Title: Self-financing atlas submodels and stock market structure
Abstract:
An Atlas submodel of a stock market is a subset of the market that has the structure of an Atlas model. An Atlas submodel is self-financing if the redistribution of capital within the submodel does not alter its total capitalization. Self-financing Atlas submodels will migrate to certain ranks within the capital distribution, and this can stabilize the market as well as induce concavity in the capital distribution curve.
Speaker: Ruoting Gong
Title: Small-time asymptotics for lévy-based jump-diffusion models
Abstract:
In recent years, small-time asymptotic methods have attracted much attention in mathematical finance. Such asymptotics are especially crucial for jump-diffusion models due to the lack of closed-form formulas and efficient valuation procedures. These methods have been widely developed and applied to diverse areas such as short-time approximations of option prices and implied volatilities, and non-parametric estimations based on high-frequency data. In this talk, I will discuss some results on the small-time asymptotic behavior of some Lévy functionals with applications in finance.
Speaker: Paolo Guasoni
Title: The limits of leverage
Abstract:
When trading incurs proportional costs, leverage can scale an asset’s return only up to a maximum multiple, which is sensitive to the asset’s volatility and liquidity. In a continuous-time model with one safe and one risky asset with constant investment opportunities and proportional transaction costs, we find the efficient portfolios that maximize long term expected returns for given average volatility. As leverage and volatility increase, rising rebalancing costs imply a declining Sharpe ratio. Beyond a critical level, even the expected return declines. For funds that seek to replicate multiples of index returns, such as leveraged ETFs, our efficient portfolios optimally trade off alpha against tracking error.
Speaker: Matt Lorig
Title: Indifference prices, implied volatilities and implied sharpe ratios
Abstract:
We consider a general local-stochastic volatility model and an investor with exponential utility. For a European-style contingent claim, we derive an explicit approximation for both the buyer’s and seller’s indifference price. Additionally, we translate indifference prices into an explicit approximation of implied volatility surfaces. We also introduce the concept of an ”implied Sharpe ratio” and derive explicit approximations for this quantity. Like implied volatility, the implied Sharpe ratio can be viewed as a measure of an option’s value. The advantage of the implied Sharpe ratio is that, unlike implied volatility, it takes into account an investor’s preferences and his alternative investment possibilities.
Speaker:
Marcel Nutz
Title:
Optimal transport and robust finance
Abstract:
Abstract: After a brief introduction to classical optimal transport, we shall focus on the so-called martingale optimal transport and its connection to finance, the problem of robust semi-static hedging. Some differences with the classical transport problem will be highlighted, in particular the failure of duality in the usual sense. We explain how to obtain a complete duality theory using notions related to Knightian uncertainty about pricing models. Based on joint work with Mathias Beiglb ̈ock and Nizar Touzi.
Speaker: Huyên Pham
Title:
Robust feedback switching control problem
Abstract:
We study a robust switching control problem where the controller only observes the evolution of the state process, and thus uses feedback (closed-loop) switching strategies, a non standard class of switching controls introduced in this work. The adverse player, which can be interpreted as the nature, chooses open-loop controls that represent Knightian uncertainty, and misspecification of the model. The (half) game switcher/nature is then formulated as a two-step (robust) optimization problem. In the first part of this talk, we present the stochastic Perron’s method for this control problem, and prove that it produces a viscosity sub and super solution to a system of variational inequalities of Hamilton-Jacobi-Bellman-Isaacs (HJBI) type, such that the value function is squeezed between them. Together with a comparison principle, this characterizes the value function of the game as the unique viscosity solution to the HJBI equation, and shows as a byproduct the dynamic programming principle for robust feedback switching control problem. The second part of the talk is devoted to the asymptotic behavior analysis for the HJBI equation, both parabolic and elliptic, arising from robust switching control problem. We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution to a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities. This is based on joint works with Erhan Bayraktar (University of Michigan) and Andrea Cosso (University Paris Diderot).
Speaker: Mykhaylo Shkolnikov
Title:
Two approaches to portfolio choice
Abstract:
We will discuss two approaches to optimal investment in equity markets. The first one is based on descriptive models of large stock markets and relies on a detailed analysis of the redistribution of capital among the listed companies. The second one relies on forward performance processes that incorporate the changes of investor’s preferences over time. In both cases there is a crucial non-linear partial differential equation in the background that gives a detailed understanding of the portfolio performance.
