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 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 Mean-Field Games with Financial Applications Machine learning in finance Commodity Modeling and Derivative Valuation Homogenization and Singular Perturbation in Finance and Insurance Real Options Valuation of financial linked insurance products You can find links to my published and working papers below.

### 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 my current students and their research interests:

 Student Research Topic Zhen Qin Commodity Models and Ambiguity Aversion David Farahany Monte Carlo-PDE nethods for Option Valuation Tianyi Jia FX markets and trading Luhui (Luke) Gan Algorithmic Trading Xuancheng (Bill) Huang Mean Field Games Ali Al-Aradi Stochastic Portfolio Theory Philippe Casgrain Partial Information in Algorithmic Trading

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

 Post Docs Research Topic Omid Namvar Gharehshiran Reinforcement learning in algorithmic trading

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 Angel Valov First Passage Times: Integral Equations, Randomization and Analytical Approximations Eddie K.H. Ng Kernel-based Copula Processes Ryan Donnelly Ambiguity Aversion in Algorithmic Trading Jason Ricci Applied Stochastic Control in High Frequency and Algorithmic Trading Yuxiang Chong Valuing and Hedging Options on Mixed Funds

### The FST Methods

The FST, mrFST and irFST method, developed in collaboration with K. Jackson and V. Surkov, for valuing a variety of options allows for efficient computation of path-dependent options for equities, commodities, and interest rate derivatives. 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/

### Publications and Working Papers

[Below you will find my working papers and publications.]

Algorithmic and High Frequency Trading, with Álvaro Cartea and Jose Penalva,

Cambridge University Press, now available!

Order here from CUP

Order here from Amazon.co.uk

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Click here for the book website where you can find data, code and other materials related to the book.

Portfolio Liquidation and Ambiguity Aversion [ PDF] with Álvaro Cartea and Ryan Donnelly

We consider an optimal execution problem where an agent holds a position in an asset which must be liquidated (using limit orders) before a terminal horizon. Beginning with a standard model for the trading dynamics, we analyse how the acknowledgement of model misspecification affects the agent's optimal trading strategy. The three possible sources of misspecification in this context are: (i) the arrival rate of market orders, (ii) the fill probability of limit orders, and (iii) the dynamics of the asset price. We show that ambiguity aversion with respect to each factor of the model has a similar effect on the optimal strategy, but the magnitude of the effect depends on time and inventory position in different ways depending on the source of uncertainty. In addition we allow the agent to employ market orders to further increase the strategy's profitability and show the effect of ambiguity aversion on the shape of the optimal impulse region. In some cases we have a closed-form expression for the optimal trading strategy which significantly enhances the efficiency in which the strategy can be executed in real time.

Hedge and Speculate: replicating option payoffs with limit and market orders [ PDF] with Álvaro Cartea and Luhui Gan

We consider an agent who takes a short position in a contingent claim and employs limit orders (LOs) and market orders (MOs) to trade in the underlying asset to maximize expected utility of terminal wealth. The agent solves a combined optimal stopping and control problem where trading has frictions: MOs (executed by the agent and other traders) have permanent price impact and pay exchange fees, and LOs earn the spread (relative to the midprice of the asset) and pay no exchange fees. We show how the agent replicates the payoff of the claim and also speculates in the asset to maximize expected utility of terminal wealth. In the strategy, MOs are used to keep the inventory on target, to replicate the payoff, and LOs are employed to build the inventory at favorable prices and boost expected terminal wealth by executing roundtrip trades that earn the spread. We calibrate the model to the E-mini contract that tracks the S\&P500 index, provide numerical examples of the performance of the strategy, and proof that our scheme converges to the viscosity solution of the dynamic programming equation.

Beating the Market: Dynamic Asset Allocation with a Market Portfolio Benchmark [ PDF] with Ali Al-Aradi

Beating the market portfolio is a problem faced by many investors. Here, we formulate and solve a dynamic allocation problem that maximizes the utility of relative wealth of the investor's portfolio to that of the market portfolio. We also allow the investor to control the deviation of the optimal allocation from the market portfolio itself, and using stochastic control techniques, provide explicit closed form expressions for the optimal allocation. In addition, we demonstrate how the optimal portfolio can be factored into five passive rule-based portfolios: (i) global minimum variance portfolio; (ii) high-growth portfolio; (iii) high-cash-flow portfolio; (iv) equal-weight portfolio; and (v) risk-parity portfolio. Finally, some numerical experiments based on calibration to real-world data are presented to illustrate the risk-reward profile of the optimal allocation in comparison to these and other commonly used strategies.

