# Fudan-Warwick Spring School

**Course 1: Simulation of Levy processes (Aleksandar Mijatovic) **15-19 April, 2019

**Time and Place:**

04.15 Monday 13:30-16:00 HGX304

04.16 Tuesday 13:30-16:00 Guanghua Main Building 1501

04.18 Thursday 9:00-11:30 Guanghua Main Building 1501

04.19 Friday 13:30-16:00 Guanghua Main Building 1801

**Abstract:** These lectures describe a novel Monte Carlo algorithm for the simulation from the joint law of the position, the running supremum and the time of the supremum of a general Levy process at an arbitrary finite time. We will show that the bias decays geometrically, in contrast to the power law for the random walk approximation (RWA). We will identify the law of the error and, inspired by the recent work of Ivanovs \cite{MR3784492} on RWA, characterise its asymptotic behaviour. We will establish a central limit theorem, construct non-asymptotic and asymptotic confidence intervals and prove that the multilevel Monte Carlo (MLMC) estimator has optimal computational complexity (i.e. of order $\epsilon^{-2}$ if the $L^2$-norm of the error is at most $\epsilon$) for locally Lipschitz and barrier-type functionals of the triplet. If the increments of the L\'{e}vy process cannot be sampled directly, we will combine our algorithm with the Asmussen-Rosi\'nski approximation by choosing the rate of decay of the cutoff level for small jumps so that the corresponding MC and MLMC estimators have minimal computational complexity. We will also describe an unbiased version of our estimator using ideas from Rhee and Glynn. New research directions based on this algorithm will be discussed.

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**Course 2: Skorokhod embeddings (David Hobson) **22-26 April, 2019

**Time and Place:**

04.22 Monday 9:00-11:30 Guanghua Main Building 1501

04.22 Monday 13:30-16:00 HGX304

04.23 Tuesday 9:00-11:30 Guanghua Main Building 1501

04.23 Tuesday 13:30-16:00 Guanghua Main Building 1501

**Abstract:** Let $X= (X_t)_{t \geq 0}$ be a stochastic process on a state space $E$ and let $\mu$ be a measure on $E$. The Skorokhod embedding problem (SEP) is to construct a stopping time (where possible) such that the law of $X_\tau$ is $\mu$. The classical case is when $X$ is Brownian motion, null at zero, $E$ is the real numbers and $\mu$ is a centred, square integrable probability measure. In this set of talks we will discuss some classical solutions to the SEP and extensions to other processes, other classes of target laws, and processes with non-trivial starting law. In addition we will discuss applications including applications to mathematical finance.

**Course schedule: **

Lecture 1: Introduction to Skorokhod embeddings:

Statement of the general problem, review of properties of Brownian motion, some first embeddings and properties of good solutions.

Lecture 2: Classical Skorokhod embeddings

The Chacon-Walsh picture, the Azema-Yor, Root, Rost and Bass solutions.

Lecture 3: Optimal Skorokhod embeddings

Solutions of the Skorokhod embedding problem with optimality properties, connections to martingale optimal transport, the Azema-Yor solution revisited.

Lecture 4: Extensions and applications of Skorokhod embeddings

Extensions to other processes and to non-trivial starting laws. Applications to the Central-Limit-Theorem and to Mathematical Finance.

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**Course 3: Martingale optimal transport with applications in finance (Dominykas Norgilas) **22-30 April 2019

**Time and Place:**

04.24 Wednesday 8:00-10:10 Guanghua Main Building 1801

04.25 Thursday 8:00-10:10 Guanghua Main Building 1501

04.26 Friday 13:30-16:00 Guanghua Main Building 1801

04.29 Monday 9:00-11:30 Guanghua Main Building 1501

**Abstract:** The theory of optimal transport has already proved to be very successful to solve many important economic issues. Recent progresses were motivated by applications to problems in mathematical finance. In this course we focus on the pricing and hedging of exotic options under model uncertainty, given the prices of vanilla options. Our approach is based on the martingale version of the Monge-Kantorovich mass transport. We will discuss the relevant mathematical tools, key results and few important examples.

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**Course 4: Risk measures and regulatory arbitrage (Martin Herdegen) **25 April-2 May 2019

**Time and Place:**

04.25 Thursday 10:20-12:30 Guanghua Main Building 1501

04.26 Friday 9:00-11:30 Guanghua Main Building 1801

04.29 Monday 13:30-16:00 HGX304

04.30 Tuesday 13:30-16:00 Guanghua Main Building 1501

**Abstract:** In this short course, we first provide a self-contained introduction to convex risk measures and their duality theory. In a second part, we look at the concept of regulatory arbitrage and study in particular the possibility of regulatory arbitrage under expected shortfall.

**Course schedule:**

Lecture 1: Monetary measures of risk and their acceptance sets

Lecture 2: Dual representation of risk measures

Lecture 3: Dual representation of Expected Shortfall

Lecture 4: Regulatory arbitrage for Expected Shortfall