Speaker: Peter Bank

Title: Limit order book models and optimal order execution

Abstract:

We present and analyze some simple diffusion based models for the stochastic dynamics of limit order books and discuss a few of their implications for optimal order placement strategies.

Speaker: Peter Carr

Title: Optionality and Volatility

Abstract:

I formalize the notion of the optionality of an option. This notion is used to define "money vol" as an arbitrage-free alternative to the implied volatility smile.

Speaker: Erik Ekström

Title: Can time-homogeneous diffusions produce any distribution?

Abstract:

Given a centered distribution, can one find a time-homogeneous martingale diffusion starting at zero which has the given law at time 1? We answer the question affirmatively if generalized diffusions are allowed, and we discuss applications in mathematical finance. This is joint work with David Hobson, Svante Janson and Johan Tysk.

Speaker: Vicky Henderson

Title:Executive Stock Options: Portfolio Effects

Abstract:

Over time, executives accumulate multiple grants of stock options with differing strike prices and times-to-maturity but face unhedgeable risk due to trading restrictions. We use CARA utility indifference pricing and solve the resulting free boundary problems numerically for the optimal exercise thresholds and shareholder costs for such portfolios. We show that the optimal exercise strategy for a risk-averse executive is much more complex than in a complete market. Exercise thresholds for a particular option can be upward-sloping and convex over time and exhibit discontinuities if the optimal exercise order changes. Shareholder costs of individual options and of portfolios can be significantly lowered relative to costs obtained by considering options on a standalone basis. The cost of a particular option depends non-monotonically on the strike price(s) of the other option(s) in the executive's portfolio. Since the unexpected grant of a new option can affect the cost of options already granted, the incremental cost of such a new grant can differ from its cost as part of the new portfolio. Joint work with Jia Sun and Elizabeth Whalley (Warwick Business School)

Speaker: Dmitry Kramkov

Title: Integral representation of martingales and endogenous completeness of financial models

Abstract:

Speaker: Arshak Petrosyan

Title: Elliptic and parabolic obstacle problems with thin and Lipschitz obstacles

Abstract:

We will discuss elliptic and parabolic obstacle problems with thin and Lipschitz (nonconvex, piecewise smooth) obstacles. Such problems appear in several contexts, including the valuation of American options on multiple assets (e.g. min-options). I will give an overview of some recent results in such problems, including the optimal regularity of the solutions and the regularity of the free boundary. One of the main technical tools is the generalization of Almgren's and Poon's frequency formula.

Speaker: Philip Protter

Title: Detecting financial bubbles in real time

Abstract:

After the 2007 credit crisis, financial bubbles have once again emerged as a topic of current concern. An open problem is to determine in real time whether or not a given asset's price process exhibits a bubble. To do this, one needs to use a mathematical theory of bubbles, which we have recently developed and will briefly explain. The theory uses the arbitrage-free martingale pricing technology. This allows us to answer this question based on the asset's price volatility. We limit ourselves to the special case of a risky asset's price being modeled by a Brownian driven stochastic differential equation. Such models are ubiquitous both in theory and in practice. Our methods use sophisticated volatility estimation techniques combined with the method of reproducing kernel Hilbert spaces. We illustrate these techniques using several stocks from the alleged internet dot-com episode of 1998 - 2001, where price bubbles were widely thought to have existed. Our results support these beliefs. We then consider the special case of the recent IPO of LinkedIn. The talk is based on several joint works with Robert Jarrow, Kazuhiro Shimbo, and Younes Kchia.

Speaker: Henrik Shahgholian

Title: Problems in mathematical finance with free boundaries

Abstract:

I shall discuss certain techniques in free boundaries related to American type contracts. Such techniques are of general nature and have the advantage of being applied to general framework in applications. Our focus shall be on the behavior of the solution function as well as the free boundary, close to initial state (American Option) or close to Dirichlet data (Convertible Bonds).

Speaker: Steven Shreve

Title: Utility maximization trading two risky assets with transaction costs

Abstract:

We consider an agent who optimally invests and consumes in the presence of proportional transaction costs. The agent invests in two types of futures contracts, modeled as correlated arithmetic Brownian motions, and in a money market account with a constant rate of interest. She also consumes and obtains utility from this consumption. The utility of consumption rate C is C to the power p, where p is between zero and one. The agent maximizes expected infinite-horizon discounted utility from consumption. A proportional transaction cost is charged for trading in the futures contracts. We compute an asymptotic expansion of the value function for positive transaction cost about the known value function for the case of zero transaction cost. The leading term in the expansion is a computable constant times the transaction cost to the 2/3 power. The method of solution when the futures are uncorrelated follows a method used previously to obtain the analogous result for one risky asset. However, when the futures are correlated, a new methodology must be developed. It is suspected in this case that the value function is not twice continuously differentiable, and this prevents application of the methodology that works for the one-dimensional model. This is joint work with Maxim Bichuch.

