S. Jaimungal Department of Statistics and Mathematical Finance Program, University of Toronto Home | Teaching | Research | Seminars | Consulting | Contact |

## Research

### Interests :

 My current interests span Mathematical Finance, Financial Engineering and Actuarial Science. In particular, I have been working on topics including: High frequency algorithmic trading Applied stochastic control in Finance Machine learning in finance Valuation of financial linked insurance products Real Options Homogenization and Singular Perturbation in Finance and Insurance Commodity Modeling and Derivative Valuation You can find links to my published and working papers below.

Together with my colleagues and PhD students, we hold regular meetings at the Field Institute for Mathematical Sciences on various topics in Mathematical Finance and Actuarial Science.

The ICIAM 2011 meeting will be held in Vancouver July 18-22, 2011. The SIAG/FM group is hosting several mini-symposia on math finance

The 2010 SIAM meeting on Financial Mathematics will be held in San Francisco Nov 19-20

The Bachelier Finance Society 6th World Congress 2010 will be held during June 22-26, 2010, Toronto

The 14th Annual International Congress on Insurance: Mathematics and Economics will be held during June 17-19, 2010, Toronto

The Fields Insitute is hosting a Thematic Program on Quantitative Finance Jan-June, 2010, Toronto

### 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, and Prof. X. S. Lin

and several Ph.D. students.

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

 Student Research Topic Yuxiang Chong Valuing and Hedging Options on Mixed Funds Gong Lan Risk Theory Panpan Wu Hedging Errors Jason Ricci High Frequency Trading Ryan Donnelly Ambiguity Aversion in Algorithmic Trading Zhen Qin TBA Dameng Tang TBA Jinyoung Yang TBA

Here is a list of our current Post Docs and their research interests:

 Post Docs Research Topic Yichuan Chi Optimal ReInsurance Damir Kinzebulatov Accelerated Share Repurchase

Here are my former Ph.D. Student's theses:

 Student Thesis Samuel Hikspoors Multi-Factor Energy Price Models and Exotic Derivatives Pricing Georg Sigloch Utility Indifference Pricing of Credit Instruments Vladimir Surkov Option Pricing Using Fourier Space Time-Stepping Framework Angelo Valov First Passage Times: Integral Equations, Randomization and Analytical Approximations Eddie K.H. Ng Kernel-based Copula Processes

### The FST Online Calculator

Our online option calculator uses the FST and mrFST method, developed in collaboration with K. Jackson and V. Surkov, for valuing a variety of options. The underlying dynamics is based on Levy models with and without regime changes and with and without mean-reversion. You can find the relevant research papers below. Matlab source code can also be found here: http://fst-framework.sourceforge.net/

### Finsurance Project

I am a core team member of the MITACS funded finsurance project. This MITACS project focuses on problems at the interface of finance and insurance - hence the name finsurance.

### Publications and Working Papers

Below you will find my working papers and publications.

The Effect of Environmental Policies and Market Uncertainty on the Oilsands Rate of Expansion [ PDF ] with Laleh Kobari and Yuri Lawryshyn

The Canadian oilsands hold a massive oil reserve. However, the extraction of the oil comes at a significant environmental cost. We develop a real-options model to evaluate the rate of oilsands expansion, while accounting for oil price uncertainty, in a multi-agent setting. As new plants come online, labour costs will increase for all oilsands companies. Thus, the optimal actions of the agents are interwoven. We consider three environmental cost scenarios: the first is based on the current government policy where we expect the environmental taxes to be increased over time to account for the true cost of environmental damage; the second is based on a policy where the true cost of environmental damage is captured at all times, and we assume that this cost will decrease as technology improves; and the third is based on a policy where the environmental tax starts at the current level, is increased at a rapid rate to capture the true environmental cost, then decreases as technology improves. Our model also accounts for full and partial carbon tax policies, where the partial policy provides tax exemptions at specified emissions levels, which may lead to agents running their plants at a reduced capacity. Our results show that a stricter environmental cost scenario delays investment, but leads to a higher rate of expansion once investment begins. Once constructed, oil prices need to drop drastically for the plants to shut down. A partial carbon tax policy leads to an increase in the number of plants coming online, sooner.

