# Stochastics and Finance Seminar

Seminars are held on Tuesdays 2:00-3:00 pm in the Access Grid Room (Carslaw Building, 8th floor, room 829), unless otherwise specified.

Please direct enquiries about this seminar to Marek Rutkowski, Ben Goldys, Anna Aksamit, Zhou Zhou.

## 2019 Semester 2

### Tuesday, November 5, 2:00 pm, AGR

Speaker: Prof Jan Obloj (University of Oxford)

Title: Robust Finance. Part II – fundamental theorems

## Abstract:

We pursue robust approach to pricing and hedging in mathematical finance. We develop a general discrete time setting in which some underlying assets and options are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. We include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. super-replication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows to quantify the impact of making assumptions. We establish suitable FTAP and Pricing-Hedging duality results which include as special cases previous results of Acciaio et al. (2013), Bouchard and Nutz (2015), Burzoni et al. (2016) as well the Dalang-Morton-Willinger theorem. Finally, we explain how to treat further problems, such as insider trading (information quantification) or American options pricing. The talk will cover a body of results developed in collaboration with A. Aksamit, M. Burzoni, S. Deng, M. Frittelli, Z. Hou, M. Maggis, X. Tan and J. Wiesel.

### Tuesday, September 24, 2:00 pm, AGR

Speaker: Prof Jan Obloj (University of Oxford)

Title: Robust finance. Part I -- proof-of-concept applications using historical time series and market option prices

## Abstract:

In this talk I introduce briefly the robust paradigm which strives to interpolate the modelling spectrum: from agnostic model-free to classical model-specific. It offers methods to walk the spectrum and quantify the impact of making assumptions and/or using market data. I explain briefly how classical fundamental notions and theorems in quantitative finance extend to the robust setting – I plan to cover this in more detail in “Part II – fundamental theorems”. In this Part I, I mostly focus on simple concrete examples. I use vanilla option prices, together with agent-prescribed bounds on key market characteristics, to drive the interval of no-arbitrage prices and the associated hedging strategies. The setting can be seen as a constrained variant of the classical optimal transportation problem and comes with a natural pricing-hedging duality. I discuss numerical methods based on discretisation and LP implementation and on a deep NN optimisation. I look at ways to coherently combine option prices data with past time series data, leading to a dynamic robust risk estimation. I explain how such non-parametric statistical estimators of key quantities (e.g., superhedging prices, 10-days V@R) superimposed with option prices can be treated as information signals. Based on joint works with Stephan Eckstein, Gaoyue Guo, Tongseok Lim and Johannes Wiesel.

### Tuesday, September 17, 2:00 pm, AGR

Speaker: Mr Guanting Liu (UNSW)

Title: On a positivity preserving numerical scheme for jump-extended CEV process: the alpha-stable case

## Abstract:

We propose a positivity preserving implicit Euler–Maruyama scheme for a jump-extended constant elasticity of variance (CEV) process where the jumps are governed by a compensated spectrally positive alpha-stable process for alpha in (1,2). Different to the existing positivity preserving numerical schemes for jump-extended CIR or CEV process, the model considered here has inﬁnite activity jumps. For this speciﬁc model, we calculate the strong rate of convergence. Jump-extended models of this type were initially studied in the context of branching processes and was recently introduced to the ﬁnancial mathematics literature to model sovereign interest rates, power and energy markets.

### Tuesday, August 20, 2:00 pm, AGR

Speaker: Dr Kristoffer Glover (UTS)

Title: Optimally Stopping a Brownian Bridge with an Unknown Pinning Time: A Bayesian Approach

## Abstract:

We consider the problem of optimally stopping a Brownian bridge with an unknown pinning time so as to maximise the value of the process upon stopping. Adopting a Bayesian approach, we allow the stopper to update their belief about the value of the pinning time through sequential observations of the process. Uncertainty in the pinning time influences both the conditional dynamics of the process and the expected (random) horizon of the optimal stopping problem. Structural properties of the optimal stopping region are shown to be qualitatively different under different prior distributions, however we provide a sufficient condition for the existence of a one-sided stopping region. Certain gamma and beta distributed priors are shown to satisfy this condition and these cases are subsequently considered in detail. A two-point prior distribution is also considered in which a richer structure emerges (with multiple optimal stopping boundaries).

