## FS Stochastische Analysis und Stochastik der Finanzmärkte

### Bereich für Stochastik

P. BANK, D. BECHERER, P.K. FRIZ, H. FÖLLMER, U. HORST, P. IMKELLER, U. KÜCHLER, A. PAPAPANTOLEON, N. PERKOWSKI

- Ort: HU Berlin,
- Institut für Mathematik,
- Johann von Naumann - Haus,
- Rudower Chaussee 25, Hörsaal 1.115
- Zeit: Donnerstag, 16 Uhr/17 Uhr c.t.

26.10.2017 (16 Uhr c.t.) |
Trading Foreign Exchange Triplets Abstract: We develop the optimal trading strategy for a Foreign Exchange (FX) broker who must liquidate a large position in an illiquid currency pair. To maximise revenues, the broker considers trading in a currency triplet which consists of the illiquid pair and two other liquid currency pairs. The liquid pairs in the triplet are chosen so that one of the pairs is redundant. The broker is risk-neutral and accounts for model ambiguity in the FX rates to make her strategy robust to model misspecification. When the broker is ambiguity neutral (averse) the trading strategy in each pair is independent (dependent) of the inventory in the other two pairs in the triplet. We employ simulations to illustrate how the robust strategies perform. For a range of ambiguity aversion parameters, we find the mean Profit and Loss (P\&L) of the strategy increases and the standard deviation of the P\&L decreases as ambiguity aversion increases. |

26.10.2017 (17 Uhr c.t.) |
Volatility and Arbitrage Abstract: The capitalization-weighted cumulative variation $\sum_{i=1}^d \int_0^\cdot \mu_i (t) \dx \langle \log \mu_i \rangle (t) $ in an equity market consisting of a fixed number $d$ of assets with capitalization weights $\mu_i (\cdot) ,$ is an observable and a nondecreasing function of time. If this observable of the market is not just nondecreasing but actually grows at a rate bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it. Joint work with Bob Fernholz and Ioannis Karatzas |

09.11.2017 (15 Uhr c.t.) |
Affine Volterra processes and models for rough volatility Abstract: Motivated by recent advances in rough volatility modeling, we introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. Nonetheless, their Fourier-Laplace functionals admit exponential-affine representations in terms of solutions of associated deterministic integral equations, extending the well-known Riccati equations for classical affine diffusions. Our findings generalize and simplify recent results in the literature on rough volatility. |

09.11.2017 (16.30 Uhr) |
Pricing and hedging with rough Heston models Abstract: It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Brownian motion, they raise new issues when it comes to derivatives pricing and hedging. Using an original link between nearly unstable Hawkes processes and fractional volatility models, we compute the characteristic function of the log-price in rough Heston models and obtain explicit hedging strategies. The replicating portfolios contain the underlying asset and the forward variance curve, and lead to perfect hedging (at least theoretically). From a probabilistic point of view, our study enables us to disentangle the infinite-dimensional Markovian structure associated to rough volatility models. |

09.11.2017 (17.30 Uhr ) |
Cautious Stochastic Choice, Optimal Stopping and Randomization Abstract: This work considers an optimal liquidation problem in the context of a Cautious Stochastic Choice (CSC) model. In the classical case the investor solves an optimal stopping problem which involves maximizing the expected value of a function of stochastic process representing the price of an asset. The optimal strategy is always a threshold strategy — to liquidate the first time the price process leaves an interval. In the CSC model context, the investor has a family of utility functions and she is concerned only about the worst case certainty equivalent. We show that the optimal strategy may be of non-threshold form and may involve randomization. In this way we show that Cautious Stochastic Choice provides a potential explanation of the use of non-threshold strategies in experimental and empirical evidence. |

23.11.2017 (16 Uhr c.t.) |
Portfolio Optimisation, Transaction Costs, Shadow Prices and Fractional Brownian Motion Abstract: While absence of arbitrage in frictionless financial markets (i.e. without transaction costs) requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this talk, I will present an overview of several results that provide a way how to use non-semimartingale price processes such as the fractional Black-Scholes model in portfolio optimisation under proportional transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. The talk is based on joint work with Walter Schachermayer. |

23.11.2017 (17 Uhr c.t.) |
Abstract: |

07.12.2017 (16 Uhr c.t.) |
A mean field game of optimal portfolio liquidation Abstract: We consider a mean field game (MFG) of optimal portfolio liquidation. We prove that the solution to the MFG can characterized in terms of a FBSDE with singular terminal condition on the backward component or, equivalently, in terms of a FBSDE with finite terminal value, yet singular driver. Extending the method of continuation to linear-quadratic FBSDE with singular driver we prove that the MFG has a unique solution. Our existence and uniqueness result allows to prove that the MFG with terminal constraint can be approximated by a sequence of MFGs without constraint. (This is joint work with Paulwin Graewe, Ulrich Horst and Alexandre Popier.) |

07.12.2017 (17 Uhr c.t.) |
A forward equation for barrier options for efficient model calibration Abstract: In this talk, we present a novel and generic calibration framework for barrier options in a large class of continuous semi-martingale models. We derive a forward equation for arbitrage-free barrier option prices in terms of Markovian projections of the instantaneous variance. This gives a Dupire-type formula for the coefficient derived by Brunick and Shreve for their mimicking diffusion and can be interpreted as the canonical extension of local volatility for barrier options. Alternatively, a forward partial-integro differential equation is deduced which yields up-and-out call prices for the complete set of strikes, barriers and maturities in one solution step. We apply this methodology to the calibration of a path-dependent volatility model (PDV) and a new Heston-type local stochastic volatility model with local vol-of-vol (LSV-LVV), using a two-dimensional particle method, for a set of EURUSD market data of vanilla and no-touch options. Finally, we conclude by extending the main Markovian projection formula to handle stochastic rates and discuss how the algorithms can be adapted at little extra computational cost. (Joint work with Matthieu Mariapragassam.) |

