Stochastic differential equations and diffusion processes

The Volatility Surface

Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? 
If we measure proximity with the usual Wasserstein distance (say), the answer is NO. Models which are similar w.r.t. Wasserstein distance may provide dramatically different information on which to base a hedging strategy. 
Remarkably, this can be overcome by considering a suitable "adapted" version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler \cite{Pf09,PfPi12,PfPi14}. It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

Adapted Wasserstein Distances and Stability in Mathematical Finance

Assume that an agent models a financial asset through a measure ℚ with the goal to price/hedge some derivative or optimise some expected utility. Even if the model ℚ is chosen in the most skilful and sophisticated way, the agent is left with the possibility that ℚ does not provide an exact description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of ℚ? If we measure proximity with the usual Wasserstein distance (say), the answer is No. Models which are similar with respect to the Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable adapted version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler (SIAM J. Optim. 20:1406–1420, 2009, SIAM J. Optim. 22:1–23, 2012, Multistage Stochastic Optimization, 2014). It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

/pdf/adapted-wasserstein-distances-and-stability-in-mathematical-1c82e5l40n.pdf

Adapted Wasserstein distances and stability in mathematical finance

The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327–3405, 2017). We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier–Strassen Theorem of Gozlan and Juillet (On a mixture of brenier and strassen theorems. arXiv:1808.02681, 2018) to general probability measures on $$\mathbb {R}^d$$ under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (‘adapted’) topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.

/pdf/existence-duality-and-cyclical-monotonicity-for-weak-4ow9axrsug.pdf

Existence, duality, and cyclical monotonicity for weak transport costs

Motivated by the approximation of Martingale Optimal Transport problems, we
study sampling methods preserving the convex order for two probability measures
$\mu$ and $\nu$ on $\mathbb{R}^d$, with $\nu$ dominating $\mu$. When
$(X_i)_{1\le i\le I}$ (resp. $(Y_j)_{1\le j\le J}$) are i.i.d. according $\mu$
(resp. $\nu$), the empirical measures $\mu_I$ and $\nu_J$ are not in the convex
order. We investigate modifications of $\mu_I$ (resp. $\nu_J$) smaller than
$\nu_J$ (resp. greater than $\mu_I$) in the convex order and weakly converging
to $\mu$ (resp. $\nu$) as $I,J\to\infty$. In dimension 1, according to Kertz
and R\"osler (1992), the set of probability measures with a finite first order
moment is a lattice for the increasing and the decreasing convex orders. From
this result, we can define $\mu\vee\nu$ (resp. $\mu\wedge\nu$) that is greater
than $\mu$ (resp. smaller than $\nu$) in the convex order. We give efficient
algorithms permitting to compute $\mu\vee\nu$ and $\mu\wedge\nu$ when $\mu$ and
$\nu$ are convex combinations of Dirac masses. In general dimension, when $\mu$
and $\nu$ have finite moments of order $\rho\ge 1$, we define the projection
$\mu\curlywedge_\rho \nu$ (resp. $\mu\curlyvee_\rho\nu$) of $\mu$ (resp. $\nu$)
on the set of probability measures dominated by $\nu$ (resp. larger than $\mu$)
in the convex order for the Wasserstein distance with index $\rho$. When
$\rho=2$, $\mu_I\curlywedge_2 \nu_J$ can be computed efficiently by solving a
quadratic optimization problem with linear constraints. It turns out that, in
dimension 1, the projections do not depend on $\rho$ and their quantile
functions are explicit, which leads to efficient algorithms for convex
combinations of Dirac masses. Last, we illustrate by numerical experiments the
resulting sampling methods that preserve the convex order and their application
to approximate Martingale Optimal Transport problems.

Sampling of probability measures in the convex order by Wasserstein projection

For µ and ν two probability measures on the real line such that µ is smaller than ν in the convex order, this property is in general not preserved at the level of the empirical measures µI = 1 I I i=1 δX i and νJ = 1 J J j=1 δY j , where (Xi) 1≤i≤I (resp. (Yj) 1≤j≤J) are independent and identically distributed according to µ (resp. ν). We investigate modifications of µI (resp. νJ) smaller than νJ (resp. greater than µI) in the convex order and weakly converging to µ (resp. ν) as I, J → ∞. According to Kertz and Rosler (1992), the set of probability measures on the real line with a finite first order moment is a complete lattice for the increasing and the decreasing convex orders. For µ and ν in this set, this enables us to define a probability measure µ ∨ ν (resp. µ ∧ ν) greater than µ (resp. smaller than ν) in the convex order. We give efficient algorithms permitting to compute µ ∨ ν and µ ∧ ν (and therefore µI ∨ νJ and µI ∧ νJ) when µ and ν have finite supports. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate martingale optimal transport problems and in particular to calculate robust option price bounds.

