Research

Interests :

My current research interests span Mathematical Finance, Financial Engineering and Actuarial Science.

I hold weekly meetings with my colleagues and Ph.D. students at the Field Institute for Mathematical Sciences.

Colleagues and Students :

The Mathematical Finance/Actuarial Science research group in the Department of Statistics consists of four professors :

Prof. A. Badescu
Prof. S. Broverman
Prof. S. Jaimungal
Prof. X. S. Lin

and several Ph.D. students.

Here is a list of our students and their current research interest:

Here is an alphabetic list of Math Finance / Financial Engineering researchers at other Departments in the University of Toronto:

Prof. Jin-Chuan Duan, Rotman School of Business

Prof. John Hull, Rotman School of Business

Prof. Kenneth Jackson, Dept. Computer Science

Prof. Tom McCurdy, Rotman School of Business

Prof. Marcel Rindisbacher, Rotman School of Business

Prof. Luis Seco, Department of Mathematics

Prof. Michael Walker, Department of Physics

Prof. Alan White, Rotman School of Business

Publications and Working Papers

Below you will find my working papers and publications in Mathematical Finance / Actuarial Science, as well as some of the work that I did in my former life as a physicist.

Consistent Functional PCA for Financial Time-Series [ PDF ] with Eddie. K. H. Ng, to appear in Proceedings of the 4th IASTED International Conference on Financial Engineering and Applications.

Functional Principal Component Analysis (FPCA) provides a powerful and natural way to model functional financial data sets (such as collections of time-indexed futures and interest rate yield curves). However, FPCA assumes each sample curve is drawn from an independent and identical distribution. This assumption is axiomatically inconsistent with financial data; rather, samples are often interlinked by an underlying temporal dynamical process. We present a new modeling approach using Vector auto-regression (VAR) to drive the weights of the principal components. In this novel process, the temporal dynamics are first learned and then the principal components extracted. We dub this method the VAR-FPCA. We apply our method to the NYMEX light sweet crude oil futures curves and demonstrate that it contains significant advantages over the conventional FPCA in applications such as statistical arbitrage and risk management.

Option Pricing with Regime Switching Levy processes using Fourier Space Time Stepping [ PDF ] with Kenneth R. Jackson and Vladimir Surkov, to appear in Proceedings of the 4th IASTED International Conference on Financial Engineering and Applications.

Although jump-diffusion and L\'evy models have been widely used in industry, the resulting pricing partial-integro differential equations poses various difficulties for valuation. Diverse finite-difference schemes for solving the problem have been introduced in the literature. Invariably, the integral and diffusive terms are treated asymmetrically, large jumps are truncated and the methods are difficult to extend to higher dimensions. We present a new efficient transform approach for regime-switching Levy models which is applicable to a wide class of path-dependent options (such as Bermudan, barrier, and shout options) and options on multiple assets.

Fourier Space Time Stepping for Option Pricing with Levy Models [ PDF ] with Kenneth R. Jackson and Vladimir Surkov.

OUR ONLINE LEVY PROCESS OPTION CALCULATOR

Jump-diffusion and Levy models have been widely used to partially alleviate some of the biases inherent in the classical Black-Scholes-Merton model. Unfortunately, the resulting pricing problem requires solving a more difficult partial-integro differential equation (PIDE) and although several approaches for solving the PIDE have been suggested in the literature, none are entirely satisfactory. All treat the integral and diffusive terms asymmetrically, truncate large jumps and are difficult to extend to higher dimensions. We present a new, efficient algorithm, based on transform methods, which symmetrically treats the diffusive and integrals terms, is applicable to a wide class of path-dependent options (such as Bermudan, barrier, and shout options) and options on multiple assets, and naturally extends to regime-switching Levy models. We present a concise study of the precision and convergence properties of our algorithm for several classes of options and Levy models and demonstrate that the algorithm is second-order in space and first-order in time for path-dependent options.

Asymptotic Pricing of Commodity Derivatives for Stochastic Volatility Spot Models [ PDF ] with Samuel Hikspoors. To appear in Applied Mathematical Finance special issue on Commodities 2008.

It is well known that stochastic volatility is an essential feature of commodity spot prices. By using methods of singular perturbation theory, we obtain approximate but explicit closed form pricing equations for forward contracts and options on single- and two-name forward prices. The expansion methodology is based on a fast mean-reverting stochastic volatility driving factor, and leads to pricing results in terms of constant volatility prices, their Delta's and their Delta-Gamma's. The stochastic volatility corrections lead to efficient calibration and sensitivity calculations.

Energy Spot Price Models and Spread Options Pricing [ PDF ] with Samuel Hikspoors, International Journal of Theoretical & Applied Finance, Nov2007, Vol. 10 Issue 7, p1111-1135.

