Sander Willems

Ph.D. in Mathematical Finance

Email: lastname dot firstname at gmail dot com

Research Interests: Mathematical finance, derivative pricing, term structure models, affine and polynomial processes, IBOR transition

Working Papers:

"A Lognormal Type Stochastic Volatility Model With Quadratic Drift" (with Peter Carr)

This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the instantaneous volatility level. The steady-state distribution of the instantaneous volatility belongs to the class of Generalized Inverse Gaussian distributions. We show that the quadratic term in the drift is crucial to avoid moment explosions and to preserve the martingale property of the stock price process. Using a conveniently chosen change of measure, we relate the model to the class of polynomial diffusions. This remarkable relation allows us to develop a highly accurate option price approximation technique based on orthogonal polynomial expansions.

Publications:

Risk Magazine, February 2021

We present a natural extension of the SABR model to price both backward and forward-looking RFR caplets in a post-Libor world. Forward-looking RFR caplets can be priced using the market standard approximations of Hagan et al. (2002). We provide closed-form effective SABR parameters for pricing backward-looking RFR caplets. These results are useful for smile interpolation and for analyzing backward and forward-looking smiles in normalized units.

International Journal of Theoretical and Applied Finance, 23(7), 2020

In this paper we propose a new model for pricing stock and dividend derivatives. We jointly specify dynamics for the stock price and the dividend rate such that the stock price is positive and the dividend rate non-negative. In its simplest form, the model features a dividend rate that is mean-reverting around a constant fraction of the stock price. The advantage of directly specifying dynamics for the dividend rate, as opposed to the more common approach of modeling the dividend yield, is that it is easier to keep the distribution of cumulative dividends tractable. The model is non-affine but does belong to the more general class of polynomial processes, which allows us to compute all conditional moments of the stock price and the cumulative dividends explicitly. In particular, we have closed-form expressions for the prices of stock and dividend futures. Prices of stock and dividend options are accurately approximated using a moment matching technique based on the principle of maximal entropy.

"A Term Structure Model for Dividends and Interest Rates" (with Damir Filipović) (SSRN)

Mathematical Finance, 40(4), 1461-1496, 2020

Over the last decade, dividends have become a standalone asset class instead of a mere side product of an equity investment. We introduce a framework based on polynomial jump-diffusions to jointly price the term structures of dividends and interest rates. Prices for dividend futures, bonds, and the dividend paying stock are given in closed form. We present an efficient moment based approximation method for option pricing. In a calibration exercise we show that a parsimonious model specification has a good fit with Euribor interest rate swaps and swaptions, Euro Stoxx 50 index dividend futures and dividend futures options, and Euro Stoxx 50 index options.

Quantitative Finance, 19(4), 605-618, 2019

In this paper we derive a series expansion for the price of a continuously sampled arithmetic Asian option in the Black-Scholes setting. The expansion is based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. We provide sufficient conditions to guarantee convergence of the series. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however we show numerically that the bias can safely be ignored in practice.

"Exact Smooth Term Structure Estimation" (with Damir Filipović) (SSRN) (errata)

SIAM Journal on Financial Mathematics, 9(3), 907-929, 2018

We present a non-parametric method to estimate the discount curve from market quotes based on the Moore-Penrose pseudoinverse. The discount curve reproduces the market quotes perfectly, has maximal smoothness, and is given in closed-form. The method is easy to implement and requires only basic linear algebra operations. We provide a full theoretical framework as well as several practical applications.

Conference and seminar talks:

• QuantMinds International, Invited Talk, Barcelona, 08-Dec-21

• WBS 4th Interest Rate Reform (Ibor Transition) Conference, Invited Talk, London, 21-Oct-21

• QuantMinds In Focus, Invited Talk, Virtual, 26-May-21

• QuantMinds International, Invited Talk, Hamburg, 05-Nov-20

• King's College Probability Seminar, Invited Talk, London, 28-Oct-19

• QuantMinds International, Invited Talk, Vienna, 15-May-19

• New York University, Quantitative Finance Weekly Seminar, New York City, 01-Nov-18

• Princeton University, Brownbag Seminar, Princeton, 24-Oct-18

• Cornell University, Young Researchers Workshop on Data-Driven Decision Making, Ithaca, 13-Oct-18

• 10th Bachelier World Congress, Contributed Talk, Dublin, 16-Jul-18

• 9th International Workshop on Applied Probability, Contributed Talk, Budapest, 21-Jun-18

• Swiss Finance Institute Research Days, Contributed Talk, Gerzensee, 05-Jun-18

• McMaster University, Brownbag Seminar, Hamilton, 24-Apr-18

• Actuarial and Financial Mathematics Conference 2018, Contributed Talk, Brussels, 09-Feb-18

• 2nd International Conference on Computational Finance, Contributed Talk, Lisbon, 07-Sep-17

• 8th General AMaMeF Conference, Contributed Talk, Amsterdam, 22-Jun-17

• School and Workshop on Dynamic Models in Finance, Invited Talk, Lausanne, 22-May-17

• Vienna Congress on Mathematical Finance, Contributed Talk, Vienna, 14-Sep-16