Speaker: Agnès Sulem
Title:
Stochastic control with nonlinear expectations
Abstract:
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Markovian stochastic control problems on a given horizon of time $T$ can typically be written as
$$
\sup_{\alpha \in \mathcal{A}}\mathbb{E}[\int_0^T g(\alpha_s,X_s^\alpha)ds+h(X_T^\alpha)],
$$
where $\mathcal{A}$ is a set of admissible control processes $\alpha_s$, and
$(X_s^\alpha)$ is a controlled jump diffusion process.
The random variable $h(X_T^\alpha)$ may represent a terminal reward and $g(\alpha_s,X_s^\alpha)$ an instantaneous reward process.
Using dynamic programming principle and under appropriate assumptions, the associated value function of the stochastic control problem can then be characterized as the solution, in a certain sense, of a Hamilton-Jacobi-Bellman equation.
We are interested here in generalizing these results to the case when the linear expectation $\mathbb{E}$ is replaced by a nonlinear expectation induced by a Backward Stochastic Differential Equation (BSDE).
Typically, such problems in the Markovian case can be formulated as
$$
\sup_{\alpha \in \mathcal{A}} {ε}_{0, T}^\alpha[h(X_T^{\alpha})],
$$
where ${ε}^\alpha$ is the nonlinear conditional expectation associated with a BSDE with jumps with controlled driver $f(\alpha_t,X_t^\alpha,y,z,k)$.
Note that when the driver $f$ does not depend on the solution of the BSDE, that is when $f(\alpha_t,X_t^\alpha,y,z,k) \equiv g(\alpha_t,X_t^\alpha)$, then we are back to the classical linear expectation case.
In this talk, we shall consider the case when there is an additional control in the form of a stopping time. We shall thus study
mixed generalized optimal control/stopping problems of the form
$$
\sup_{\alpha \in \mathcal{A}} \sup_{\tau \in \mathcal{T}} {ε}_{0, \tau}^\alpha[h(X_\tau^{\alpha})],
$$
where $\mathcal{T}$ denotes the set of stopping times.
We first establish a dynamic programming principle (DPP)
for our mixed problem with $f$-expectation. This requires some specific techniques of stochastic analysis and BSDEs to handle measurability and other issues due to the nonlinearity of the expectation.
The value function of the problem is not a solution of a reflected BSDE and
hence the DPP dos not follow from the flow property for
reflected BSDEs.
The method we propose allows us to handle the case when the value function is not measurable, thus leading to a weak dynamic programming principle.
Using this principle and properties of reflected BSDEs, we then prove that the value function of our mixed problem is a viscosity solution of
a generalized Hamilton-Jacobi-Bellman (HJB) variational inequality.
Uniqueness of the viscosity solution is obtained under additional assumptions. Illustrating examples in mathematical finance are provided.
\
Joint work with Roxana Dumitrescu and Marie-Claire Quenez
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Speaker: Martin Keller-Ressel
Title: Implied volatilities from strict local martingales
Abstract:
Several authors have proposed to model price bubbles in stock markets by specifying a strict local martingale for the risk-neutral stock price process. Such models are consistent with absence of arbitrage (in the NFLVR sense) while allowing fundamental prices to diverge from actual prices and thus modeling investors’ exuberance during the appearance of a bubble. We show that the strict local martingale property as well as the “distance to a true martingale” can be detected from the asymptotic behavior of implied option volatilities for large strikes, thus providing a model-free asymptotic test for the strict local martingale property of the underlying. This talk is based on joint work with Antoine Jacquier.
Speaker: Kavita Ramanan
Title:
A free boundary problem arising in order book dynamics
Abstract:
We describe and analyze a free boundary problem that arises in a model of order book dynamics with strategic agents proposed by Rosu. We show that this problem can be reformulated in terms of a more classical free boundary problem, and use special features of our problem to establish uniqueness and additional regularity properties that do not follow from the general theory.
The free boundary also admits a stochastic interpretation as the locus of certain points associated with a certain reflected Brownian motion. This makes its study of independent interest. This is joint work with Lukasz Kruk.
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: Ram Sharan Adhikari
Title:
A weak simpson method for a class of stochastic differential equation and numerical stability results
Abstract:
We propose a novel weak Simpson method for numerical solution for a class of stochastic differential equations. We show that such a method has weak order of convergence three, which improves the weak convergence order of two for the weak trapezoid method developed in Anderson and Mattingly (2011). This work also aims to determine mean-square stability region of the weak Simpson method for linear stochastic differential equations with multiplicative noises. In this work, a mean-square stability region of the weak Simpson scheme is identified, and step-sizes for the numerical method where errors propagation are under control in well-defined sense are given. The main results are illustrated with the numerical examples.This is a joint work with professors Chao Zhu and Bruce A. Wade.
