Abstracts and Slides

Maxim Bichuch

Title. Robust XVA.

Abstract. We introduce an arbitrage-free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a defaultable counterparty, but has incomplete information about her credit quality. By constraining the actual default intensity of the counterparty within an uncertainty interval, we derive both upper and lower bounds for the XVA process. We show that these bounds may be recovered as solutions of nonlinear ordinary differential equations. The presence of collateralization and closeout payoffs introduce fundamental differences relative to classical credit risk valuation. The superhedging value cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the close-out value throughout the life of the transaction. This is a joint work with Agostino Capponi and Stephan Sturm.

Agostino Capponi

Title. The term structure of liquidity.

Abstract. We study the price policy and welfare implications of temporary liquidation pressure imposed on the intermediation sector. We show that, if the institutional investor reveals the duration of his liquidation to a group of competitive high frequency traders (HFTs), then he chooses to sell at a 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 benefits of revealing duration are even stronger if the investor strategically liquidates the position, fully cognizant of the optimal response of market makers. The Stackelberg game model of strategic interactions explains several empirical facts: order duration and participation rate correlate negatively, and price pressure subsides before execution ends. Competition between market makers raises aggregate welfare relative to a monopolistic market making environment. Joint work with Albert Menkveld and Hongzhong Zhang.

Peter Carr

Title. Interpreting Implied Volatility.

Abstract. We provide a probabilistic and financial Interpretation of a factor arbitrage-free implied volatility smile. In our family of models, the Black Merton Scholes implied variance rate captures the rate of increase of the quadratic variation of the P&L from a dynamic trading strategy in the options' underlying asset and in an always delta-neutral co-terminal straddle.

Jerome Detemple

Title. Optimal Investment Strategies for Power Generation: the Value of Green Energy.

Abstract. This paper examines the investment in and the valuation of power generation projects under uncertainty. The analysis incorporates the possibility of producing from alternative types of fuels, such as renewables (wind) or fossil fuels (gas), hence alternative types of plants/technologies, and accounts for endogenous operating decisions of plants. We show that the optimal investment decision can be characterized by two boundaries that satisfy a system of coupled integral equations of Fredholm type and propose a numerical algorithm for resolution. The option to produce power from renewables, as opposed to fossil fuels, increases the value of power generation projects, can postpone the optimal investment time and reduces the likelihood of investments in fossil fuels technologies. The optimal delay can be substantial even if the values of the individual projects are both large. The premium from renewables can be positive, even when the current value of fossil fuel power plants exceeds that of renewables power plants. Green power generation emerges as a significant source of value creation.

Ibrahim Ekren

Title. Equilibrium option price with competing market makers.

Abstract. In this talk, we present a market equilibrium between $N$ option market makers that compete for the orders of their clients. The market makers can offset their risk in the stock market. However they face market illiquidity when trading the stock and manage their inventory optimally. In this framework, we exhibit a Nash equilibrium for the interaction of the market makers and compute the equilibrium price of the option.

This is an ongoing work with Sergey Nadtochiy and Yavor Stoev.

Jose Figueroa-Lopez

Title. Optimal Kernel Estimation of Spot Volatility of Stochastic Differential Equations.

Abstract. A feasible method of bandwidth and kernel selection for spot volatility kernel estimators is proposed, under some mild conditions on the volatility process, which not only cover classical Brownian motion driven dynamics but also some processes driven by long-memory fractional Brownian motions. We characterize the leading order terms of the mean squared error, which in turn enables us to determine an explicit formula for the leading term of the optimal bandwidth. Central limit theorems for the estimation error are also obtained. A feasible plug-in type bandwidth selection procedure is then proposed, for which, as a sub-problem, a new estimator of the volatility of volatility is developed. The optimal selection of the kernel function is also investigated. For Brownian Motion type volatilities, the optimal kernel turns out to be an exponential function, while, for fractional Brownian motion type volatilities, numerical results to compute the optimal kernel are devised. Simulation studies further confirm the good performance of the proposed methods. This is based on joint work with Cheng Li and Bei Wu.