Optimal Execution with a Price Limiter [ PDF ] with Damir Kinzebulatov

Agents often wish to limit the price they pay for an asset. If they are acquiring a large number of shares, they must balance the risk of trading slowly (to limit price impact) with the risk of future uncertainty in prices. Here, we address the optimal acquisition problem for an agent who is unwilling to pay more than a specified price for an asset while they are subject to market impact and price uncertainty. The problem is posed as an optimal stochastic control and we provide an analytical closed form solution for the perpetual case as well as a dimensional reduced PDE for the general case. The optimal seed of trading is found to no longer be deterministic and instead depends on the fundamental price of the asset. Moreover, we demonstrate that a price limiter constraint significantly reduces the conditional tail expectation of the securities costs.

Risk Metrics and Fine Tuning of High Frequency Trading Strategies[ PDF ] with Alvaro Cartea, Mathematical Finance, Forthcoming

We propose risk measures to assess the performance of High Frequency (HF) trading strategies that seek to maximize profits from making the realized spread where the holding period is extremely short (fractions of a second, seconds or at most minutes). The HF trader is risk-neutral and maximizes expected terminal wealth but is constrained by both capital and the amount of inventory that she can hold at any time. The risk measures enable the HF trader to fine tune her strategies by trading off different measures of inventory risk, which also proxy for capital risk, against expected profits. The dynamics of the midprice of the asset are driven by information flows which are impounded in the midprice by market participants who update their quotes in the limit order book. Furthermore, the midprice also exhibits stochastic jumps as a consequence of the arrival of market orders that have an impact on prices which can give rise to market momentum (expected prices to trend up or down).

Valuing GWBs with Stochastic Interest Rates and Volatility [ PDF ] with Dmitri Rubisov and Ryan Donnelly

Guaranteed withdrawal benefits (GWBs) are long term contracts which provide investors with equity participation while guaranteeing them a secured income stream. Due to the long investment horizons involved, stochastic volatility and stochastic interest rates are important factors to include in their valuation. Moreover, investors are typically allowed to participate in a mixed fund composed of both equity and fixed-income securities. Here, we develop an efficient method for valuing these path-dependent products through re-writing the problem in the form of an Asian styled claim and a dimensionally reduced PDE. The PDE is then solved using an Alternating Direction Implicit (ADI) method. Furthermore, we derive an analytical closed form approximation and compare this approximation with the PDE results and find excellent agreement. We illustrate the various effects of the parameters on the valuation through numerical experiments and discuss their financial implications.

Buy Low Sell High: A High Frequency Trading Perspective [ PDF ] with Alvaro Cartea and Jason Ricci

We develop a High Frequency (HF) trading strategy where the HF trader uses her superior speed to process information and to post limit sell and buy orders. We introduce a multi-factor self-exciting process which allows for feedback effects in market buy and sell orders and the shape of the limit order book (LOB). The model accounts for arrival of market orders that influence activity, trigger one-sided and two-sided clustering of trades, and induce temporary changes in the shape of the LOB. The resulting strategy outperforms the Poisson strategy where the trader does not distinguish between influential and non-influential events.

Irreversible Investments and Ambiguity Aversion [ PDF ]

Real-option valuation traditionally is concerned with investment under conditions of project-value uncertainty, while assuming that the agent has perfect confidence in a specific model. However, agents generally do not have perfect confidence in their models, and this ambiguity affects their decisions. Moreover, real investments are not spanned by tradable assets and generate inherently incomplete markets. In this work, we account for an agent's aversion to model ambiguity and address market incompleteness through the notation of robust indifference prices. We derive analytical results for the perpetual option to invest and the linear complementarity problem that the finite time problem satisfies. We find that ambiguity aversion has dual effects that are similar to, but distinct from, those of risk aversion. In particular, agents are found to exercise options earlier or later than their ambiguity-neutral counterparts, depending on whether the ambiguity stems from uncertainty in the investment or in a hedging asset.

The Generalized Shiryaev's Problem and Skorohod Embedding [ PDF ] with Alex Kreinin and Angel Valov, Probability Theory and its Applications, Forthcoming

In this paper we consider a connection between the famous Skorohod embedding problem and the Shiryaev inverse problem for the first hitting time distribution of a Brownian motion: given a probability distribution, F, find a boundary such that the first hitting time distribution is F. By randomizing the initial state of the process we show that the inverse problem becomes analytically tractable. The randomization of the initial state allows us to significantly extend the class of target distributions in the case of a linear boundary and moreover allows us to establish connection with the Skorohod embedding problem.