Speaker: Mihai Sirbu

Title: Optimal investment in a hedge-fund and multiple correlated assets

Abstract:

We consider the problem of optimally investing in a hedge-fund charging high-watermark fees and a number of correlated assets. We include interest rates and hurdles in the model. We solve the stochastic control problem using dynamic programming. They key ingredient is a careful identification of the two-dimensional state process using Skorohod representation. The presentation is based on joint work with G. Brunick and K. Janecek.

Speaker: Johan Tysk

Title: Boundary behavior of densities for non-negative diffusions

Abstract:

It is well-known that the transition density of a di ffusion process solves the corresponding Kolmogorov forward equation. If the state space has finite boundary points, then naturally one also needs to specify appropriate boundary conditions when solving this equation. However, many processes in finance have degenerating diffusion coefficients, and for these processes the density may explode at the boundary. We describe a simple symmetry relation for the density that transforms the forward equation into a backward equation, the boundary conditions of which being much more straightforward to handle. This relation allows us to derive new results on the precise asymptotic behavior of the density at boundary points where the diffusion degenerates. This is joint work with Erik Ekström

Speaker: Andrew Barnes

Title: Conditional expected default rate calculations for credit risk applications

Abstract:

Calculation of portfolio loss distributions is an important part of credit risk management in all large banking institutions. Mathematically, this calculation is tantamount to efficiently computing the probability distribution of the sum of a very large number of correlated random variables. Typical Monte Carlo aggregation models apply brute force computation to this problem and suffer from two main drawbacks: lack of speed and lack of transparency for further credit risk analysis.
We use an asymptotic probabilistic model based on the Central Limit Theorem for solving the portfolio risk aggregation problem for credit risky portfolios. We then prove a theorem that enables us to efficiently compute the conditional expectation of the default rate for any subportfolio, conditioned on the total portfolio loss. This calculation is used in a variety of stress testing and multi-period risk analyses. The theorem also allows us to solve the capital allocation problem (using expected shortfall as the risk measure) without resorting to Monte Carlo simulation. The approach is very efficient, even for portfolios with several million positions, as is typical for the portfolios of large financial institutions.

Speaker: Maxim Bichuch

Title: Pricing a contingent claim liability using asymptotic analysis for optimal investment in finite time with transaction costs

Abstract:

We price a contingent claim liability using the utility indifference argument. We consider an agent who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time $T$ in the presence of a proportional transaction cost $\varepsilon>0$ in two cases with and without a contingent claim liability. The utility function is of the form $\U (x)=\Util-e^{-\gamma x},~x\in\R, ~\gamma>0$. Using the computations from the heuristic argument in Whalley \& Wilmott, we provide a rigorous derivation of the asymptotic expansion of the value function in powers of $\varepsilon^{1/3}$ in both cases with and without a contingent claim liability. Additionally, using utility indifference method we derive the price of the contingent claim liability up to order $\varepsilon$. In both cases, we also obtain a ``nearly optimal" strategy, whose utility asymptotically matches the leading terms of the value function.

Speaker: Gerard Brunick

Title: Weak uniqueness for a class of degenerate diffusions with continuous covariance

Abstract:

Motivated by the problem of calibrating linear pricing rules to the market prices of options, we provide a new weak uniqueness result for degenerate diffusions. In particular, we consider path-dependent stochastic differential equations in which the diffusion coefficient is a function of both the current location of the process and the running integral of the process, and we show that uniqueness holds for continuous, strictly positive-definite diffusion coefficients. These results combine tools from the theory of singular integrals on Lie groups with the localization machinery of Stroock and Varadhan.

Speaker: Ionut Florescu

Title: Numerical solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market

Abstract:

We present two numerical algorithms to calculate the solution of an integro-differential parabolic problem coming from a model with jumps and stochastic volatility. One of the algorithm builds on our proof of the theorem of existence of such solutions and the other is a more traditional finite element scheme. The algorithms are implemented in PDE2D a general purpose PDE solver. Joint work with Maria C. Mariani and Granville Sewell from University of Texas at El Paso

Speaker: Andreas Hamelg

Title: A set-valued approach to utility maximization in markets with transaction costs

Abstract:

Transaction costs, liquidity constraints and other market frictions may have the effect that some portfolios are not comparable with others. Therefore, it does not make sense to work with a scalar utility function which assumes a total preference relation. Even if vector-valued utility functions are used the lack of an applicable duality theory frustrates the solution of the vector utility maximization problem. Instead, we will use a new set-valued Lagrangian and a corresponding set-valued duality theorem to give a complete solution for the case of a finite probability space. Surprisingly, the simple set-valued functions which replace the dual variables have a precise economical interpretation. This is a joint work with Sophie Qingzhen Wang from Princeton University.

