Fractional and Volterra processes in Finance

15h

Empirical studies indicate the presence of memory and strong inter-temporal dependence across various phenomena in the fields of finance and economics. The Brownian motion, characterized by independent increments, is not suitable for modeling such phenomena. In this course, we will consider Stochastic Volterra processes: a class of processes which extends the standard Brownian motion to include memory; the fractional Brownian motion constitutes a special case.

In the first part of the course, we will develop the mathematical tools needed to deal with these (non-standard) Volterra integral equations that go beyond the standard stochastic calculus theory of Markovian processes and semimartingales.

In the second part, we will explore the modeling flexibility of such equations in introducing memory in a broad range of financial problems, including:

• Interest rates with short and long memory: fast pricing of interest rates products.
• Rough stochastic volatility models: fast pricing and calibration via Fourier inversion techniques; analytic solutions for the Markowitz portfolio allocation problem with multilple assets with rough volatilities.
• Optimal execution and liquidation with transient market impact.

Bibliography

• [1] Abi Jaber, E., Larsson, M., & Pulido, S. (2019). Affine Volterra processes. The Annals of Applied Probability, 29(5), 3155-3200.
• [2] El Euch, O., & Rosenbaum, M. (2019). The characteristic function of rough Heston models. Mathematical Finance, 29(1), 3-38.
• [3] Gatheral, J., Jaisson, T., & Rosenbaum, M. (2018). Volatility is rough. Quantitative finance, 18(6), 933-949.
• [4] Gripenberg, G., Londen, S. O., & Staffans, O. (1990). Volterra integral and functional equations (No. 34). Cambridge University Press.
• [5] Mandelbrot, B. B., & Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM review, 10(4), 422-437.