43 research outputs found
From (Martingale) Schrodinger bridges to a new class of Stochastic Volatility Models
Following closely the construction of the Schrodinger bridge, we build a new class of Stochastic Volatility Models exactly calibrated to market instruments such as for example Vanillas, options on realized variance or VIX options. These models differ strongly from the well-known local stochastic volatility models, in particular the instantaneous volatility-of-volatility of the associated naked SVMs is not modified, once calibrated to market instruments. They can be interpreted as a martingale version of the Schrodinger bridge. The numerical calibration is performed using a dynamic-like version of the Sinkhorn algorithm. We finally highlight a striking relation with Dyson non-colliding Brownian motions
Some Results on Skorokhod Embedding and Robust Hedging with Local Time
In this paper, we provide some results on Skorokhod embedding with local time
and its applications to the robust hedging problem in finance. First we
investigate the robust hedging of options depending on the local time by using
the recently introduced stochastic control approach, in order to identify the
optimal hedging strategies, as well as the market models that realize the
extremal no-arbitrage prices. As a by-product, the optimality of Vallois'
Skorokhod embeddings is recovered. In addition, under appropriate conditions,
we derive a new solution to the two-marginal Skorokhod embedding as a
generalization of the Vallois solution. It turns out from our analysis that one
needs to relax the monotonicity assumption on the embedding functions in order
to embed a larger class of marginal distributions. Finally, in a full-marginal
setting where the stopping times given by Vallois are well-ordered, we
construct a remarkable Markov martingale which provides a new example of fake
Brownian motion
A numerical algorithm for a class of BSDEs via branching process
We generalize the algorithm for semi-linear parabolic PDEs in
Henry-Labord\`ere (2012) to the non-Markovian case for a class of Backward SDEs
(BSDEs). By simulating the branching process, the algorithm does not need any
backward regression. To prove that the numerical algorithm converges to the
solution of BSDEs, we use the notion of viscosity solution of path dependent
PDEs introduced by Ekren, Keller, Touzi and Zhang (2012) and extended in Ekren,
Touzi and Zhang (2013).Comment: 31 page
Unbiased simulation of stochastic differential equations
We propose an unbiased Monte-Carlo estimator for , where is a diffusion process defined by a
multi-dimensional stochastic differential equation (SDE). The main idea is to
start instead from a well-chosen simulatable SDE whose coefficients are updated
at independent exponential times. Such a simulatable process can be viewed as a
regime-switching SDE, or as a branching diffusion process with one single
living particle at all times. In order to compensate for the change of the
coefficients of the SDE, our main representation result relies on the automatic
differentiation technique induced by Bismu-Elworthy-Li formula from Malliavin
calculus, as exploited by Fourni\'e et al.(1999) for the simulation of the
Greeks in financial applications. In particular, this algorithm can be
considered as a variation of the (infinite variance) estimator obtained in
Bally and Kohatsu-Higa [Section 6.1](2014) as an application of the parametrix
method