Speaker: Xuedong He

Title: Optimal Insurance Design under Rank Dependent Utility

Abstract:

We derive explicitly the optimal insurance contract for an individual behaving according to rank dependent utility as described in Quiggin (1993). The utility function is assumed to be concave and the probability distortion is reversed S-shaped. We apply the technique of quantile formulation to solve this problem thoroughly, and we show that the optimal contract fully insures small losses and insures large losses above a deductible. We finally compare, both analytically and numerically, our result with those of models having convex or concave distortions. This is a joint work with Carole Bernard at University of Waterloo, Jia-an Yan at Chinese Academy of Science, and Xunyu Zhou at Oxford and CUHK

Speaker: Abdul Khaliq

Title: Efficient numerical schemes for pricing exotic path-dependent American options with transaction cost

Abstract:

Financial markets are becoming more and more complex with trading not only of stocks, but also of numerous types of financial derivatives. Options and derivative securities account for more than half the modern market and the basic tools for risk hedging in any portfolio management. The development of mathematical models to understand the relationship among complicated financial instruments has enabled the proliferation of these instruments which enhance the efficiency of worldwide capital markets. With the rapid increase in sophisticated quantitative and computational techniques employed in financial firms, it then follows that development of new mathematical and computational techniques for the accurate evaluation of complex financial models have considerable financial worth in addition to constituting cutting-edge research. Valuation of exotic options, such as options with multiple strike prices, complex digital options and barrier options is particularly challenging for traditional computational techniques which can perform inaccurately due to the discontinuities in the payoff functions or its derivatives. Large errors may also occur in estimating the hedging parameters such as delta and gamma values, even though the prices appear to be correct. The non-smooth data can further lead to serious degradation in the convergence of the numerical schemes. We present new efficient numerical methods for solving a non-linear Black-scholes PDE model with transaction cost for pricing and hedging exotic path dependent options. The strong stability of methods can successfully deal with high volatility as well discontinuities in the payoffs.

Speaker: Philibert Kongtcheu

Title: Method for optimizing inequality and equality constrained resources allocation problems in industrial applications

Abstract:

In this talk I will be presenting a new algorithm for solving inequality and equality constrained optimization algorithms that fall within the Karush KuhnTucker Theorem applicability conditions with exact solutions. It has been published previously as a patent application which has just been allowed for issue as a patent (See attached). It is an important method that will be particularly useful for finance practitioners dealing with portfolio optimization and portfolio hedging issue.

Speaker: Kasper Larsen

Title: Unspanned endowment and face-lifting

Abstract:

We study the relation between optimizing over finite additive and countable additive probability measures. Standard arguments show that the optimizer is always attained in the finite additive class. We will then discuss and derive the influence a possible singular component has on the value function close to maturity. More specifically, in general incomplete Brownian settings, we explicitly identify the face-lift (boundary layer) in the value function. This is joint with Gordan Zitkovic.

Speaker: Tim Leung

Title: Derivatives purchase timing under risk-neutral and risk-averse pricing rules

Abstract:

We study the problem of optimal timing to buy/sell derivatives by risk-averse and risk-neutral agents in incomplete markets. In the risk-averse case, we adopt the indifference pricing mechanism to investigate the investor's timing of trades. This leads to a stochastic control and optimal stopping problem that combines the market price dynamics and the agent's risk preferences. In the zero risk-aversion limit, we observe a phenomenon where the investor and the market are pricing under different equivalent martingale measures. This motivates a formulation of price discrepancy in a general incomplete market. We introduce the delayed purchase premium and study the associated variational inequalities. The optimal timing is illustrated numerically for both equity and credit derivatives.

Speaker: Matthew Lorig

Title: Pricing derivatives on multiscale diffusions: simplicity through spectral theory

Abstract:

Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a derivative-asset. The payoff of the derivative-asset may be path-dependent. Additionally, the process underlying the derivative may exhibit killing (i.e. jump to default) as well as combined local/nonlocal stochastic volatility. The nonlocal component of volatility is multiscale, in the sense that it is driven by one fast-varying and one slow-varying factor. The flexibility of our modeling framework is contrasted by the simplicity of our method. We reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once the eigenvalue equation is solved, the approximate price of a derivative can be calculated formulaically. To illustrate our method, we calculate the approximate price of a call option in the jump-to-defualt CEV model of Carr and Linetsky with multiscale volatility.