Jim Gatheral

Title. Diamonds: A quant's best friend.

Abstract. We use the Alòs Itô Decomposition Formula to express certain conditional expectations as exponentials of forests of trees. Each tree represents iterated applications of a new diamond operator. As one application, we compute an exact formal expression for the leverage swap for any stochastic volatility model expressed in forward variance form. As another, we show how to extend the Bergomi-Guyon expansion to all orders in volatility of volatility. Finally, we compute exact expressions under rough volatility, obtaining in particular the fractional Riccati equation for the rough Heston characteristic function. As a corollary, we compute a closed-form expression for the leverage swap in the rough Heston model.

Paul Glasserman

Title. Information-Driven Price and Volatility Cycles.

Abstract. How do small shocks get amplified into market crashes? We propose and analyze a dynamic model in which stock market prices and volatility shift between higher and lower levels based on changes in the information available to investors about future dividends. The model evolves through overlapping generations. In each generation, investors decide whether to acquire information at a cost. The precision of available information is subject to exogenous shocks; in addition, more information becomes available as more investors choose to become informed. The model gives rise to multiple fixed points for the fraction of informed investors, corresponding to different price and volatility regimes. Investors rationally anticipate potential transitions between regimes. We calibrate the model to historical data and find that the transitions can generate large price drops accompanied by large increases in volatility. This pattern may play a role in the onset of a financial crisis and in responses to changes in information disclosure. The model isolates the role of information dynamics in producing these effects. This is joint work with Harry Mamaysky and Yiwen Shen.

Ruoting Gong

Title. Small-time Asymptotics for Levy-Based Models.

Abstract. In recent years, small-time asymptotic methods have attracted much attention in mathematical finance. They have been widely used as model selection and calibration tools based on near expiration option prices. Such asymptotic methods are especially useful to study complex models with jumps and stochastic volatility due to the lack of tractable formulas and efficient numerical procedures. In this talk, I will discuss some recent advances in the area and illustrate their broad relevance in financial mathematics.

Dmitry Kramkov

Title. An optimal transport problem with backward martingale constraints motivated by insider trading.

Abstract. Given a probability measure $\nu$ on $\R^2$, we want to

\begin{equation}

\label{eq:1}

\text{minimize} \quad \int c(x,y)d\gamma \quad \text{over} \quad

\gamma\in \Gamma(\nu)

\end{equation}

for the \emph{covariance-type} cost function $c(x,y) = (y_1-x_1)(y_2-x_2)$, where $\Gamma(\nu)$ is the family of probability measure $\gamma$ on $\R^2\times \R^2$, that have $\nu$ as their $y$-marginal and make a martingale from the canonical two-dimensional process $(x,y)$. Problem~\eqref{eq:1} belongs to the class of optimal

transport problems with \emph{backward} martingale constraints, in the sense that the initial $x$-marginal is part of the solution. The motivation comes from a version of Kyle's equilibrium with insider. Our main result states that a probability measure $\gamma\in \Gamma(\nu)$ is optimal if and only if there is a maximal monotone set $G\subset\R^2$ such that (1) it supports the $x$-marginal of $\gamma$, and (2) $c(x,y)=\min_{z\in G}c(z,y)$ for every $(x,y)\in \supp\gamma$. Furthermore, if $\nu$ is continuous, then the solution is uniquely determined by the subdifferential of the concave function $u_G(y)=\inf_{z\in G}\{c(z,y)-y_1y_2\}$. The presentation is based on a joint paper with Yan Xu.

Daniel Lacker

Title. Competitive optimal investment under relative performance criteria.

Abstract. We analyze a family of portfolio management problems under relative performance criteria, for fund managers having CARA or CRRA utilities and trading in a common investment horizon in log-normal markets. We construct explicit equilibrium strategies for both the N-agent games and the corresponding mean field games, which we show are unique in the class of constant equilibria--as in the classical Merton problem, agents invest a constant fraction (CRRA) or amount (CARA) of wealth in the risky asset. In the CARA case, competition drives agents to invest more in the risky asset than they would otherwise, while in the CRRA case competitive agents may over- or under-invest, depending on their levels of risk tolerance. Joint work with Thaleia Zariphopoulou.