Valuing Clustering in Catastrophe Derivatives [ PDF ] with Yuxiang Chong, Forthcoming in Quantitive Finance

The role that clustering in activity and/or severity plays in catastrophe modeling and derivative valuation is a key aspect that has been overlooked in the recent literature. Here, we propose two marked point processes to account for these features. The first approach assumes the points are driven by a stochastic hazard rate modulated by a Markov chain while marks are drawn from a regime specific distribution. In the second approach, the points are driven by a self-exciting process while marks are drawn from a fixed distribution. Within this context, we provide a unified approach to efficiently value catastrophe options -- such as those embedded in catastrophe bonds -- and show that our results are within the 95% confidence interval computed using Monte Carlo simulations. Our approach is based on deriving the valuation PIDE and utilizes transforms to provide semi-analytical closed form solutions. This contrasts with most prior works where the valuation formulae require computing several infinite sums together with numerical integration.

Incorporating Managerial Information into Real Option Valuation [ PDF ] with Yuri Lawryshyn

Real options analysis (ROA) is widely recognized as a superior method for valuing projects with managerial flexibilities. Yet, its adoption remains limited due to varied difficulties in its implementation. In this work, we propose a real options approach that utilizes managerial cash-flow estimates to value early stage project investments. By introducing a sector indicator process which drives the project-value we are able to match arbitrary managerial cash-flow distributions. This sector indicator allows us to value managerial flexibilities and obtain hedges in an easy to implement manner. Our approach to ROA is consistent with financial theory, requires minimal subjective input of model parameters, and bridges the gap between theoretical ROA frameworks and practice.

Modeling Asset Prices for Algorithmic and High Frequency Trading [ PDF ] with Alvaro Cartea, Applied Mathematical Finance, Forthcoming

Algorithmic Trading (AT) and High Frequency (HF) trading, which are responsible for over 70% of US stocks trading volume, have greatly changed the microstructure dynamics of tick-by-tick stock data. In this paper we employ a hidden Markov model to examine how the intra-day dynamics of the stock market have changed, and how to use this information to develop trading strategies at ultra-high frequencies. In particular, we show how to employ our model to submit limit-orders to profit from the bid-ask spread and we also provide evidence of how HF traders may profit from liquidity incentives (liquidity rebates). We use data from February 2001 and February 2008 to show that while in 2001 the intra-day states with shortest average durations were also the ones with very few trades, in February 2008 the vast majority of trades took place in the states with shortest average durations. Moreover, in 2008 the fastest states have the smallest price impact as measured by the volatility of price innovations.

Real Option Valuation with Uncertain Costs [ PDF ] with Max O. de Souza and Jorge P. Zubelli. European J. Finance, Forthcoming

In this work we are concerned with valuing the option to invest in a project when the project value and the investment cost are both mean-reverting. Previous works on stochastic project and investment cost concentrate on geometric Brownian motions (GBMs) for driving the factors. However, when the project involved is linked to commodities, mean-reverting assumptions are more meaningful. Here, we introduce a model and prove that the optimal exercise strategy is not a function of the ratio of the project value to the investment V/I -- contrary to the GBM case. We also demonstrate that the limiting trigger curve as maturity approaches traces out a non-linear curve in the (V,I) plan and derive its explicit form. Finally, we numerically investigate the finite-horizon problem using the Fourier space time-stepping algorithm of Jaimungal & Surkov (2009). Numerically, the optimal exercise policies are found to be approximately linear in V/I; however, contrary to the GBM case they are not described by a curve of the form V*/I* = c(t). The option price behavior as well as the trigger curve behavior nicely generalize earlier one-factor model results.

Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models [ PDF ] with Jean-Pierre Fouque and Matthew Lorig. SIAM J. Financial Mathematics (2) pp. 665-691

Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in order to demonstrate the versatility of our method. These include: European options, up-and-out options, double-barrier knock-out options, and options which pay a rebate upon hitting a boundary. For European options, our method is shown to produce option price approximations which are equivalent to those developed in Fouque, Papanicolaou, and Sircar (2000).

Valuing Early Exercise Interest Rate Options with Multi-Factor Affine Models [ PDF ][ Matlab ] with Vladimir Surkov.

Multi-factor interest rate models are widely used in practice. Quite often, contingent claims with earlier exercise features are valued by resorting to trees, finite-difference schemes and Monte Carlo simulations. However, when jumps are present these methods are less accurate and/or efficient. In this work we develop an algorithm based on a sequence of measure changes coupled with Fourier transform solutions of the pricing partial-integro differential equation to solve the pricing problem. The method, coined the irFST method, also neatly computes option sensitivities. Furthermore, we develop closed form formulae for accrual swaps and accrual range notes under our multi-factor jump-diffusion model. We demonstrate the versatility and precision of the method through numerical experiments on European, Bermudan and callable bond options, (accrual) swaps and range notes.