In this article, we construct forward price curves and value a class of two asset exchange options for energy commodities. We model the spot prices using an affine two-factor mean-reverting process with and without jumps. Within this modeling framework, we obtain closed form results for the forward prices in terms of elementary functions. Through measure changes induced by the forward price process, we further obtain closed form pricing equations for spread options on the forward prices. For completeness, we address both an Actuarial and a risk-neutral approach to the valuation problem. Finally, we provide a calibration procedure and calibrate our model to the NYMEX Light Sweet Crude Oil spot and futures data, allowing us to extract the implied market prices of risk for this commodity.

On Valuing Equity-Linked Insurance and Reinsurance Contracts : V2 [ PDF ] with Suhas Nayak.

Through the issuance of equity-linked insurance policies, insurance companies are increasingly facing losses that have heavy exposure to capital market risks. In this paper, we determine the continuous premium rate that an insurer exposed to such risks charges via the principle of equivalent utility. Using exponential utility, we obtain the resulting premium rate in terms of an expectation under the unique minimal martingale measure and perform a perturbation expansion around a risk-neutral investor. Within the same consistent framework, we address the problem of pricing of a double-trigger reinsurance contract, taking into account counter-party risk. The indifference price is found to satisfy a non-linear and nonlocal PDE. This price is further expanded around the risk-neutral price resulting in closed form solutions in the form of risk-neutral expectations. Finally, we recast the pricing PDE as a linear stochastic control problem and provide an implicit-explicit finite-difference scheme for solving the PDE numerically.

Catastrophe Options with Stochastic Interest Rates and Compound Poisson Losses [ PDF ] with Tao Wang, Insurance: Mathematics and Economics, vol 38, issue 3, pg. 469-483 (2006).

We analyze the pricing and hedging of catastrophe put options under stochastic interest rates with losses generated by a compound Poisson process. Asset prices are modeled through a jump-diffusion process which is correlated to the loss process. We obtain explicit closed form formulae for the price of the option, and the hedging parameters Delta, Gamma and Rho. The effects of stochastic interest rates and variance of the loss process are illustrated through numerical experiments. Furthermore, we carry out a simulation analysis of the Delta-Gamma-Rho hedging scheme and illustrate that accounting for stochastic interest rates, through Rho hedging, can significantly reduce the expected conditional loss of the hedged portfolio.

Pricing and Hedging Equity Indexed Annuities with Variance Gamma Deviates [ PDF ]

The author analyzes the pricing and hedging problem for Equity Indexed Annuities (EIAs) with underlying risky assets following a geometric Variance-Gamma process. This model allows accurate and parsimonious replication of the implied volatility smiles observed in the financial market. I argue that this model produces consistency in pricing and hedging between the financial and insurance markets. Closed form expressions for prices of Point-to-Point and Cliquet instruments are developed and used to investigate the break-even participation rates. Furthermore, I derive the hedging parameters - Delta, Gamma and Vega - for the Cliquet design. Mortality risk is incorporated through the Actuarial present value principal and I use numerical experiments to investigate the effects of the model parameters.

Pricing Equity Indexed Pure Endowments with Risky Assets that follow Levy Processes [ PDF ] with V. R. Young Insurance: Mathematics and Economics, vol 36, issue 2, pg. 329-346 (2005)

We investigate the pricing problem for pure endowment contracts whose life contingent payment is linked to the performance of a tradable risky asset or index. The heavy tailed nature of asset return distributions is incorporated into the problem by modeling the price process of the risky asset as a finite variation L´evy process. We price the contract through the principle of equivalent utility. Under the assumption of exponential utility, we determine the optimal investment strategy and show that the indifference price solves a non-linear partial-integro-differential equation (PIDE). We solve the PIDE in the limit of zero risk aversion, and obtain the unique risk-neutral equivalent martingale measure dictated by indifference pricing. In addition, through an explicit-implicit finite difference discretization of the PIDE we numerically explore the effects of the jump activity rate, jump sizes and jump skewness on the pricing and the hedging of these contracts.

A Two State-Jump Model [ PDF ] with C. Albanese and D Rubisov, Quantitative Finance, vol 3, issue 2, pg. 145-154 (Apr. 2003)

We introduce a pricing model for equity options in which sample paths follow a variance-gamma (VG) jump model whose parameters evolve according to a two-state Markov chain process. As in GARCH type models, jump sizes are positively correlated to volatility. The model is capable of justifying the observed implied volatility skews for options at all maturities. Furthermore, the term structure of implied VG kurtosis is an increasing function of the time to maturity, in agreement with empirical evidence. Explicit pricing formulae, extending the known VG formulae, for European options are derived. In addition, a resummation algorithm, based on the method of lines, which greatly reduces the algorithmic complexity of the pricing formulae, is introduced. This algorithm is also the basis of approximate numerical schemes for American and Bermudan options, for which a state dependent exercise boundary can be computed.