## 2019 Semester 1

### Tuesday, July 30, 2:00 pm, AGR

Speaker: Prof Antoine Ayache (Université de Lille)

Title: Almost sure approximations in Hölder norms of a general stochastic process defined by a Young integral

## Abstract:

We focus on a stochastic process Y defined by a pathwise Young integral of a general form. Thanks to the Haar basis, we connect the classical method of approximation of Y through Euler scheme and Riemann-Stieltjes sums with a new approach consisting in the use of an appropriate series representation of Y. This representation is obtained through a general compactly supported orthonormal wavelet basis. An advantage offered by the new approach with respect to the classical one is that a better almost sure rate of convergence in Hölder norms can be derived, under a general chaos condition. Also, this improved rate turns out to be optimal in some situations; typically, when the integrand and integrator associated to Y are independent fractional Brownian motions with appropriate Hurst parameters. Joint work with Céline Esser and Qidi Peng.

### Thursday, July 25, 2:00 pm, AGR

Speaker: Dr Alexandru Hening (Tufts University)

Title: The competitive exclusion principle in stochastic environments

## Abstract:

The competitive exclusion principle states that a number of species competing for a smaller number of resources cannot coexist. Even though this is a fundamental principle in ecology, it has been observed empirically that in some settings it will fail. One example is Hutchinson's `paradox of the plankton'. This is an instance where a large number of phytoplankton species coexist while competing for a very limited number of resources. Both experimental and theoretical studies have shown that in some instances (deterministic) temporal fluctuations of the environment can facilitate coexistence for competing species. Hutchinson conjectured that one can get coexistence because nonequilibrium conditions would make it possible for different species to be favored by the environment at different times. In this talk I will look at how environmental noise interacts with competitive exclusion. I will show that, contrary to Hutchinson's explanation, one can switch between two environments in which the same species is favored and still get coexistence.

### Tuesday, July 23, 2:00 pm, AGR

Speaker: Dr Alexandru Hening (Tufts University)

Title: Harvesting of populations in stochastic environments

## Abstract:

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold \(x^*>0\) such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate. We also show that, if the yield function is \(C^2\) and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls.

### Tuesday, May 28, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Edward Kim (University of Sydney, School of Mathematics and Statistics)

Title: BSDEs driven by discontinuous martingales

## Abstract:

We prove some new results on BSDEs, reflected BSDEs, and doubly reflected BSDEs driven by a multi-dimensional discontinuous martingale. We generalize the setups studied by Peng and Xu (2009) and Quenez, Grigorova, Sulem and Dumitrescu (2018) who dealt with BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump. Our results are not covered by existing results on BSDEs driven by a Brownian motion and a Poisson random measure.

### Tuesday, May 21, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Prof Ben Goldys (University of Sydney, School of Mathematics and Statistics)

Title: On the existence of stochastic flows

## Abstract:

An important question in the theory of stochastic (ordinary and partial) differential equations is about the existence and regularity of the stochastic flow associated to solutions of the equation. It has been studied by many authors, including Elworthy, Flandoli, Malliavin, Bismut, Ikeda and Watanabe, and Kunita. It is still a major open problem in the case of stochastic PDEs and even for ordinary stochastic differential equations many questions remain unanswered. In this talk I will discuss the difficulties arising in the case of stochastic PDEs and present some partial answers. This is a joint work with Szymon Peszat.

### Tuesday, May 14, 3:10 pm, Wilkinson Seminar Room 231

Speaker: James Yang (University of Sydney, School of Mathematics and Statistics)

Title: Singular Perturbation of Zero-sum Linear-Quadratic Stochastic Differential Games

## Abstract:

We develop a framework for zero-sum linear-quadratic stochastic differential games on a finite time horizon governed by multiscale state equations. The multiscale nature of the problem can be leveraged to reformulate the associated generalised Riccati equation in terms of a deterministic singular perturbation problem. In doing so, we can show that, for small enough \(\epsilon\), the existence of solution to the associated generalised Riccati equation is guaranteed by the existence of a solution to a decoupled pair of differential and algebraic Riccati equations with a reduced order of dimensionality. Furthermore, we construct two asymptotic estimates to the closed-loop value of the game. The first is by constructing an approximate closed-loop strategy and the second is by observing the limiting value as \(\epsilon\) goes towards zero. This is a joint work with Ben Goldys and Zhou Zhou.

### Tuesday, April 30, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Dr Anna Aksamit (University of Sydney, School of Mathematics and Statistics)

Title: The Robust Pricing–Hedging Duality for American Options in Discrete Time Financial Markets (Part II)

## Abstract:

We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry-Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing-hedging duality for American options. This is joint work with Shuoqing Deng, Jan Obłój and Xiaolu Tan.