Trading algorithms with learning in latent alpha models [ PDF ] with Philippe Casgrain

Alpha signals for statistical arbitrage strategies are often driven by latent factors. This paper analyses how to optimally trade with latent factors that cause prices to jump and diffuse. Moreover, we account for the effect of the trader's actions on quoted prices and the prices they receive from trading. Under fairly general assumptions, we demonstrate how the trader can learn the posterior distribution over the latent states, and explicitly solve the latent optimal trading problem. To illustrate the efficacy of the optimal strategy, we demonstrate its performance through simulations and compare it to strategies which ignore learning in the latent factors.

Mixing LSMC and PDE Methods to Price Bermudan Options [ PDF with David Farahany and Ken Jackson

We develop a mixed least square Monte Carlo-partial differential equation (LSMC-PDE) method for pricing Bermudan style options with assets whose dynamics are driven by latent variables. The algorithm is formulated for an arbitrary number of assets and volatility processes, and its probabilistic convergence is established. Our numerical examples focus on the Heston model and we compare our hybrid algorithm with a full LSMC approach. Using Fourier methods we are able to derive efficient FFT based solutions, and we demonstrate that our algorithm greatly reduces the variance in the computed prices and optimal exercise boundaries. We also compare the early exercise boundaries and prices computed by our hybrid algorithm with those produced by finite difference methods and find excellent agreement.

Speculative Trading of Electricity Contracts in Interconnected Locations [ PDF ] with  Álvaro Cartea and Zhen Qin

We derive an investor's optimal trading strategy of electricity contracts traded in two locations joined by an interconnector. The investor employs a price model which includes the impact of her own trades. The investor's trades have a permanent impact on prices because her trading activity affects the demand of contracts in both locations. Additionally, the investor receives prices which are worse than the quoted prices as a result of the elasticity of liquidity provision of contracts. Furthermore, the investor is ambiguity averse, so she acknowledges that her model of prices may be misspecified and considers other models when devising her trading strategy. We show that as the investor's degree of ambiguity aversion increases, her trading activity decreases in both locations, and thus her inventory exposure also decreases. Finally, we show that there is a range of ambiguity aversion parameters where the Sharpe ratio of the trading strategy increases when ambiguity aversion increases.

Liquidating Baskets of Co-Moving Assets[ PDF ] with  Álvaro Cartea and Luhui Gan

We show how to execute a basket consisting of a subset of co-moving assets and demonstrate how the information carried in other traded assets, which are not in the basket, improves execution performance. Market orders (MOs) from all participants, including the agent's orders to execute her basket, have permanent price impact on the assets, i.e. executions in a single asset affect prices of all assets. Furthermore, we assume the agent's MOs are executed at worse than midprices (by walking the LOB) through a temporary price impact. The execution problem is posed as an optimal stochastic control one and we reduce the dynamic programming equation to a system of coupled partial differential equations, which reduces to a coupled system of Riccati equations when other agents' order flow are deterministic. We use data of five stocks traded in the Nasdaq exchange to estimate the model parameters and use simulations to illustrate the performance of the strategy. As an example, the agent liquidates a portfolio consisting of shares in INTC and SMH. We show that including the information provided by three additional assets (FARO, NTAP, ORCL) considerably improves the strategy's performance -- for the portfolio we execute, it outperforms the multi-asset version of Almgren-Chriss by approximately 2 to 4 basis points per share.

Enhancing Trading Strategies using Order Book Signals [ PDF ] with  Álvaro Cartea and Ryan Donnelly

We use high-frequency data from the Nasdaq exchange to build a measure of volume order imbalance in the limit order book (LOB). We show that our measure is a good predictor of the sign of the next market order (MO), i.e. buy or sell, and also helps to predict price changes immediately after the arrival of an MO. Based on these empirical findings, we introduce and calibrate a Markov chain modulated pure jump model of price, spread, LO and MO arrivals, and order imbalance. As an application of the model, we pose and solve a stochastic control problem for an agent who maximizes terminal wealth, subject to inventory penalties, by executing roundtrip trades using LOs. We use in-sample-data (January to June 2014) to calibrate the model to ten equities traded in the Nasdaq exchange, and use out-of-sample data (July to December 2014) to test the performance of the strategy. We show that introducing our volume imbalance measure into the optimisation problem considerably boosts the profits of the strategy. Profits increase because employing our imbalance measure reduces adverse selection costs and positions LOs in the book to take advantage of favorable price movements.