Speaker: Maxim Bichuch
Title: Arbitrage-free pricing of XVA
Abstract:
We introduce a framework for computing the total valuation adjustment (XVA) of an European claim accounting for funding costs, counterparty risk, and collateral mitigation. Based on no-arbitrage arguments, we derive the nonlinear backward stochastic differential equations (BSDEs) associated with the replicating portfolios of long and short positions in the claim. This leads to defining buyer and seller’s XVAs which in turn identify a no-arbitrage band. When borrowing and lending rates coincide we provide a fully explicit expression for the uniquely determined price of XVA. When they differ, we derive the semi-linear PDEs associated with the non-linear BSDEs. We use them to conduct a numerical analysis showing high sensitivity of the no-arbitrage band and replicating strategies to funding spreads and collateral levels. This is joint work with Agostino Capponi (Columbia) and Stephan Sturm (WPI).
Speaker: Ovidiu Calin
Title: Transience of Brownian motion with constraints
Abstract:
The transience and recurrence properties of Brownian motion have been extensively studied on Riemanian manifolds. However, these type of problems are still open in the case of sub-Riemannian manifolds. In this case the diffusion is degenerate and moves along a non-integrable distribution, which is defined by some non-holonomic constraints. We shall discuss the transience of Brownian motion that is constraint to move along the Heisenberg, Grushin, and Martinet distributions.
Speaker: Dan Pirjol
Title: Explosive behavior in discrete time log-normal interest rate models
Abstract:
Interest rates models with log-normally distributed rates simulated in discrete time display explosive behavior, which is different and more complex than that of the better studied continuous time case. This phenomenon is studied in detail for the expectation of the money market account, and the Eurodollar futures prices in the Black-Derman-Toy model. The conditions under which the explosions occur are presented.
Speaker: Birgit Rudloff
Title: Measures of systemic risk
Abstract:
Systemic risk refers to the risk that the financial system is susceptible to failures due to the characteristics of the system itself. The tremendous cost of this type of risk requires the design and implementation of tools for the efficient macroprudential regulation of financial institutions. We propose a novel approach to measuring systemic risk. Key to our construction is a rigorous derivation of systemic risk measures from the structure of the underlying system and the objectives of a financial regulator. Systemic risk is measured by the set of allocations of additional capital that lead to acceptable outcomes. We explain the conceptual framework and the definition of systemic risk measures, provide an algorithm for their computation, and illustrate their application in numerical case studies. We apply our methodology to systemic risk aggregation extending Chen, Iyengar & Moallemi (2013) and to network models as suggested in the seminal paper of Eisenberg & Noe (2001) and their generalizations as in Cifuentes, Shin & Ferrucci (2005). This is joint work with Zach Feinstein and Stefan Weber.
Speaker: Konstantinos Spiliopoulos
Title: Indifference pricing for contingent claims: large deviations effects
Abstract:
We study utility indifference prices and optimal purchasing quantities for a non-traded contingent claim in an incomplete semi-martingale market with vanishing hedging errors, making connections with the theory
of large deviations. We concentrate on sequences of semi-complete markets where for each n the claim hn admits the decomposition hn = Dn +Yn where Dn is replicable and Yn is completely unhedgeable in that the indifference price of Yn for an exponential investor is its certainty equivalent. Under broad conditions, we may assume that Yn vanishes in accordance with a large deviations principle as n grows. In this setting, we identify limiting indifference prices as the position size becomes large, and show the prices typically are not the unique arbitrage free price in the limiting market. Furthermore, we show that optimal purchase quantities occur at the large deviations scaling, and hence large positions endogenously arise in this setting. This is joint work with Scott Robertson.
Speaker: Pierre Garreau
Title: A consistent spectral element framework for option pricing under general Lévy processes
Abstract:
We derive a consistent spectral element framework to compute the price of vanilla derivatives when the dynamic of the underlying follows a general Lévy process. The representation of the solution with Legendre polynomials allows to naturally approximate the convolution integral with high order quadratures. We use a third order implicit/explicit approximation to integrate in time. The method is spectrally accurate in space for the solution and the greeks, and third order accurate in time. The spectral element framework does not require the approximation of the Lévy measure nor the lower truncation of the convolution integral as commonly seen in Finite Difference schemes.