### Tuesday, April 23, 3:10 pm, Your favourite beach or park

Easter break

## Abstract:

Partly cloudy. Medium (50%) chance of showers in the afternoon and evening. Light winds.### Tuesday, April 16, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Prof Marek Rutkowski (University of Sydney, School of Mathematics and Statistics)

Title: Valuation under Credit Risk, Margins and Funding Costs (Part II)

## Abstract:

We present a unified valuation theory that incorporates credit risk (defaults), collateralization and funding costs, by expanding the replication approach to a generality that has not yet been studied previously and reaching valuation when replication is not assumed. We show that this unifying theoretical framework clarifies the relationship between the two valuation approaches: the so-called adjusted cash flows approach and the classic replication approach. In particular, results of this work cover most previous papers where the authors studied specific replication-based models. This is joint work with Damiano Brigo, Cristin Buescu, Marco Francischello and Andrea Pallavicini.

### Tuesday, April 9, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Prof Marek Rutkowski (University of Sydney, School of Mathematics and Statistics)

Title: Valuation under Credit Risk, Margins and Funding Costs (Part I)

## Abstract:

We develop a unified valuation theory that incorporates credit risk (defaults), collateralization and funding costs, by expanding the replication-based approach to a generality that has not yet been studied previously and reaching valuation when replication is not assumed. We first discuss practical issues motivating our research and we stress that the classic approach is not suitable when dealing with over-the-counter collateralized contracts under differential funding costs. This is joint work with Damiano Brigo, Cristin Buescu, Marco Francischello and Andrea Pallavicini.

### Tuesday, April 2, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Dr Anna Aksamit (University of Sydney, School of Mathematics and Statistics)

Title: The Robust Pricing–Hedging Duality for American Options in Discrete Time Financial Markets (Part I)

## Abstract:

We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality. We show that it is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. This is joint work with Shuoqing Deng, Jan Obłój and Xiaolu Tan.

### Tuesday, March 26, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Dr Zhou Zhou (University of Sydney, School of Mathematics and Statistics)

Title: Transport Plans with Domain Constraints (Part II)

## Abstract:

This talk is the second part of my presentation commenced on March 12. I will first recall the main results presented in the first part. Subsequently, I will focus on the optimal transport problem with constraints and obtain the Kantorovich duality. A corollary of this result is a monotonicity principle, which gives us a geometric way of identifying the optimizer.

### Tuesday, March 19, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Prof Marek Musiela (University of Oxford and Oxford-Man Institute of Quantitative Finance)

Title: Multivariate Fractional Brownian Motion and Generalizations of SABR Model

## Abstract:

SABR is a generalization of the Black and Scholes model which was adopted as the market standard for quoting cap and swaption volatilities. At the time it fitted the market relatively well. Its success was driven by the approximate formula for implied volatility developed by Pat Hagan. More recently, SABR was applied to price equity and FX options. This generated new challenges for the SABR framework and suggested a new class of models inspired by SABR, where the process defining the noise is a bivariate fractional Brownian motion with parameter \((1/2, H)\). In the classical SABR, the correlation parameter between the two Brownian motions determines many of its mathematical properties. In the modified SABR, a Brownian motion defining the dynamics of stochastic volatility is replaced by a fractional Brownian motion. This leads to the question how one should define the dependence structure between the Brownian motion driving the asset dynamics and the fractional Brownian motion used to define the volatility. In this talk, I propose dependence structure implicit to the definition of multivariate fractional Brownian motion with multivariate self-similarity parameter \(H\). I also consider the consequences and suggest further modifications.

### Tuesday, March 12, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Dr Zhou Zhou (University of Sydney, School of Mathematics and Statistics)

Title: Transport Plans with Domain Constraints (Part I)

## Abstract:

Let \(\Omega\) be one of \(\mathbb{X}^{N+1},C[0,1],D[0,1]\): product of Polish spaces, space of continuous functions from \([0,1]\) to \(\mathbb{R}^d\), and space of RCLL (right-continuous with left limits) functions from \([0,1]\) to \(\mathbb{R}^d\), respectively. We first consider the existence of a probability measure \(P\) on \(\Omega\) such that \(P\) has the given marginals \(\alpha\) and \(\beta\) and its disintegration \(P_x\) must be in some fixed \(\Gamma(x)\subset \mathfrak{P}(\Omega)\), where \(\mathfrak{P}(\Omega)\) is the set of probability measures on \(\Omega\). The main application we have in mind is the martingale optimal transport problem in mathematical finance when the martingales are assumed to have bounded volatility/quadratic variation. We show that such probability measure exists if and only if the \(\alpha\) average of the so-called \(G\)-expectation of bounded continuous functions with respect to the measures in \(\Gamma\) is less than their \(\beta\) average. As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times. This is joint work with Erhan Bayraktar and Xin Zhang.