Randomized First Passage Times [ PDF ] with Alex Kreinin and Angelo Valov.

In this article we study a problem related to the first passage and inverse first passage time problems for Brownian motions originally formulated by Jackson, Kreinin and Zhang (2009). Specifically, define $\tau_X = \inf\{t>0:W_t + X \le b(t) \}$ where $W_t$ is a standard Brownian motion, then given a boundary function $b:[0,\infty) \to \RR$ and a target measure $\mu$ on $[0,\infty)$, we seek the random variable $X$ such that the law of $\tau_X$ is given by $\mu$. We characterize the solutions, prove uniqueness and existence and provide several key examples associated with the linear boundary.

Kernel-Based Copula Processes [ PDF ] with Eddie K.H. Ng, ECML-PKDD 2009, LNAI 5781, pp. 628-643, 2009.

Kernel-based Copula Processes (KCPs), a new versatile tool for analyzing multiple time-series, are proposed here as a unifying framework to model the interdependency across multiple time-series and the long-range dependency within an individual time-series. KCPs build on the celebrated theory of copula which allows for the modeling of complex interdependence structure, while leveraging the power of kernel methods for efficient learning and parsimonious model specification. Specifically, KCPs can be viewed as a generalization of the Gaussian processes enabling non-Gaussian predictions to be made. Such non Gaussian features are extremely important in a variety of application areas. As one application, we consider temperature series from weather stations across the US. Not only are KCPs found to have modeled the heteroskedasticity of the individual temperature changes well, the KCPs also successfully discovered the interdependencies among different stations. Such results are beneficial for weather derivatives trading and risk management, for example.

Incorporating Risk Aversion and Model Misspecification into a Hybrid Model of Default [ PDF ] with G. Sigloch. Mathematical Finance, Vol. 22 (1), pp. 57-81, 2012

It is well known that purely structural models of default cannot explain short term credit spreads, while purely intensity based models of default lead to completely unpredictable default events. Here we introduce a hybrid model of default in which a firm enters distress upon a non-tradable credit worthiness index (CWI) hitting a critical level. Upon distress, the firm defaults at the next arrival of a Poisson process. To value defaultable bonds and CDSs we introduce the concept of robust indifference pricing which differs from the usual indifference valuation paradigm by the inclusion of model uncertainty. To account for model uncertainty, the embedded optimization problems are modified to include a minimization over a set of candidate measures equivalent to the estimated reference measure. With this new model and pricing paradigm, we succeed in determining corporate bond spreads and CDS spreads and find that model uncertainty plays a similar, but distinct, role to risk aversion. In particular, model uncertainty allows for significant short term spreads.

Integral Equations and the First Passage Time of Brownian Motions [ PDF ] with A. Kreinin and A. Valov.

The first passage time problem for Brownian motions hitting a barrier has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the barrier itself have appeared. Most interestingly, Peskir(2002b) demonstrates that a master integral equation can be used to generate a countable number of new equations via differentiation or integration by parts. In this article, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of Volterra type integral equations of the first kind and prove uniqueness for a subclass. Furthermore, through the integral equations, we demonstrate how certain functional transforms of the boundary affect the density function. Finally, we demonstrate a fundamental connection between the Volterra integral equations and a class of Fredholm integral equations.

An Insurance Risk Model with Stochastic Volatility [ PDF ] with Yichun Chi and Sheldon X. Lin. Insurance: Mathematics and Economics.46(1), pg. 52-66.

In this paper, we extend the Cramer-Lundberg insurance risk model perturbed by diffusion to incorporate stochastic volatility and study the resulting Gerber-Shiu expected discounted penalty(EDP) function. Under the assumption that volatility is driven by an underlying Ornstein-Uhlenbeck (OU) process, we derive the integro-differential equation which the EDP function satisfies. Not surprisingly, no closed-form solution exists; however, assuming the driving OU process is fast mean-reverting, we apply singular perturbation theory to obtain an asymptotic expansion of the solution. Two integro-differential equations for the first two terms in this expansion are obtained and explicitly solved. When the claim size distribution is of phase-type, the asymptotic results simplify even further and we succeed in estimating the error of the approximation. Hyper-exponential and mixed-Erlang distributed claims are considered in some detail.