Speaker: Alexander Shklyarevsky
Title: Certain developments in ODE, SDE, PDE, PIDE and related analytical approaches and their applications to physics and quantitative finance and insurance
Abstract:
We discuss certain latest developments in methodology and approaches to solve ordinary differential equations (ODE), stochastic differential equation (SDE), partial differential equations (PDE), partial integrodifferential equations (PIDE) and related objects analytically. These approaches are used in both Physics and Quantitative Finance and Insurance both theoretically and in practical applications. An additional advantage is that the approaches developed in Physics could be often applied in Quantitative Finance and Insurance and vice versa. In our presentation, we will show that these analytical methodologies are making both research and its implementation in Physics and research and its implementation in Quantitative Finance and Insurance much more efficient and are critical to substantial advances in both Physics and Quantitative Finance and Insurance.
Speaker: Agnes Tourin
Title: A dynamic model for pairs trading strategies
Abstract:
The profitability of pairs trading strategies has been extensively studied in the empirical literature. However, there are still relatively few attempts to model dynamic pairs trading strategies. In this talk, I will present a family of models based on the theory of stochastic control that lead to analytical formulae for the optimal trading strategies. Some preliminary experiments in the stock and bitcoin markets suggest that the computed strategies perform well and are comparable to those used by practitioners.
Speaker: Yu Gu
Title: Local versus non-local forward equations for option prices
Abstract:
When the underlying asset is a continuous martingale, call option prices solve the Dupire equation, a forward parabolic PDE in the maturity and strike variables. By contrast, when the underlying asset is described by a discontinuous semimartingale, call price solve a partial integro-differential equation (PIDE), containing a nonlocal integral term. We show that the two classes of equations share no common solution: a given set of option prices is either generated from a continuous martingale ("diffusion") model or from a model with jumps, but not both. In particular, our result shows that Dupire's inversion formula for reconstructing local volatility from option prices does not apply to option prices generated from models with jumps.
Speaker: Kim Weston
Title: Stability of utility maximization in nonequivalent markets
Abstract:
Stability of the utility maximization problem with random endowment and indifference prices is studied for a sequence of financial markets in an incomplete Brownian setting. Our novelty lies in the nonequivalence of markets, in which the volatility of asset prices (as well as the drift) varies. Degeneracies arise from the presence of nonequivalence. In the positive real line utility framework, a counterexample is presented showing that the expected utility maximization problem can be unstable. A positive stability result is proven for utility functions on the entire real line.
Speaker: Andrea Karlova
Title: Volatility surfaces generated by stable distributions
Abstract:
In the talk we present approximate formulas for volatility smiles generated by stable laws. We discuss its properties and quality of numerical implementation.
Speaker: Mackenzie Wildman
Title: A gaussian markov alternative to fractional brownian motion for pricing financial derivatives
Abstract:
Replacing Black-Scholes’ driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls because its implementation allows for the occurrence of arbitrage in the financial market. I will discuss the development, testing, and implementation of a simplified alternative to using fractional Brownian motion for pricing derivatives. By relaxing the assumption of past independence of Brownian motion but retaining the Markovian property, we are developing a competing model that retains the mathematical simplicity of the standard Black-Scholes model but also has the improved accuracy of allowing for past dependence. This is achieved by replacing Black-Scholes’ underlying process, Brownian motion, with the Dobric-Ojeda process. This is joint work with Daniel Conus and Vladimir Dobric.
Speaker: Siyan Zhang
Title: Option pricing under SABR model with mean reversion
Abstract:
SABR model is a stochastic volatility model that is widely used in financial industry. However, it may be more reasonable to impose a mean reversion term to volatility. In this talk, I’ll discuss this revised SABR model and price European call option under this model. Though there’s no evidence showing that we can write down the exact solution, in this talk I’ll give an exact formula of an approximation solution by ideas of semigroup and Duhamel’s formula.
Speaker: Oleksii Mostovyi
Title: The University of Texas at
Austin
Abstract:
In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale, an explicit first-order expansion formula for the power investor’s value function - seen as a function of the underlying market price of risk
process - is provided and its second-order error is quantified. The numerical examples illustrating the accuracy of the method are also given. This talk is based on the joint work with Kasper Larsen and Gordan Zitkovic.