Trading Strategies within the Edges of No-Arbitrage [ PDF ] with Álvaro Cartea and Jason Ricci

We develop a trading strategy which employs limit and market orders in a multi-asset economy where the assets are not only correlated, but can also be structurally dependent. To model the structural dependence, the midprice processes follow a multivariate reflected Brownian motion on the closure of a no-arbitrage region which is dictated by the assets' bid-ask spreads. We provide a formal framework for such an economy and solve for the value function and optimal control for an investor who takes positions in these assets. The optimal strategy exhibits two dominant features which depend on how far the vector of midprices is from the no-arbitrage bounds. When midprices are sufficiently far from the no-arbitrage edges, the strategy behaves as that of a market maker who posts buy and sell limit orders. And when the midprice vector is close to the edge of the no-arbitrage region, the strategy executes a combination of market orders and limit orders to profit from statistical arbitrages. Moreover, we discuss a numerical scheme to solve for the value function and optimal control, and perform a simulation study to discuss the main characteristics of the optimal strategy.

Algorithmic Trading of Co-Integrated Assets [ PDF ] with Álvaro Cartea, Int. J. Theoretical and Applied Finance, Forthcoming

We assume that the drift in the returns of asset prices consists of an idiosyncratic component and a common component given by a co-integration factor. We analyze the optimal investment strategy for an agent who maximizes expected utility of wealth by dynamically trading in these assets. The optimal solution is constructed explicitly in closed-form and is shown to be affine in the co-integration factor. We calibrate the model to three assets traded on the Nasdaq exchange (Google, Facebook, and Amazon) and employ simulations to showcase the strategy's performance.

Foreign Exchange Markets with Last Look [ PDF ] with Álvaro Cartea

We examine the Foreign Exchange (FX) spot price spreads with and without Last Look on the transaction. We assume that brokers are risk-neutral and they quote spreads so that losses to latency arbitrageurs (LAs) are recovered from other traders in the FX market. These losses are reduced if the broker can reject, ex-post, loss-making trades by enforcing the Last Look option which is a feature of some trading venues in FX markets. For a given rejection threshold the risk-neutral broker quotes a spread to the market so that her expected profits are zero. When there is only one venue, we find that the Last Look option reduces quoted spreads. However, if there are two venues we show that the market reaches an equilibrium where traders have no incentive to migrate. The equilibrium can be reached with both venues coexisting, or with only one venue surviving. Moreover, when one venue enforces Last Look and the other one does not, counterintuitively, it may be the case that the Last Look venue quotes larger spreads.

Model Uncertainty in Commodity Markets [ PDF ] with Álvaro Cartea and Zhen Qin, SIAM Journal of Financial Mathematics, Forthcoming

Agents who acknowledge that their models are incorrectly specified are said to be ambiguity averse, and this affects the prices they are willing to trade at. Models for prices of commodities attempt to capture three stylized features: seasonal trend, moderate deviations (a diffusive factor) and large deviations (a jump factor) both of which mean-revert to the seasonal trend. Here we model ambiguity by allowing the agent to consider a class of models absolutely continuous w.r.t. their reference model, but penalize candidate models that are far from it. The buyer (seller) of a forward contract introduces a negative (positive) drift in the dynamics of the spot price, and enhances downward (upward) jumps so the prices they are willing to trade at are lower (higher) than that of the forward price under P. When ambiguity averse buyers and sellers employ the same reference measure they cannot trade because the seller requires more than what the buyer is willing to pay. Finally, we observe that when ambiguity averse agents price options written on the commodity forward, the effect of ambiguity aversion is strongest when the option is at-the-money, and weaker when it is deep in-the-money or deep out-of-the-money.

Irreversible Investments and Ambiguity Aversion [ PDF ] with Álvaro Cartea

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.

Mean-Field Game Strategies for Optimal Execution [PDF ] with Bill Huang and Mojtaba Nourian

Algorithmic trading strategies for execution often focus on the individual agent who is liquidating/acquiring shares. When generalized to multiple agents, the resulting stochastic game is notoriously difficult to solve in closed-form. Here, we circumvent the difficulties by investigating a mean-field game framework containing (i) a major agent who is liquidating a large number of shares, (ii) a number of minor agents (high-frequency traders (HFTs)) who detect and trade against the liquidator, and (iii) noise traders who buy and sell for exogenous reasons. Our setup accounts for permanent price impact stemming from all trader types inducing an interaction between major and minor agents. Both optimizing agents trade against noise traders as well as one another. This stochastic dynamic game contains couplings in the price and trade dynamics, and we use a mean-field game approach to solve the problem. We obtain a set of decentralized feedback trading strategies for the major and minor agents, and express the solution explicitly in terms of a deterministic fixed point problem. For a finite $N$ population of HFTs, the set of major-minor agent mean-field game strategies is shown to have an $\epsilon_N$-Nash equilibrium property where $\epsilon_N\to0$ as $N\to\infty$.