Kasper Larsen

Title. Smart TWAP trading in continuous-time equilibria.

Abstract. This paper presents a continuous-time equilibrium model of TWAP trading and liquidity provision in a market with multiple strategic investors with heterogeneous intraday trading targets. We solve the model in closed-form and show there are infinitely many equilibria. We compare the competitive equilibrium with different non-price-taking equilibria. In addition, we show intraday TWAP benchmarking reduces market liquidity relative to just terminal trading targets alone. The model is computationally tractable, and we provide a number of numerical illustrations. An extension to stochastic VWAP targets is also provided. Joint work with Jin Hyuk Choi and Duane J. Seppi.

Roger Lee

Title. Cumulant Formulas for Implied Volatility.

Abstract. Expressing option prices as Black-Scholes implied volatilities reveals features of the underlying probability distribution. We prove a formulation of this idea in a near-Gaussian asymptotic regime where we relate the shape of the implied volatility skew to the cumulants of the underlying distribution.

Zongxi Li

Title. Portfolio Feedback.

Abstract. We study portfolio optimization problem with feedback effect. The feedback effect is introduced through a market clearing condition, which is the supply and demand equation. We characterize an endogenous market equilibrium, including the optimal investment strategy and the dynamics of the market price, using coupled nonlinear PDEs. Furthermore, we prove the existence of an equilibrium and discuss its uniqueness. Our work provides details on solving endogenous equilibrium in power utility maximization problem under finite time horizon.

Alexander Lipton

Title. Stochastic Processes in Domains with Boundaries and Some of Their Financial Applications.

Abstract. In this talk we consider two connected problems. First, we study the classical problem of the first passage hitting density of an Ornstein-Uhlenbeck process. We give two complementary (forward and backward) formulations of this problem and provide semi-analytical solutions for both. The corresponding problems are comparable in complexity. By using the method of heat potentials, we show how to reduce these problems to linear Volterra integral equations of the second kind. For small values of t we solve these equations analytically by using Abel equation approximation; for larger t we solve them numerically. We also provide a comparison with other known methods for finding the hitting density of interest, and argue that our method has considerable advantages and provides additional valuable insights. Second, we study the non-linear diffusion equation associated with a particle system where the common drift depends on the rate of absorption of particles at a boundary. We provide an interpretation as a structural credit risk model with default contagion in a large interconnected banking system. Using the method of heat potentials, we derive a coupled system of Volterra integral equations for the transition density and for the loss through absorption. An approximation by expansion is given for a small interaction parameter. We also present a numerical solution algorithm and conduct computational tests. Joint work with V. Kaushansky and C. Reisinger, University of Oxford.

Wiejie Pang

Title. XVA Valuation under market illiquidity.

Abstract. Before the 2008 financial crisis, most option pricing methods ignored the effects of counterparties’ default and funding illiquidity. Recently models were proposed to compute the total valuation adjustment (XVA) of a European claim, including funding costs, counterparty credit risk and collateralization. However, those models abstract from an important fact: the repo market froze during the 2008 financial crisis, because of the rarity of general collateral and loss of confidence in other collaterals. The frozen repo market led to a shutdown of short trades in stock. Thus, it’s very important to include the different behavior of repo and stock market in normal and financial crisis status. In our research we describe the switching between two financial status by an alternating renewal process, which switches between zero and one with inter-arrival times following two exponential distributions. We develop a framework for pricing the XVA of a European claim in this state-dependent framework. We show the existence of a unique classical solution to the pricing BSDE based on a martingale decomposition theorem on a space generated by not-independent increment stochastic processes.

Andrew Papanicolaou

Title. Statistics of VIX futures and their applications to trading volatility exchange-traded products.