Jumping in Line [ PDF ] with C. Albanese and D RubisovRisk Magazine, Feb. 2001.

We introduce an efficient numerical method to price derivative claims assuming the underlying follows a jump process of the variance Gamma (VG) type. The algorithm is based on the method of lines and involves the solution of ordinary differential equations and a Richardson extrapolation method. The method is applied to equity, Bermudan and barrier options with VG jumps.

The model of lines for option pricing with jumps [ PDF ] with C. Albanese and D Rubisov

This article reviews a pricing model, suitable for variance-gamma jump processes, based on the method of lines. The method accuracy is studied using European style calls as a benchmark. Implementation details for continuously and discretely monitored barrier options, and American and Bermudan options are given.

Coulomb Gas Representation of the QCD Effective Lagrangian, with A.R.Zhitnitsky, Proceedings of Lightcone QCD and nonperturbative Hadron Physics Adelaide 1999, pg. 269-275.

A novel Coulomb gas (CG) description of low energy $QCD_4$, based on the dual transformation of the QCD effective chiral Lagrangian, is constructed. The CG is found to contain several species of charges, one of which is fractionally charged and can be interpreted as instanton-quarks. The creation operator which inserts a pseudo-particle in the CG picture is explicitly constructed and demonstrated to have a non-zero vacuum expectation value.

Phase Transition In Quantum Gravity? with V. Husain, Mod. Phys. Lett. A14 (199) p:1079-1082

A fundamental problem with attempting to quantize general relativity is its perturbative non-renormalizability. However, this fact does not rule out the possibility that non-perturbative effects can be computed, at least in some approximation. We outline a quantum field theory calculation, based on general relativity as the classical theory, which implies a phase transition in quantum gravity. The order parameters are composite fields derived from spacetime metric functions. These are massless below a critical energy scale and become massive above it. There is a corresponding breaking of classical symmetry.

Topological Holography, with V. Husain, Phys. Rev. D60 (1999)

We study a topological field theory in four dimensions on a manifold with boundary. A bulk-boundary interaction is introduced through a novel variational principle rather than explicitly. Through this scheme we find that the boundary values of the bulk fields act as external sources for the boundary theory. Furthermore, the full quantum states of the theory factorize into a single bulk state and an infinite number of boundary states labeled by loops on the spatial boundary. In this sense the theory is purely holographic. We show that this theory is dual to Chern-Simons theory with an external source. We also point out that the holographic hypothesis must be supplemented by additional assumptions in order to take into account bulk topological degrees freedom, since these are apriori invisible to local boundary fields.

Universality In Effective Strings with G.W. Semenoff, K. Zarembo JETP Lett. 69 (1999) p:509-515

We demonstrate that, due to the finite thickness of domain walls, and the consequent ambiguity in defining their locations, the effective string description obtained by integrating out bulk degrees of freedom contains ambiguities in the coefficients of the various geometric terms. The only term with unambiguous coefficient is the zeroth order Nambu-Goto term. We argue that fermionic ghost fields which implement gauge-fixing act to balance these ambiguities. The renormalized string tension, obtained after integrating out both bulk and world-sheet degrees of freedom, can be defined in a scheme independent manner; and we compute the explicit finite expressions, to one-loop, for the case of compact quantum electrodynamics and phi-4 theory.

Wilson Loops, Bianchi Constraints and Duality In Abelian Lattice Models, Nucl. Phys. B542 (1999) p:441-470

We introduce new modified Abelian lattice models, with inhomogeneous local interactions, in which a sum over topological sectors are included in the defining partition function. The dual models, on lattices with arbitrary topology, are constructed and they are found to contain sums over topological sectors, with modified groups, as in the original model. The role of the sum over sectors is illuminated by deriving the field-strength formulation of the models in an explicitly gauge-invariant manner. The field-strengths are found to satisfy, in addition to the usual local Bianchi constraints, global constraints. We demonstrate that the sum over sectors removes these global constraints and consequently softens the quantization condition on the global charges in the system. Duality is also used to construct mappings between the order and disorder variables in the theory and its dual. A consequence of the duality transformation is that correlators which wrap around non-trivial cycles of the lattice vanish identically. For particular dimensions this mapping allows an explicit expression for arbitrary correlators to be obtained.