21.12.2017 (16 Uhr c.t.) |
Semi-static and sparse variance-optimal hedging Abstract: We consider hedging of a contingent claim by a 'semi-static' strategy composed of a dynamic position in one asset and a static (buy-and-hold) position in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance-optimality and provide tractable formulas using Fourier-integration in case of the Heston model. We also consider the problem of optimally selecting a sparse semi-static hedging strategy, i.e. a strategy which only uses a small subset of available hedging assets. The developed methods are illustrated in an extended numerical example where we compute a sparse semi-static hedge for a variance swap using European options as static hedging assets. This is joint work with Paolo Di Tella and Martin Haubold. |

21.12.2017 (17 Uhr c.t.) |
Equilibrium Returns with Transaction Costs Abstract: We study how trading costs are reflected in equilibrium returns. To this end, we develop a tractable continuous-time risk-sharing model, where heterogeneous mean-variance investors trade subject to a quadratic transaction cost. The corresponding equilibrium is characterized as the unique solution of a system of coupled but linear forward-backward stochastic differential equations. Explicit solutions are obtained in a number of concrete settings. The sluggishness of the frictional portfolios makes the corresponding equilibrium returns mean-reverting. Compared to the frictionless case, expected returns are higher if the more risk-averse agents are net sellers or if the asset supply expands over time. The talk is based on joint work with Bruno Bouchard, Masaaki Fukasawa and Johannes Muhle-Karbe. |

18.01.2018 (16 Uhr c.t.) |
On the Optimal Management of Public Debt: a Singular Stochastic Control Problem Abstract: Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as a singular stochastic control problem over an infinite time-horizon. In a general not necessarily Markovian framework, we first show by probabilistic arguments that the optimal debt reduction policy can be expressed in terms of the optimal stopping rule of an auxiliary optimal stopping problem. We then exploit such link to characterise the optimal control in a two-dimensional Markovian setting in which the state variables are the level of the debt-to-GDP ratio and the current inflation rate of the country. The latter follows uncontrolled Ornstein-Uhlenbeck dynamics and affects the growth rate of the debt ratio. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio under an inflation-dependent ceiling. This curve is given in terms of the solution of a nonlinear integral equation arising in the study of a fully two-dimensional optimal stopping problem. |

18.01.2018 (17 Uhr c.t.) |
Rough mean field equations Abstract: We provide in this work a robust solution theory for random rough differential equations of mean field type $$ dX_t = V\big( X_t,{\mathcal L}(X_t)\big)dt + \textrm{F}\bigl( X_t,{\mathcal L}(X_t)\bigr) dW_t, $$ where $W$ is a random rough path and ${\mathcal L}(X_t)$ stands for the law of $X_t$, with mean field interaction in both the drift and diffusivity. Propagation of chaos results for large systems of interacting rough differential equations are obtained as a consequence, with explicit convergence rate. The development of these results requires the introduction of a new rough path-like setting and an associated notion of controlled path. We use crucially Lions' approach to differential calculus on Wasserstein space along the way. This is a joint work with I. Bailleul and R. Catellier. |

01.02.2018 (16 Uhr c.t.) |
On concavity of the principal's profit maximization facing agents who respond nonlinearly to prices Abstract: A monopolist wishes to maximize her profits by finding an optimal price menu. After she announces a menu of products and prices, each agent will choose to buy that product which maximizes his own utility, if positive. The principal's profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal's price menu. In this talk, we describe a necessary and sufficient condition for the convexity or concavity of the principal's problem, assuming each agent's disutility is a strictly increasing but not necessarily affine (i.e. quasilinear) function of the price paid. Concavity when present, makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the principal's strategy in particular. Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity, by establishing conditions which are not only sufficient but necessary, and by requiring fewer hypotheses on the agents' preferences. This talk represents joint work with my supervisor Robert McCann. |

01.02.2018 (17 Uhr c.t.) |
Storing, hedging, and speculating in commodities markets: a dynamic model Abstract: I will present a dynamic model for a commodity market. At every time, two markets are open, a physical one where the commodity is traded and a financial one where futures are traded. The commodity arrives in uncertain supply, but storers can transfer from the preceding period and industrial users have to commit for the next period. We show that there is a rational equilibrium which is a stationary strategy for all agents, and we derive some stylised facts. This is joint work with Delphine Lautier and Bertrand Villeneuve. |

15.02.2018 (16 Uhr c.t.) |
Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications Abstract: We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of the paper is a verification theorem which provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the value functions and the optimal strategies of the two players. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a Nash equilibrium and provide explicit expressions for the optimal strategies and the value functions. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further non-symmetric examples where a Nash equilibrium is found numerically. Link to the paper: https://arxiv.org/abs/1605.00039 |

15.02.2018 (17 Uhr c.t.) |
The Term Structure of Liquidity: A Liquidation Game Approach Abstract: We analyze a dynamic liquidation game where both liquidity demand and supply are endogenous. A large uninformed investor strategically liquidates a position, fully cognizant of the optimal response of competitive market makers. The Stackelberg game solution shows that, if the investor reveals the duration of the trade to the intermediation sector, then he chooses to sell at higher intensity when he has less time to trade. This enables market makers to predict when execution ends, which helps them provide liquidity and thus reduces the liquidity premium they charge. The model explains several empirical facts: order duration and participation rate correlate negatively, and price pressure subsides before execution ends.(Joint work with Albert Menkveld and Hongzhong Zhang) |

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Telefon: 2093 5811

Telefax: 2093 5848http://www.qfl-berlin.de/tags/stochastic-analysis-and-stochastic-finance-seminar