Tag - Mathematical finance
There has been a recent dramatic increase in trading on exchanges of short-dated options, i.e., options with very short time to expiration. This workshop covers non-parametric methods for extracting information from these options. Formal non-parametric econometric analysis of derivatives data has proved difficult. The complications arise from the highly non-linear dependence of option prices on state variables and parameters as well as the possible dependence in the option observation errors. The short-dated options allow to aggregate option data in ways that facilitate the practical application of asymptotic expansions for option maturities approaching zero.
We first start by introducting various model-free measures of spot volatility. These measures separate true spot volatility from the price jump component (and its pricing) as well as the volatility mean-reversion effects present in option prices. Following that, we introduce measures of risk-neutral jump variation and jump tails as well as methods for studying anticipated event risks. Empirical illustrations of the methods will be presented along with various applications for studying volatility forecasting, return predictability via option measures and analysis of risk premia.
In recent years, the field of optimal transport has attracted the attention of many high-profile mathematicians with a wide range of applications. In this talk we will discuss some of its recent applications in financial mathematics, particularly on the problems of model calibration, robust finance and portfolio optimisation. Classical topological duality results are extended to probabilistic settings, connecting stochastic control problems with non-linear partial differential equations and providing interesting practical interpretations in finance. We will also look at how numerical methods, including machine learning algorithms, can be implemented to solve these problems.
For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. Here, we propose a new approach based on convex duality instead of martingale measures duality: our prices will be expressed using Fenchel conjugate and biconjugate.
This naturally leads to a weak condition of absence of arbitrage opportunity, called Absence of Immediate Profit (AIP), which asserts that the price of the zero claim should be zero. We study the link between (AIP), (NA) and the no-free lunch condition. We show in a one step model that, under (AIP), the super-hedging cost is just the payoff's concave envelop and that (AIP) is equivalent to the non-negativity of the super-hedging prices of some call option.
In the multiple-period case, for a particular, but still general setup, we propose a recursive scheme for the computation of a the super-hedging cost of a convex option. We also give some numerical illustrations.
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