Mathematics Department - Mathematical Finance, Probability and PDE Seminar - Spring 2017

Mathematical Finance, Probability and PDE Seminar - Spring 2017



Organizer(s)

Paul Feehan, Kihun Nam, Daniel Ocone, Triet Pham

Archive

Website

http://www.finmath.rutgers.edu/index.php?d=seminars&p=msmfseminar&year=2012&sem ster=1&type=0




Past Talks


Tuesday, April 25th

Oleksii Mostovyi, University of Connecticut

"SENSITIVITY ANALYSIS OF THE UTILITY MAXIMIZATION PROBLEM WITH RESPECT TO MODEL PERTURBATIONS"

Time: 11:45 AM
Location: Hill 705
Abstract: We study the sensitivity of the expected utility maximization problem in a continuous semimartingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled with a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and construct trading strategies that match the indirect utility function up to the second order. f a risk-tolerance wealth process exists, using it as a numeraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita-Watanabe decomposition.

This talk is based on the joint work with Mihai Sirbu.


Tuesday, April 18th

Jie Zhong, University of Rochester

"The Parametrix method for skew diffusions"

Time: 11:45 AM
Location: Hill 705
Abstract: In this talk, we introduce the parametrix method and study the existence and the regularity properties of the density of a skew diffusion and provide a Gaussian upper bound. This expansion leads to a probabilistic representation.


Tuesday, April 11th

Yavor Stoev, University of Michigan

"Martingale optimal transport with stopping"

Time: 11:45 AM
Location: Hill 705
Abstract: We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in ArXiv: 1507.02651 allows us to obtain an equivalent infinite-dimensional controller-stopper problem. We use the stochastic Perron's method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions for a class of cost functions.

Joint work with Erhan Bayraktar (UM) and Alex Cox (Bath)


Tuesday, April 4th

Hyungbin Park, Worcester Polytechnic Institute

"Sensitivity analysis of long-term cash flows"

Time: 11:45 AM
Location: Hill 705
Abstract: This talk discusses a sensitivity analysis of long-term cash flows, which is given as a pricing operator of a Markov diffusion. We study how much the cash flows is vulnerable to small perturbations of the underlying Markov diffusion. The main tool is the Hansen-Scheinkman decomposition, which is a technique expressing the cash flow by the eigenvalue and eigenfunction of the pricing operator. By combining the results of Fournie et al., we conclude that the sensitivities of long-term cash flows can be represented in simple forms of the eigenvalue and the eigenfunction.


Tuesday, March 28th

Allen Cheng, Columbia University

"Modeling wealth dynamics under central clearing"

Time: 11:45 AM
Location: Hill 705
Abstract: We provide a flexible, scalable, and analytically tractable model for participants of a centrally cleared market. By exploiting real-world features of clearing practices, we providing a unifying framework that models, jointly and explicitly, financial institutions' business operations, trading activities, allocation of capital, and the resulting wealth dynamics. We find that (i) trading is more capital intensive for large clearing members, (ii) incomplete information can significantly distort measured concentration, (iii) market collateral demand is positively correlated with wealth concentration, (iv) wealth concentration has the inherent tendency to build up, (v) there is a potential tradeoff between wealth and business diversity, and (vi) a concentration charge on collateralizing margins can be used to steer concentration.


Tuesday, March 7th

Hyungbin Park, Worcester Polytechnic Institute

"Sensitivity analysis of long-term cash flows"

Time: 11:45 AM
Location: Hill 705
Abstract: This talk discusses a sensitivity analysis of long-term cash flows, which is given as a pricing operator of a Markov diffusion. We study how much the cash flows is vulnerable to small perturbations of the underlying Markov diffusion. The main tool is the Hansen-Scheinkman decomposition, which is a technique expressing the cash flow by the eigenvalue and eigenfunction of the pricing operator. By combining the results of Fournie et al., we conclude that the sensitivities of long-term cash flows can be represented in simple forms of the eigenvalue and the eigenfunction.


Tuesday, February 28th

Sebastian Herrmann, University of Michigan

"Hedging with Uncertainty-Averse Preferences"

Time: 11:45 AM
Location: Hill 705
Abstract: We study the pricing and hedging of derivative securities with uncertainty about the volatility of the underlying asset. Rather than taking all models from a prespecified class equally seriously, we penalize less plausible ones based on their "distance" to a reference local volatility model. In the limit for small uncertainty aversion, this leads to explicit formulas for prices and hedging strategies in terms of the security's cash gamma. If the reference model is a Black-Scholes model which is dynamically recalibrated to the market price of a liquidly traded vanilla option, delta-vega hedging is asymptotically optimal. The corresponding indifference price corrections are then determined by the disparity between the vegas, gammas, vannas, and volgas of the non-traded and the liquidly traded options.


Tuesday, February 21st

Chihoon Lee, Stevens Institute of Technology

"Transform Analysis for Markov Processes and its Applications in Finance"

Time: 11:45 AM
Location: Hill 705
Abstract: We present a transform analysis of one-dimensional Markov processes through their infinitestimal generators. More precisely, we characterize the joint double (Laplace) transforms of additive functionals of Markov processes and the terminal values, which extends the work of Cai et al. (2015) computing Asian options. We also establish a simple duality relationship between continuous positive additive functionals of processes and their right inverses. Through this duality, we obtain the double transforms of inverses of additive functionals of Markov processes.


Tuesday, February 7th

Dan Pirjol, JP Morgan

"Infinite sums of the geometric Brownian motion and generalizations"

Time: 11:45 AM
Location: Hill 705
Abstract: The infinite sum of a geometric Brownian motion (gBM) sampled on a sequence of uniformly spaced times appears in problems of actuarial science and theoretical probability. For example this appears when considering the present value of a perpetuity: a fixed recurring payment made in perpetuity from an initial deposit of stock, assumed to follow a geometric Brownian motion. The talk studies the distributional properties of the infinite sum of the gBM. This can be characterized as the stationary distribution of a linear stochastic recursion. Tail asymptotics are derived, and the distribution is found numerically by solving an integral equation. Similar results are obtained for the sum of the gBM with a geometrically distributed stopping time. The results can be generalized further by replacing the gBM with an exponential Levy process.


Tuesday, January 24th

Po–Keng Cheng, Stony Brook

"An Interactive Agent-Based Model"

Time: 11:45 AM
Location: Hill 705
Abstract: We develop and examine a simple heterogeneous agent model, where the distribution of returns generated from the model have stylized facts in financial markets, such as fat tails and volatility clustering. Our results indicate that the risk tolerance of fundamentalists and the relative funding rate of positive-feedback traders versus fundamentalists are key factors determining the path of price fluctuations. Fundamentalists are more able to dominate the market when they are more willing than positive-feedback traders to take risks. In addition, more crises occur as positive-feedback traders face higher funding costs compared to fundamentalists. Our model suggests that fundamentalists cause heavy tails, and positive-feedback traders cause the formation of speculative bubbles.

We introduce a heterogeneous agent mechanism extending from the model. We add one more key factor, length of evaluations on performances between strategies, which also have significant influence on price fluctuations. We also introduce Markov transition matrix, Perron-Frobenius transition matrix, and Inertia to investigate the transitions among states. Our results show the stickiness of states switching from one to another, and the longer length of evaluations on performances would generate more complex dynamic price fluctuations.

We then estimate key parameters in our model. Our empirical results indicates that traders’ attitudes towards risk vary across time and market. The generally low level of risk bearing by fundamentalists could explain the frequent occurrence of bubbles.


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