Speaker: Mustapha Pemy
Title:
Optimal algorithms for trading large positions
Abstract:
In this paper, we are concerned with the problem of efficiently trading a large position on the market place. If the execution of a large order is not dealt with appropriately this will certainly break the price equilibrium and result in large losses. Thus, we consider a trading strategy that breaks the order into small pieces and execute them over a predetermined period of time so as to minimize the overall execution shortfall while matching or exceeding major execution benchmarks such as the volume-weighted average price (VWAP). The underlying
problem is formulated as a discrete-time stochastic optimal control problem with resource constraints. The value function and optimal trading strategies are derived in closed-form. Numerical simulations with market data are reported to illustrate the pertinence of these results.
Speaker: Andrey Sarantsev
Title:
Infinite Atlas Model
Abstract:
Consider an infinite system of Brownian particles
on the real line. The (currently) bottom particle moves
as a Brownian motion with drift one. All other particles
move as driftless Brownian motions. Pal and Pitman (2008)
proved that the gaps between adjancet particles have
stationary distribution which is product of exponentials with
rates two. We prove that if we start with the gaps
stochastically larger than this stationary distribution,
then the gap process weakly converges to this stationary distribution.
Speaker: Asaf Cohen
Title:
Risk Sensitive Control of the Lifetime Ruin Problem
Abstract:
We study a risk sensitive control version of the lifetime ruin probability
problem. We consider a sequence of investments problems in Black-Scholes
market that includes a risky asset and a riskless asset. We present a
differential game that governs the limit behavior. We solve it explicitly
and use it in order to find an asymptotically optimal policy.(joint work with Erhan Bayraktar)
Speaker: Oleksii Mostovyi
Title: An approximation of utility maximization in incomplete markets
Abstract:
The implementation of utility-maximization methods for the
optimal portfolio choice rely on proper calibration of the model, and,
in particular, on the correct estimation of the parameters of the
stock-price dynamics. We analyze the effect of a misspecification in
the parametric description of the stock-price evolution on the value
function of utility-maximization problem for rational economic agent,
whose preferences are described by a utility function of the "power"
type, with p < 0. In the framework of an incomplete financial market
where the stock price is modeled by a continuous semimartingale, we
perform an asymptotic analysis of the value function with respect to a
small perturbation of the finite-variation part of the price process.
We establish a first-order expansion formula and bound the error of our
approximation. The implications of our result, such as an approximation
of the less tractable models by the more tractable ones, are
illustrated by specific examples. The talk is based on the joint work with
Gordan Zitkovic.
Speaker: Asaf Cohen
Title:
Risk Sensitive Control of the Lifetime Ruin Problem
Abstract:
We study a risk sensitive control version of the lifetime ruin probability
problem. We consider a sequence of investments problems in Black-Scholes
market that includes a risky asset and a riskless asset. We present a
differential game that governs the limit behavior. We solve it explicitly
and use it in order to find an asymptotically optimal policy.(joint work with Erhan Bayraktar)
Speaker: Kihun Nam
Title:
Multidimensional quadratic BSDEs which are related to stochastic differential game
Abstract:
In general, multidimensional quadratic BSDE do not have a solution. In this talk, I
will present the existence and uniqueness result for multidimensional quadratic
BSDEs of special structure. This types of BSDEs often appears in stochastic
differential games. Using Girsanov transform and results from FBSDE and one
dimensional quadratic BSDE, I will show how one can construct the solutions. This is
a joint work with Patrick Cheridito.
Speaker: Alexandre Roch
Title: Term structure of interest rates with liquidity risk
Abstract:
We develop an arbitrage pricing theory for liquidity risk and price impacts on fixed income markets. We define a liquidity term-structure of interest rates by hypothesizing that liquidity costs arise from the quantity impact of trading of bonds with different maturities on the interest rates and the associated risk-return premia. We derive no arbitrage conditions which gives a number of theoretical relation satisfied between the impact on risk premia and the volatility structure of the term structure and prices. We calculate the quantity impact of trading a zero-coupon on prices of zero-coupons of other maturities and represent this quantity as a supermartingale. We give conditions under which the market is complete, and show that the replication cost of an interest rate derivative is the solution of a quadratic backward stochastic differential equation. Joint work with Robert Jarrow.