T-Duality in Lattice Regularized Sigma Models, Phys. Lett. B441 (1998) p:147-154

It is shown that when the underlying sigma model of bosonic string theory is written in terms of single-valued fields, which live in the covering space of the target space, Abelian T-duality survives lattice regularization of the world-sheet. The projection onto the target-space is implemented through a sum over cohomology, which bears resemblance to summing over topological sectors in Yang-Mills theories. In particular, the case of string theory on a circle is shown to be explicitly self-dual in the lattice regulated model and automatically forbids vortex excitations which would otherwise destroy the duality. For other target spaces a generalized notion of T-duality is observed in which the target space and the cohomology coefficient group are interchanged under duality. Specific examples show that the fundamental group of the target space may not be preserved in the T-dual theory. Generalized models which exhibit T-duality behaviour, with dynamical variables that live on the k-dimensional cells of (p+1)-dimensional world-volumes, are also constructed. These models correspond to gauge theories, and higher-dimensional analogues, in which one sums over various topological sectors of the theory.

Loops, Surfaces and Grassmann Representation in Two-Dimensional and Three-Dimensional Ising Models, with C.R. Gattringer and G.W. Semenoff, Int. J. Mod. Phys. A14 (1999) p:4549-4574

Starting from the known representation of the partition function of the 2- and 3-D Ising models as an integral over Grassmann variables, we perform a hopping expansion of the corresponding Pfaffian. We show that this expansion is an exact, algebraic representation of the loop- and surface expansions (with intrinsic geometry) of the 2- and 3-D Ising models. Such an algebraic calculus is much simpler to deal with than working with the geometrical objects. For the 2-D case we show that the algebra of hopping generators allows a simple algebraic treatment of the geometry factors and counting problems, and as a result we obtain the corrected loop expansion of the free energy. We compute the radius of convergence of this expansion and show that it is determined by the critical temperature. In 3-D the hopping expansion leads to the surface representation of the Ising model in terms of surfaces with intrinsic geometry. Based on a representation of the 3-D model as a product of 2-D models coupled to an auxiliary field, we give a simple derivation of the geometry factor which prevents overcounting of surfaces and provide a classification of possible sets of surfaces to be summed over. For 2- and 3-D we derive a compact formula for 2n-point functions in loop (surface) representation.

Topology and Duality in Ableian Lattice Theories, with C.R. Gattringer and G.W. Semenoff, Phys. Lett. B425 (1998) p:282-290

We show how to obtain the dual of any lattice model with inhomogeneous local interactions based on an arbitrary Abelian group in any dimension and on lattices with arbitrary topology. It is shown that in general the dual theory contains disorder loops on the generators of the cohomology group of a particular dimension. An explicit construction for altering the statistical sum to obtain a self-dual theory, when these obstructions exist, is also given. We discuss some applications of these results, particularly the existence of non-trivial self-dual 2-dimensional Z_N theories on the torus. In addition we explicitly construct the n-point functions of plaquette variables for the U(1) gauge theory on the 2-dimensional g-tori.

Theta Sectors and Thermodynamics of a Classical Adjoint Gas, with L.D. Paniak, Nucl.Phys.B517 (1998) p:622-637

The effect of topology on the thermodynamics of a gas of adjoint representation charges interacting via 1+1 dimensional SU(N) gauge fields is investigated. We demonstrate explicitly the existence of multiple vacua parameterized by the discrete superselection variable k=1,...,N. In the low pressure limit, the k dependence of the adjoint gas equation of state is calculated and shown to be non-trivial. Conversely, in the limit of high system pressure, screening by the adjoint charges results in an equation of state independent of k. Additionally, the relation of this model to adjoint QCD at finite temperature in two dimensions and the limit of large N are discussed.

Gauged Yukawa Matrix Models and Two-Dimensional Lattice Theories, with H. Hamidian, G.W. Semenoff, P. Suranyi and L.C.R. Wijewardhana, Phys. Rev. D53 (1996) p:5886-5890

We argue that chiral symmetry breaking in three dimensional QCD can be identified with N\'eel order in 2-dimensional quantum antiferromagnets. When operators which drive the chiral transition are added to these theories, we postulate that the resulting quantum critical behavior is in the universality class of gauged Yukawa matrix models. As a consequence, the chiral transition is typically of first order, although for a limited class of parameters it can be second order with computable critical exponents.

What can a Relativistic Quark Model Tell Us About Charmed Mesons?, with M. Sutherland, B. Holdom, and Randy Lewis, Phys. Rev. D51 (1995) p:5053-5063

A relativistic quark model is extended to incorporate chiral and gauge symmetries. We obtain the DD*pi and DD*gamma couplings and find that the ratio Gamma(D*0->D0pi0)/Gamma(D*0->D0gamma) constrains the charm-quark mass close to 1.45 GeV. Large 1/mc corrections appear in the heavy-quark contribution to the DD*gamma coupling. The model is extended further to describe seven excited D meson states. We find that semileptonic B decays into the ground and excited D meson states do not account for the total semileptonic decay width of the B0. The nonresonant contributions to the processes B0->D(*)pi+ell+nu appear to be large enough to account for the discrepancy.