Speaker: Oleksii Mostovyi

Title: Necessary and sufficient conditions in the problem of optimal consumption from investment in incomplete markets

Abstract:

We consider the problem of maximizing the expected utility of consumption and terminal wealth in the framework of a general incomplete semimartingale model of a financial market. Our goal is to find minimal conditions on the model and the utility stochastic field for the validity of several key assertions of the theory to hold true. We show that the necessary and sufficient conditions on both, the utility stochastic field and the model, are that the value functions of the primal and dual problems are finite.

Speaker: Michael Okelola

Title: Group analysis of exotic options

Abstract:

Exotic options are derivatives which have features that makes them more complex than commonly traded products - thus finding their fair value is not an always easy task. Due to their exotic and high return (and associated high risk) nature, these options are unarguably the fastest growing segment of the options market today. We will study one of these exotic options viz. the power options. This is a class of exotic options similar to the European option, but with an additional feature in which the pay-off at expiry is related to a factor ?, of the stock price S. We shall consider the particular case of this problem with boundary conditions V (S, T ) = 0; providing new solutions to this problem via the group analysis approach.

Speaker: Camelia Pop

Title: Mimicking theorem for generalized Heston-like processes

Abstract:

We prove existence, uniqueness and regularity results for a certain class of degenerate elliptic partial differential equations. We use these results to build generalized Heston-like processes which match the 1-dimensional marginal distributions of a certain class of Itô processes. The mimicking process is the unique weak solution to a stochastic differential equation and it possesses the strong Markov property. This is joint work with P. Feehan.

Speaker: Birgit Rudloff

Title: Superhedging in markets with transaction costs

Abstract:

In this talk, we will show how to calculate the set of initial endowments that allow to superhedge a European option in a d-asset market with transaction costs, as well as a method to calculate superhedging strategies. This leads to a sequence of linear vector optimization problems solved by Benson's algorithm. We will show that the representation of the scalar superhedging price is related to the set-valued problem by geometric duality.

Speaker: Hasanjan Sayit

Title: A note on consistent price systems

Abstract:

In a recent paper Guasoni, Rasonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008), pp. 491-520] introduced the conditional full support property (CFS) and showed that for strictly positive unbounded continuous processes the CFS property implies the existence of consistent price systems (CPSs). In this note we generalize this result and show that the CFS property implies CPSs for general (bounded and unbounded) continuous processes. We also give equivalent and elementary formulations of the CFS property. (Joint work with Razvan Maris and Eric Mbakop.)

Speaker: Alexander Shklyarevsky

Title: Analytical approaches to the solution of ODEs, PDEs and PIDEs and their application in physics and quantitative finance

Abstract:

We would like to present an overview of our latest developments in analytical approaches to the solution of ordinary differential equations (ODEs), partial differential equations (PDEs) and partial integro-differential equations (PIDEs) in light of their application in physics and quantitative finance, specifically to pricing and risking derivative securities and their portfolios and giving examples of structured and non-structured products. These analytical approaches include using certain integral transforms, non-integral transforms, operator theory and other functional analysis methodologies

Speaker: Jian Song

Title: A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution

Abstract:

In this paper, we establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with spatially correlated noise. For a class of stochastic heat equations, we study the Hölder continuity of the solutions, and get an explicit expression for the Malliavin derivatives of the solutions by using the Feynman-Kac formula. Based on the above result and the result from the Malliavin calculus, we obtain the smooth property of the solutions.

Speaker: Guan Jun Wang

Title: Explaining the forward rate bias puzzle

Abstract:

The relationship between forward exchange rates and subsequent future spot rate has received extensive theoretical and empirical attention in the literature in the past three decades. General consensus is that forward rates are not very good predictors of future spot rates. This is widely referred to forward bias puzzle. Numerous scholars attempt to explain this important puzzle, however there is no widely accepted solution. In this paper, we first discuss the forward rate and spot rate determination and the rationale of the forward exchange rate unbiasedness hypothesis, and then we examine the methodologies used in the existing studies on forward exchange rate unbiasedness hypothesis testing. We point out the serious flaws in the existing methodologies and suggest better testing methods by using use Jensen’s Inequality. We propose an alternative test method for forward exchange rate unbiasedness hypothesis. Our results from these tests indicate that the foreign exchange market is efficient despite decisive clear rejections of forward exchange rate unbiasedness hypothesis using the conventional approach.

Speaker: Olympia Hadjiliadis

Title: Preventing market crashes through insuring the speed of drawdowns

Abstract:

In this work we derive analytical formulas for the joint distribution of the drawdown, the last visit time of the maximum of a process preceding the drawdown and the maximum of the process under general diffusion dynamics. The motivation for this work arises in the financial risk management of drawdowns. Drawdowns measure the first time the current drop of an investor’s wealth from its historical maximum reaches a pre-specified level. Therefore a quantity related to the drawdown is the duration of time between the drawdown and the last time at which the maximum was achieved. This quantity is studied here.