Speaker: Hasanjan Sayit
Title: Absence of arbitrage in a general framework
Abstract:
Cheridito (Finance Stoch. 7: 533-553, 2003) studies a financial market that consists of a money market account and a risky asset driven by a fractional Brownian motion (fBm). It is shown that arbitrage pos-sibilities in such markets can be excluded by suitably restricting the class of allowable trading strategies. In this note, we show an analogous result in a multi-asset market where the discounted risky asset prices follow more general non-semimartingale models. In our framework, investors are allowed to trade between a risk-free asset and multiple risky assets by following simple trading strategies that require a minimal deterministic waiting time between any two trading dates. We present a condition on the discounted risky asset prices that guarantee absence of arbitrage in this setting. We give examples that satisfy our condition and study its invariance under certain transformations.
Speaker: Alexander Shklyarevsky
Title: Mutual benefit of ODE, PDE, PIDE and related analytical approaches developed in and applied to physics and quantitative finance
Abstract:
We present a comprehensive methodology and approach to tackle ordinary differential equations (ODE), partial differential equations (PDE), partial integro-differential equations (PIDE) and related topics analytically. These approaches are used in both physics and quantitative finance with mutual benefit, both theoretically and practically. In our presentation, we will show that these analytical methodologies are making both research in physics and research and its implementation in quantitative finance much more efficient and are critical to substantial advances in physics and quantitative finance, as well as assure a trading and risk optimization success across asset classes.
Speaker: Jian Song
Title: Feynman-Kac Formula for SPDE driven by fractional Brownian motion.
Abstract:
In the talk, I will introduce the Feynman-Kac Formula for SPDE driven by fractional Brownian motion, and show its long term behavior.
Speaker: Stephan Sturm
Title: Optimal incentives for delegated portfolio optimization
Abstract:
We study the problem of an investor who hires a fund manager to manage his wealth. The latter is paid by an incentive scheme based on the performance of the fund. Manager and investor have different risk aversions; the manager may invest in a financial market to form a portfolio optimal for his expected utility whereas the investor is free to choose the incentives -- taking only into account that the manager is paid enough to accept the managing contract. We discuss the problem of existence of optimal incentives in general semimartingale models and give an assertive answer for some classes of incentive schemes. This is joint work with Maxim Bichuch (Worcester Polytechnic Institute).
Speaker: Gu Wang
Title: Consumption in Incomplete Markets
Abstract:
An agent maximizes isoelastic utility from consumption with infinite horizon in an incomplete market, in which state variables are driven by diffusions. We first provide a general verification theorem, which links the solution of the Hamilton-Jacobi-Bellman equation to the optimal consumption and investment policies. To tackle the intractability of such problems, we propose approximate policies, which admit an upper bound, in closed-form for their utility loss. The approximate policies have closed form solutions in common models, and become optimal if the market is complete, or utility is logarithmic.
Speaker: Tao Wu
Title: Pricing and hedging the smile with SABR: Evidence from the interest rate caps market
Abstract:
This is the first comprehensive study of the SABR (Stochastic Alpha-Beta-Rho) model (Hagan et. al (2002)) on the pricing and hedging of interest rate caps. We implement several versions of the SABR interest rate model and analyze their respective pricing and hedging performance using two years of daily data with seven different strikes and ten different tenors on each trading day. In-sample and out-of-sample tests show that the fully stochastic version of the SABR model exhibits excellent pricing accuracy and more importantly, captures the dynamics of the volatility smile over time very well. This is further demonstrated through examining delta hedging performance based on the SABR model. Our hedging result indicates that the SABR model produces accurate hedge ratios that outperform those implied by the Black model.
Speaker: Tianyao Yue
Title: Pricing binary options and their sensitivities under CEV using spectral element method
Abstract:
Because binary option has its unique property of discontinuous payoff at maturity, classical finite difference method (FDM) produces oscillation in the numerical solutions especially for the Greeks. Spectral element method (SEM) is introduced to solve the partial differential equation (PDE) of the option to achieve high convergence rate and avoid such oscillation phenomenon around discontinuous points. A European binary option under constant elasticity of variance (CEV) is studied and computed with this approach. The numerical results of the price and Greeks show the spectral element method is an efficient alternative method for exotic options with discontinuous payoffs.