Levy Based Cross-Commodity Models and Derivative Valuation [ PDF ][ Matlab ] with Vladimir Surkov. SIAM Journal on Financial Mathematics (2) pp.464-487

Energy commodities, such as oil, gas and electricity, lack the liquidity of equity markets, have large costs associated with storage, exhibit high volatilities and can have significant spikes in prices. Furthermore, and possibly most importantly, commodities tend to revert to long run equilibrium prices. Many complex commodity contingent claims exist in the markets, such as swing and interruptible options; however, the current method of valuation relies heavily on Monte Carlo simulations and tree based methods. In this article, we develop a new framework for dealing with mean-reverting jump-diffusion (and pure jump) models by working in Fourier space. The method is based on the Fourier space time stepping algorithm of Jackson, Jaimungal, and Surkov (2008), but is tailored for mean-reverting models. We demonstrate the utility of the method by applying it to the valuation of European, American and barrier options on a single underlier, European and Bermudan spread options on two-dimensional underliers, and swing options.

Stepping Through Fourier Space [ PDF ][ Matlab ] with Vladimir Surkov. Risk, July, 2009, p78-83.

Diverse finite-difference schemes for solving pricing problems with Levy underliers have been used in the literature. Invariably, the integral and diffusive terms are treated asymmetrically, large jumps are truncated, the methods are difficult to extend to higher dimensions and cannot easily incorporate regime switching or stochastic volatility. We present a new efficient approach which switches between Fourier and real space as time propagates backwards. We dub this method Fourier Space Time-Stepping (FST). The FST method applies to regime switching Levy models and is applicable to a wide class of path-dependent options (such as Bermudan, barrier, shout and catastrophe linked options) and options on multiple assets.

Fourier Space Time Stepping for Option Pricing with Levy Models [ PDF ][ Matlab ] with Kenneth R. Jackson and Vladimir Surkov. Journal of Computational Finance, Vol 12 Issue 2, p1-29.

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. Applied Mathematical Finance, vol 15 Issue 5&6, p449-447.

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.

Consistent Functional PCA for Financial Time-Series [ PDF ] with Eddie. K. H. Ng, 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, 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 L´evy models which is applicable to a wide class of path-dependent options (such as Bermudan, barrier, and shout options) and options on multiple assets.

Energy Spot Price Models and Spread Options Pricing[ PDF ] with Samuel Hikspoors, International Journal of Theoretical and Applied Finance, vol 10(7), pg. 1111-1135.

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.

Catastrophe options with stochastic interest rates and compound Poisson losses[ PDF ] with Tao Wang, Insurance Mathematics and Economics (2006) vol 38 (3) 469-483

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 on the options price are illustrated through numerical experiments. Furthermore, we carry out a simulation analysis to hedge a short position in the catastrophe put option by using a DeltaGammaRho neutral self-financing portfolio. We find that accounting for stochastic interest rates, through Rho hedging, can significantly reduce the expected conditional loss of the hedged portfolio.

Pricing Equity Linked Pure Endowments with Risky Assets that follow Levy Processes.[ PDF ] with Virginia Young, Insurance Mathematics and Economics (2005) vol 36 (3) 329-346

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 Levy 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 Claudio Albanese and Dmitri .H. Rubisov, Quantitative Finance (2003) vol 3(2) 145-154

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 Claudio Albanese and Dmitri .H. Rubisov, RISK, Feb. issue, pg. 65-70

Coulomb Gas Representation of Low Energy QCD and Monopoles [ PDF ] with Ariel 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. The Wilson loop operator as well as the creation operator for the domain wall in the CG representation is also constructed.

Persistent Currents in Mesoscopic Rings and Boundary Conformal Field Theory [ PDF ] with M.H. Amin and G. Rose, International Journal of Modern Physics (1999) B 13 pg. 3171-3181

A tight-binding model of electron dynamics in mesoscopic normal rings is studied using boundary conformal field theory. The partition function is calculated in the low energy limit and the persistent current generated as a function of an external magnetic flux threading the ring is found. We study the cases where there are defects and electron–electron interactions separately. The same temperature scaling for the persistent current is found in each case and the functional form can be fitted, with a high degree of accuracy, to experimental data.

Phase transition in quantum gravity [ PDF ] with Viqar Husain, Modern Physics Letters A (1999) Volume: 14, Issue: 16(1999) pp. 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 nonperturbative 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 space-time metric functions. These are massless below a critical energy scale and become massive above it. There is a corresponding breaking of classical symmetry.