Abstract. In this paper, we study the dynamics of Chicago Board Options Exchange volatility index (VIX) futures and exchange-traded notes (ETNs)/exchange-traded funds (ETFs). We find that, unlike classical commodities, the VIX and VIX futures exhibit high volatility and skewness, consistent with the absence of cash-and-carry arbitrage. The constant-maturity futures (CMFs) term structure can be modeled as a stationary stochastic process in which the most likely state is contango with VIX \approx 12% and a long-term futures price VIX^\infty \approx 20%. We analyze the behavior of ETFs and ETNs based on constant-maturity rolling futures strategies, such as VXX, XIV and VXZ, assuming stationarity, and through a multifactor model calibrated to historical data.We find that buy-and-hold strategies consisting of shorting ETNs that roll long futures, or buying ETNs that roll short futures, will produce theoretically sure profits if it is assumed that CMFs are stationary and ergodic. To quantify further, we estimate a two-factor lognormal model with mean-reverting factors for VIX and CMF historical data from 2011 to 2016. The results confirm the profitability of buy-and-hold strategies but also indicate that the latter have modest Sharpe ratios, of the order of SR \leq 0.5, and high variability over one-year horizon simulations. This is due to the surges in VIX and CMF backwardations that are experienced sporadically, but also inevitably, in the volatility futures market.

Mykhaylo Shkolnikov

Title. Particles interacting through the hitting times: neuron firing, supercooling and systemic risk.

Abstract. I will discuss a class of particle systems that serve as models for supercooling in physics, neuron firing in neuroscience and systemic risk in finance. The interaction between the particles falls into the mean field framework pioneered by McKean and Vlasov in the late 1960s, but many new phenomena arise due to the singularity of the interaction. The most striking of them is the loss of regularity of the particle density caused by the self-excitation of the system. In particular, while initially the evolution of the system can be captured by a suitable Stefan problem, the following irregular behavior necessitates a more robust probabilistic approach. Extensions to the setting where the interaction takes place on networks will be also discussed. Based on joint works with Sergey Nadtochiy.

Florian Stebegg

Title. Fine Properties of Skorokhod Embedding.

Abstract. Optimal Skorokhod Embedding has been established as a tractable approach to continuous-time optimal transport with martingale constraints. The currently used formulations of the dual problem fail to provide existence of dual optimizers. We will show that it cannot hold in general and provide an alternative formulation for the dual problem that recovers dual existence. This allows us to extend the proof of duality from semi-continuous to general measurable payoffs. (joint work with Mathias Beiglböck and Marcel Nutz)

Stanislav Uryasev

Title. Risk Management with POE, VaR, CVaR, and bPOE: Applications in Finance.

Abstract. Joint paper with Giorgi Pertaia. This paper compares four closely related probabilistic measures: Probability of Exceedance (POE), Value-at-Risk (VaR) which is a quantile, Conditional Value-at-Risk (CVaR), and Buffered Probability of Exceedance (bPOE). The Probability of Exceedance (POE) is frequently used to measure uncertainties in outcomes. For instance, POE is used to estimate probability that assets of a company fall below liabilities. POE measures only the frequency of outcomes and ignores magnitude of outcomes. POE counts outcomes exceeding the threshold, and it “does not worry” about the amount by which each outcome exceeds the threshold. POE is lumping together all threshold exceedance events, potentially “hiding” quite large and very troublesome outcomes. Moreover, POE has poor mathematical properties when used to characterize discrete distributions of random values (e.g., when distributions are defined by observed historical data). POE for discrete distributions is a discontinuous function of control variables, making it difficult to analyze and optimize. POE is used for defining financial ratings of companies and financial derivative instruments (such as CDO). This presentation discusses a new probabilistic characteristic called Buffered Probability of Exceedance (bPOE). With bPOE, it is possible to count outcomes close to a threshold value, rather than only outcomes exceeding the threshold. To be more precise, bPOE counts tail outcomes averaging to some specific threshold value. For instance, 4% of land-falling hurricanes in US have cumulative damage exceeding $50 billion (i.e., POE = 0.04 for threshold=$50 billion). It is estimated, that the average damage from the worst 10% of hurricanes is $50 billion. In terms of bPOE, we say bPOE=0.1 for the threshold=$50 billion. bPOE shows that the largest damages having magnitude around $50 billion have frequency 10%. bPOE can be considered as an important supplement to POE. We think that bPOE should be routinely calculated together with POE. This example shows that bPOE exceeds POE, which is why it is called Buffered Probability of Exceedance. The positive difference, bPOE-POE, can be interpreted as some “buffer.” bPOE is an inverse function of Conditional Value-at-Risk (CVaR); it inherits a majority of exceptional mathematical properties of CVaR (which is a so called “coherent measure of risk”). Similar to CVaR, minimization of bPOE can be reduced to Convex and Linear Programming. We will discuss applications of bPOE concept in finance. In particular, we will consider a Cash Matching Problem of a Bond Portfolio and so called Buffered Financial Ratings.