Speaker: Wenhua Zou
Title: A unified treatment of derivative pricing and forward
decision problems within HJM framework
Abstract:
We study the HJM approach which was originally introduced in the fixed income market by David Heath, Robert Jarrow and Andrew Morton and later was implemented in the case of European option market by Martin Schweizer, Johannes Wissel, Rene Carmona and Sergey Nadtochiy. The main contribution of this thesis is to apply HJM philosophy to the American option market. We derive the absence of arbitrage by a drift condition and compatibility between long and short rate by a spot consistency condition. In addition, we introduce a forward stopping rule which is significantly different from the classical stopping rule which requires backward induction. When It\^{o} stochastic differential equation are used to model the dynamics of underlying asset, we discover that the drift part instead of the volatility part will determine the value function and stopping rule. As counterpart to the forward rate for the fixed income market and implied forward volatility and local volatility for the European option market, we introduce the forward drift for the American option market.
Speaker: Michael O. Okelola
Title: Solving a PDE associated with the pricing of power options with time dependent parameters
Abstract:
In recent times, the Lie group approach has been employed in the solution of time dependent PDEs. This method proves successful in providing exact solutions to these PDEs - even in cases where solutions did not previously exist. In this presentation, we look at the particular case of the PDE which models the power option. Using Lie symmetry analysis, we obtain the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. We not only present results obtained via this approach for the constant parameter scenario but we also employ the approach for the solution of the time dependent parameter case. (Joint with K. S. Govinder and J. G. O'Hara.)
Speaker: Gordon Ritter
Multi-period portfolio choice and Bayesian dynamic models
Abstract:
We describe a novel approach to the study of multi-period portfolio selection problems with time
varying alphas, trading costs, and constraints. We show that, to each multi-period portfolio
optimization problem, one may associate a ``dual'' Bayesian dynamic model. The dual model is
constructed so that the most likely sequence of hidden states is the trading path which optimizes
expected utility of the portfolio. The existence of such a model has numerous implications, both
theoretical and computational. Sophisticated computational tools developed for Bayesian state
estimation can be brought to bear on the problem, and the intuitive theoretical structure attained
by recasting the problem as a hidden state estimation problem allows for easy generalization to
other problems in finance. We discuss optimal hedging for derivative contracts as a special case. (Joint with Petter Kolm.)
Speaker: Triet Pham
Two person zero-sum game under feedback controls and path dependent Bellman-Isaacs equations
Abstract:
We introduce the feedback control setting to study two person zero-sum stochastic differential games. In standard literature, the open-loop setting is typically used, which requires the game to be set up under the strategy versus control framework. The main drawback of this approach is the asymmetry of information between the two players. The feedback control allows us to consider the game under the control versus control setting, which preserves the symmetry. Under natural conditions, we show the game value exists. We also allow for non-Markovian structure, and thus the game value is a random process. We characterize the value process as the unique viscosity solution of the corresponding path dependent Bellman-Isaacs equation, a notion recently introduced by Ekren-Keller-Touzi-Zhang and Ekren-Touzi-Zhang. This is joint work with Jianfeng Zhang.
Speaker: tba
tba
Abstract:
Speaker: Ruihua Liu
Title: A tree method for option pricing in switching models with state dependent switching rates
Abstract:
We present a tree approach for option pricing in switching diffusion mod- els where the rates of switching are assumed to depend on the underlying asset price process. The models generalize many existing models in the literature and in particular, the Markovian regime-switching models. The proposed trees grow linearly as the number of tree steps increases. Con- ditions on the choices of key parameters for the tree design are provided that guarantee the positivity of branch probabilities. Numerical results are provided and compared with results reported in the literature for the Markovian regimeswitching cases. The reported numerical results for the state-dependent switching models are new and can be used for comparison in the future.
Speaker: Lakshithe Wagalath
Title: Institutional investors and the dependence structure of asset returns
Abstract:
We propose a model of a financial market with multiple assets, which takes into account the impact of a large institutional investor rebalancing its positions, so as to maintain a fixed allocation in each asset. We show that feedback effects can lead to significant excess realized correlation between asset returns and modify the principal component structure of the (realized) correlation matrix of returns. Our study naturally links, in a quantitative manner, the properties of the realized correlation matrix – correlation between assets, eigenvectors and eigenvalues – to the sizes and trading volumes of large institutional investors. In particular, we show that even starting with uncorrelated ’fundamentals’, fund rebalancing endogenously generates a correlation matrix of returns with a first eigenvector with positive components, which can be associated to the market, as observed empirically. Finally, we show that feedback effects flatten the differences between assets’ expected returns and tend to align them with the returns of the institutional investor’s portfolio, making this benchmark fund more difficult to beat, not because of its strategy but precisely because of its size and market impact.