/pdf/sampling-of-one-dimensional-probability-measures-in-the-52ymq2sq0i.pdf

Sampling of one-dimensional probability measures in the convex order and computation of robust option price bounds

Motivated by the approximation of Martingale Optimal Transport problems, we study sampling methods preserving the convex order for two probability measures $\mu$ and $\nu$ on $\mathbb{R}^d$, with $\nu$ dominating $\mu$. When $(X_i)_{1\le i\le I}$ (resp. $(Y_j)_{1\le j\le J}$) are i.i.d. according $\mu$ (resp. $\nu$), the empirical measures $\mu_I$ and $\nu_J$ are not in the convex order. We investigate modifications of $\mu_I$ (resp. $\nu_J$) smaller than $\nu_J$ (resp. greater than $\mu_I$) in the convex order and weakly converging to $\mu$ (resp. $\nu$) as $I,J\to\infty$. In dimension 1, according to Kertz and R\"osler (1992), the set of probability measures with a finite first order moment is a lattice for the increasing and the decreasing convex orders. From this result, we can define $\mu\vee\nu$ (resp. $\mu\wedge\nu$) that is greater than $\mu$ (resp. smaller than $\nu$) in the convex order. We give efficient algorithms permitting to compute $\mu\vee\nu$ and $\mu\wedge\nu$ when $\mu$ and $\nu$ are convex combinations of Dirac masses. In general dimension, when $\mu$ and $\nu$ have finite moments of order $\rho\ge 1$, we define the projection $\mu\curlywedge_\rho \nu$ (resp. $\mu\curlyvee_\rho\nu$) of $\mu$ (resp. $\nu$) on the set of probability measures dominated by $\nu$ (resp. larger than $\mu$) in the convex order for the Wasserstein distance with index $\rho$. When $\rho=2$, $\mu_I\curlywedge_2 \nu_J$ can be computed efficiently by solving a quadratic optimization problem with linear constraints. It turns out that, in dimension 1, the projections do not depend on $\rho$ and their quantile functions are explicit, which leads to efficient algorithms for convex combinations of Dirac masses. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate Martingale Optimal Transport problems.

Sampling of probability measures in the convex order and approximation of Martingale Optimal Transport problems

We provide explicit conditions on the distribution of risk-neutral log-returns which yield sharp asymptotic estimates on the implied volatility smile. We allow for a variety of asymptotic regimes, including both small maturity (with arbitrary strike) and extreme strike (with arbitrary bounded maturity), extending previous work of Benaim and Friz [Math. Finance 19 (2009), 1-12]. We present applications to popular models, including Carr-Wu finite moment logstable model, Merton's jump diffusion model and Heston's model.

General smile asymptotics with bounded maturity

A new risk measure, lambda value at risk (), has been recently proposed as a generalization of value at risk (VaR). appears attractive for its potential ability to solve several problems of VaR. This paper provides the first study on the backtesting of . We propose three nonparametric tests which exploit different features. Two tests are based on simple results of probability theory. One test is unilateral and is more suitable for small samples of observations. A second test is bilateral and provides an asymptotic result. A third test is based on simulations and allows for a more accurate comparison among computed with different assumptions on the asset return distribution. Finally, we perform a backtesting exercise that confirms a higher performance of in respect to VaR especially when it is estimated with distributions that better capture tail behavior.

/pdf/backtesting-lambda-value-at-risk-36ser91m31.pdf

Jacopo Corbetta

Papers

Sampling of probability measures in the convex order by Wasserstein projection

Sampling of one-dimensional probability measures in the convex order and computation of robust option price bounds

Sampling of probability measures in the convex order and approximation of Martingale Optimal Transport problems

General smile asymptotics with bounded maturity

Backtesting lambda value at risk