Asymptotic Expansions for Pricing and Hedging

ABSTRACT:

In this lectures we shall present recently developed asymptotic approaches for obtaining explicit approximate formulas for option prices and hedging strategies in imperfect financial markets. 

 

Lecture 1.  ASYMPTOTIC METHODS FOR INDIFFERENCE PRICING AND UNCERTAINTY QUANTIFICATION IN EXPONENTIAL LÉVY MODELS

(Thursday March 9, 09:00-11:00, Lecture Room 341)

Lecture 2. ASYMPTOTIC METHODS FOR MARKET WITH FRICTIONS: LOWER BOUNDS

(Thursday March 16, 09:00-11:00, Lecture Room 342)

Lecture 3. ASYMPTOTIC METHODS FOR MARKET WITH FRICTIONS: FEEDBACK STRATEGIES

(Thursday March 23, 09:00-11:00, Lecture Theatre 130)

In the first lecture, we shall discuss indifference pricing and model uncertainty quantification in exponential Lévy models in the asymptotic regime where the effect of market incompleteness is small. Approximate indifference prices can then be represented as the sum of the Black-Scholes complete market price and explicit correction terms, and the spread due to market incompleteness and model uncertainty may be explicitly quantified.

In the last two lectures we shall focus on markets with transaction costs or market impact and show that portfolio optimization problems in such markets may be reformulated as the problems of tracking a stochastic target, where the aim of the controller is to minimize both deviation from the target and tracking efforts. We shall then establish the existence of  asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the time-average control of Brownian motion, which is characterized as a deterministic linear programming problem. Next, we shall provide a comprehensive list of examples where the time average control for the Brownian motion can be solved explicitly, the lower bound is sharp and is attained by an explicit feedback strategy. In the last part, applications to various control problems arising in mathematical finance (option hedging in discrete time, utility maximization with transaction costs) will be discussed. We shall first review the approaches developed in the literature for the specific situations and then show how that our methods provide a unified framework for dealing with these problems. 

References:

*Clément Ménassé and Peter Tankov, Asymptotic indifference pricing in exponential Lévy models, Applied Mathematical Finance, 23(3), 2016.
* Jiatu Cai, Mathieu Rosenbaum and Peter Tankov. Asymptotic Lower Bounds for Optimal Tracking: a Linear Programming Approach, arxiv preprint  arXiv:1510.04295, to appear in Annals of Applied Probability (2017)
* Jiatu Cai, Mathieu Rosenbaum and Peter Tankov. Asymptotic Optimal Tracking: Feedback Strategies, arxiv preprint  arXiv:1603.09472, to appear in Stochastics (2017)

 

Professor Tankov visit is co-funded by an ICL-CNRS Fellowship in Mathematics.

More information about the scheme can be found here: www.imperial.ac.uk/mathematics/research/opportunities/icl-cnrs-fellowships