Speaker: Mingxin Xu
Title: Forward stopping rule within HJM framework.
Abstract:
We revisit the optimal stopping problem using Heath-Jarrow-Morton (HJM) approach. The HJM method was originally introduced to model the fixed-income market by Heath et al. (1992). More recently, it was implemented in equity market models by Schweizer and Wissel (2008), and Carmona and Nadtochiy (2008,2009). Prior work has mainly focused on European derivative pricing, while in this paper we apply the HJM philosophy to American derivative pricing with a focus toward solving optimal stopping problems in general. As a counterpart to forward rate for the fixed-income market and forward volatility for the equity market, we introduce forward drift for the optimal stopping problem. The standard results for HJM-type models are confirmed for the forward drift dynamics: the drift condition and the spot consistency condition. More interestingly, we discover a forward stopping rule that is fundamentally different from the classical stopping rule based on backward induction. We illustrate this difference in two benchmark models: a binomial example for American option pricing and a Black-Sholes example for the optimal time to sell a stock. In addition to the minimal optimal stopping time, we characterize the maximal optimal stopping time in the forward approach. (Joint with Wenhua Zou.)
Speaker: Duy Nguyen
Title: Numerical schemes for pricing asian options under state-dependent regime-switching jump-diffusion models
Abstract:
We study the pricing problem of Asian options when the underlying asset price follows a very general state-dependent regime-switching jump-diffusion process via a partial differential equation approach. Under this model, the price of the option can be obtained by solving a highly complex system of coupled two-dimensional parabolic partial integro-differential equations (PIDEs). We prove existence of the solution to this system of PIDEs by the method of upper and lower solutions via constructing a monotonic sequence of approximating solutions whose limit is a strong solution of the PIDE system. We then propose several numerical schemes for solving the system of PIDEs. One of the proposed schemes is built upon the constructive proof, hence its results are provably convergent to the solution of the system of PIDEs. We illustrate the accuracy of the proposed methods by several numerical examples. (Joint with D.M. Dang and G. Sewell).
Speaker: Zhixin Yang
Title: Evaluation of risk based premium of pension benefit guaranty corporation with regime switching
Abstract:
This work studies the defined pension plan supported by pension benefit guaranty corporation (PBGC). Our work generalizes Chen ’s work in (2011) and (2014) by taking the market regime switching into consideration. Both the premature of pension fund and distress termination of sponsor asset are analyzed, a closed-form solution for the risk based premiums are achieved.
Speaker: Paolo Guasoni
Title: Spending and investment for shortfall-averse endowments
Abstract:
A dynamic spending and investment model allows for spending shortfall
aversion through a utility function that entails scaling of spending by
a fractional power of past peak spending. The past peak spending is the
current reference or target spending. Under the closed form solution the
wealth to target ratio follows a diffusion process. At the lowest levels
of the ratio, up to a point, the spending rate and weight of the risky
asset are fixed fractions of wealth, as prescribed by Merton. Beyond
that point, at the higher levels of wealth to target ratio, spending is
constant and the weight of the risky asset increases with wealth. Wealth
to spending ratio has an upper bound at which increases in the spending
rate (and the target) offset wealth increases. (Joint work with Gur
Huberman and Dan Ren)
Speaker: Christian Keller
Title: Pathwise viscosity solutions of stochastic PDEs
Abstract:
We present a notion of pathwise viscosity solutions for fully nonlinear stochastic PDEs and establish well-posedness for a large class of equations. We operate in the framework of rough path theory. Thus we can study stochastic PDEs in a pathwise manner. This is crucial for proving our main results since we can then circumvent very difficult problems regarding null sets. Moreover, ideas from our previous work on path-dependent PDEs play an important role.
This is joint work with Rainer Buckdahn, Jin Ma, and Jianfeng Zhang.
Speaker: Michael Spector
Title: SABR spreads its wings
Abstract:
The stochastic alpha beta rho (SABR) model introduced in Hagan, Lesniewski & Woodward (2001) and Hagan et al (2002) is widely used by practitioners to capture the volatility skew and smile effects of interest rate options. Traditional methods for the stochastic alpha beta rho model tend to focus on expansion approximations that are inaccurate in the long maturity ‘wings’. However, if the Brownian motions driving the forward and its volatility are uncorrelated, option prices are analytically tractable. In the correlated case, model parameters can be mapped to a mimicking uncorrelated model for accurate option pricing. (Joint with Alexander Antonov and Michael Konikov.)