Gu Wang

Title. High-water Mark Fees with Stochastic Benchmark.

Abstract. A hedge fund manager invests the fund in a constant investment opportunity, and receives high-water mark fees when the fund reaches a new maximum relative to a stochastic benchmark, aiming to maximize the expected power utility from fees in the long run. The manager's optimal portfolio includes a Merton component with the risk aversion parameter shifted towards one, and a hedging component against the risk in the benchmark, both of which depend on how the fund investment opportunity compares to the benchmark. A stochastic benchmark in the calculation of high-water mark fees introduces the moral hazard of risky fund investment with hedging motives, but also provides a possibility of regulating the manager's risk taking with a carefully chosen benchmark.

Kevin Webster

Title. Leland Strategy with both market and limit orders.

Abstract. The Leland strategy proposes a discretization of the Black and Scholes delta-hedging strategy for European options under the presence of transaction costs. The talk revisits this problem in a market where limit orders and market orders are used simultaneously to trade. Adverse selection of the limit orders by other market participants plays a crucial role. Adverse selection drives the profitability of limit orders down, while providing hedging benefits for negative gamma options. This insight leads to an explicit execution strategy for delta-hedging after buying options.

Jiongmin Yong

Title. Recursive Utility, Dynamic Risk Measure, and Backward Stochastic Volterra Integral Equations.

Abstract. Continuous-time recursive utility process (or stochastic differential utility) for a given future payoff can be represented by adapted solutions to backward stochastic differential equations (BSDEs, for short). At the same time, dynamic risk measure for a terminal payoff can also be represented by the adapted solution to BSDEs. Now, for a financial position process (which is not just a random variable and not necessary adapted), we would also like to define the recursive utility process and/or dynamic risk measure process. This can be successfully done by introducing the so-called backward stochastic Volterra integral equations (BSVIEs, for short). It turns that with BSVIEs, we obtain the equilibrium recursive utility process and equilibrium dynamic risk measure process, which are all time-consistent.

Lan Zhang

Title. The Five Trolls Under the Bridge: Principal Component Analysis with Asynchronous and Noisy High Frequency Data.

Abstract. We develop a principal component analysis (PCA) for high frequency data. As in Northern fairly tales, there are trolls waiting for the explorer. The first three trolls are market microstructure noise, asynchronous sampling times, and edge effects in estimators. To get around these, a robust estimator of the spot covariance matrix is developed based on the Smoothed TSRV (Mykland et al (2017)). The fourth troll is how to pass from estimated time-varying covariance matrix to PCA. Under finite dimensionality, we develop this methodology through the estimation of realized spectral functions. Rates of convergence and central limit theory, as well as an estimator of standard error, are established. The fifth troll is high dimension on top of high frequency, where we also develop PCA. With the help of a new identity concerning the spot principal orthogonal complement, the high-dimensional rates of convergence have been studied after eliminating several strong assumptions in classical PCA. As an application, we show that our first principal component (PC) closely matches but potentially outperforms the S&P 100 market index, while three of the next four PCs are cointegrated with two of the Fama-French non-market factors. From a statistical standpoint, the close match between the first PC and the market index also corroborates this PCA procedure and the underlying S-TSRV matrix. (Joint with Dachuan Chen and Per Mykland).