture notes)

Fabio Mercurio: Pricing Inflation-Indexed Options with Stochastic Volatility

There is nowadays an increasing interest for derivatives whose payoffs are based on the inflation rates observed in predefined periods. These derivatives, like inflation-indexed swaps and caps, are typically priced by a foreign-currency analogy, as in Jarrow and Yildrim (2003). However, one can consider an alternative approach where the evolution of forward consumer-price indices is modeled under a suitable measure. In this talk, we extend this approach by introducing stochastic volatility, with the purpose to recover smile-consistent prices for inflation-indexed caps and floors. Under volatility dynamics as in Heston (1993), closed-form formulas for inflation-indexed caplets and floorlets based on the Carr and Madan (1998) Fourier transform approach are then derived. An example of calibration to market data and numerical details concerning our pricing procedure are finally given.
References
[1] Fabio Mercurio (2005), Pricing Inflation-Indexed Derivatives (preprint, appeared also as Quantitative Finance 5(3), 289-302)
[2] Fabio Mercurio and Nicola Moreni (2005), Pricing Inflation-Indexed Options With Stochastic Volatility (preprint)
[3] Fabio Mercurio (2006), Pricing Inflation-Indexed Options With Stochastic Volatility (slides of the lecture)

Short lectures

Otto van Hemert: Dynamic portfolio and mortgage choice for homeowners

We investigate the impact of owner-occupied housing on financial portfolio and mortgage choice under stochastic inflation and real interest rates. To this end we develop a dynamic framework in which investors can invest in stocks and bonds with different maturities. We use a continuous-time model with CRRA preferences and calibrate the model parameters using data on inflation rates and equity, bond, and house prices. For the case of no short-sale constraints, we derive an implicit solution and identify the main channels through which the housing to total wealth ratio and the horizon affect financial portfolio choice. This solution is used to interpret numerical results that we provide when the investor has short-sale constraints. We also use our framework to investigate optimal mortgage size and type. A moderately risk-averse investor prefers an adjustable-rate mortgage (ARM), while a more risk-averse investor prefers a fixed-rate mortgage (FRM). A combination of an ARM and an FRM further improves welfare. Choosing a suboptimal mortgage leads to utility losses up to 6%.
References
[1] Brennan, Michael J., and Yihong Xia (2002), Dynamic asset allocation under inflation, Journal of Finance, 57 (3), 1201-1238.
[2] Otto van Hemert (2005), Life-cycle housing and portfolio choice with bond markets, Working Paper.
[3] Otto van Hemert, Frank de Jong and Joost Driessen (2005), Dynamic portfolio and mortgage choice for homeowners
[4] Otto van Hemert (2006), Dynamic portfolio and mortgage choice for homeowners (slides of the lecture)

Ralph Koijen: Labor Income and the Demand for Long-Term Bonds

The riskless nature in real terms of inflation-linked bonds has led to the conclusion that inflation-linked bonds should constitute a substantial part of the optimal investment portfolio of long-term investors. This conclusion is reached in models where investors do not receive labor income during the investment period. Since such an income stream is often indexed with inflation, labor income in itself constitutes an implicit holding of real bonds. As such, the optimal investment in inflation-linked bonds is substantially reduced. By extending recently developed simulation-based techniques, we are able to determine the optimal portfolio choice among inflation-linked bonds, nominal bonds, and stocks for investors endowed with an indexed stream of income. We find that the fraction invested in inflation-linked bonds is much smaller than reported in the literature, the duration of the optimal nominal bond portfolio is lengthened, and the utility gains of having access to inflation- linked bonds are substantially reduced. We investigate as well the robustness of our results to time-variation in bond risk premia, the riskiness of labor income, and correlation between labor income risk and financial risks. We find that especially accounting for time-variation in bond risk premia and correlation between labor income risk and financial risks is important for both optimal portfolios and the utility gains of having access to inflation-linked bonds.
References
[1] Ralph S.J. Koijen, Theo Nijman and Bas J. M. Werker (2005), Labor Income and the Demand for Long-Term Bonds
[2] M.W. Brandt, A. Goyal, P. Santa-Clara and J.R. Stroud (2005), A Simulation Approach to Dynamic Port- folio Choice with an Application to Learning About Return Predictability
[3] Ralph S.J. Koijen (2006), Labor income and the Demand for Long-Term Bonds (slides of the lecture)

Roger Laeven: On the tail probability for discounted sums of heavy-tailed losses

The discounted sum of losses within a finite or infinite time period can be described as a randomly weighted sum of a sequence of independent random variables. These independent random variables represent the amounts of losses in successive development years, while the weights represent the stochastic discount factors. In this talk, we investigate the asymptotic behavior of the tail probability for this weighted sum in the case when the losses have heavy-tailed distributions and the discount factors are mutually dependent.
References
[1] Laeven, Roger J.A., Marc J. Goovaerts & Tom Hoedemakers (2005), Some asymptotic results for sums of dependent random variables, with actuarial applications, Insurance: Mathematics and Economics 37 (2), 154-172
[2] Goovaerts, Marc J., Rob Kaas, Roger J.A. Laeven, Qihe Tang & Raluca Vernic (2005), The tail probability of discounted sums of Pareto-like losses in insurance, Scandinavian Actuarial Journal 6, 446-461 (to appear), preliminary version.
[3] Roger Laeven (2006), On the Tail Probability for Discounted Sums of Heavy-tailed Losses (slides of the lecture)

Roger Lord: Pricing baskets, Asians and swaptions within general models

A vast number of articles deal with the pricing of discretely sampled arithmetic Asian options in the Black-Scholes model. For more general models PIDE formulations, upper bounds and moment-matching approximations have to some extent been covered. In this paper we consider general claims on an arithmetic average and only assume the knowledge of the characteristic function of the return. We demonstrate how to write the highly accurate lower bound of Curran [1992] and Rogers and Shi [1995] as a Fourier integral. It turns out that for basket options any model is allowed, whereas for Asian options, rate of return guarantees, interest rate and credit default swaptions, we have to restrict ourselves to exponentially affine Lévy models. In numerical examples for Asian options and swaptions it is demonstrated that the lower bound is the most accurate approximation to date.
As background reading material, I suggest the article of Rogers and Shi [1] for the lower bound approach, my article on the approximation of Asians and baskets in a Black-Scholes world [2], the article of Carr and Madan [3] for pricing when the characteristic function is known, and of course a preliminary version of the slides for my presentation [4], in which you will (in the last slides) find an exhaustive list of background material.
References
[1] Rogers, L.C.G. and Z. Shi (1995). The value of an Asian option, Journal of Applied Probability, no. 32, pp. 1077-1088.
[2] Lord, R. (2005). Partially exact and bounded approximations for arithmetic Asian options, submitted.
[3] Carr, P. and D. Madan (1999). Option valuation using the Fast Fourier Transform, Journal of Computational Finance, vol. 2, no. 4, pp. 61-73.
[4] Lord, R. (2006). Condition and conquer - Pricing of baskets, Asians and swaptions in general models (slides of the lecture).