Incorporating Order-Flow into Optimal Execution [ PDF ] with Álvaro Cartea, Mathematics and Financial Economics, Forthcoming

We provide an explicit closed-form strategy for an investor who executes a large order when market order-flow from all agents, including the investor's own trades, has a permanent price impact. The strategy is found in closed-form when the permanent and temporary price impacts are linear in the market's and investor's rates of trading. We do this under very general assumptions about the stochastic process followed by the order-flow of the market. The optimal strategy consists of an Almgren-Chriss execution strategy adjusted by a weighted-average of the future expected net order-flow (given by the difference of the market's rate of buy and sell market orders) over the execution trading horizon and proportional to the ratio of permanent to temporary linear impacts. We use historical data to calibrate the model to five Nasdaq traded stocks (FARO, SMH, NTAP, ORCL, INTC) and use simulations to show how the strategy performs.

Order-Flow and Liquidity Provision [ PDF ] with Álvaro Cartea

We show how to optimally take positions in the limit order book by placing limit orders at-the-touch when the midprice of the asset is affected by the trading activity of the market. The midprice dynamics have a short-term-alpha component which reflects how instantaneous net order-flow, the difference between the number of market buy and market sell orders, affects the asset's drift. If net-order flow is positive (negative), so short-term-alpha is positive (negative), the strategy may even withdraw from the sell (buy) side of the limit order book to take advantage of inventory appreciation (depreciation) and to protect the trading strategy from adverse selection costs.

A Closed-Form Execution Strategy to Target VWAP [ PDF ] with Álvaro Cartea, SIAM J. Financial Mathematics, Forthcoming

We provide two explicit closed-form optimal execution strategies to target VWAP. We do this under very general assumptions about the stochastic process followed by the volume traded in the market, and the agent's orders have both temporary and permanent impact on the midprice. The strategies that target VWAP are found in closed-form. One strategy consists of TWAP adjusted upward by a fraction of instantaneous order-flow and adjusted downward by the average order-flow that is expected over the remaining life of the strategy. The other strategy consists of the Almgren-Chriss execution strategy adjusted by the expected volume and net order-flow during the remaining life of the strategy. We calibrate model parameters to five stocks traded in Nasdaq (FARO, SMH, NTAP, ORCL, INTC) and use simulations to show that the strategies target VWAP very closely and on average outperform the target by between 0.10 and 8 basis points.

Optimal Execution with Limit and Market Orders [ PDF ] with Álvaro Cartea, Quantitative Finance, Vol. 15, No. 8, 1279–1291

We develop an optimal execution policy for an investor seeking to execute a large order using limit and market orders. The investor solves the optimal policy considering different restrictions on volume of both types of orders and depth at which limit orders are posted. As a particular example we show how the execution policies perform when targeting the volume schedule of the time-weighted-average-price (TWAP). The different strategies considered by the investor always outperform TWAP with an average savings per share of about two to three times the spread. This improvement over TWAP is due to the strategies benefiting from the optimal mix of limit orders, which earn the spread, and market orders, which keep the investor's inventory schedule on target.

How to Value a Gas Storage Facility [ PDF ] with Álvaro Cartea and James Cheeseman, Chapter in Handbook of Multi-Commodity Markets and Products: Structuring, Trading and Risk Management, (The Wiley Finance Series)

We show how to value a storage facility using Least Squares Monte Carlo (LSMC). We present a toy model to understand how to employ the LSMC algorithm and then show how to incorporate realistic constraints in the valuation including: the maximum capacity of the storage, injection and withdrawal rates and costs, and market constraints such as bid-ask spread in the spot market and transaction costs.

Algorithmic Trading with Learning [ PDF ] with Álvaro Cartea and Damir Kinzebulatov, Int. J. Theoretical and Applied Finance, Forthcoming

We propose a model where an algorithmic trader takes a view on the distribution of prices at a future date and then decides how to trade in the direction of her predictions using the optimal mix of market and limit orders. As time goes by, the trader learns from changes in prices and updates her predictions to tweak her strategy. Compared to a trader that cannot learn from market dynamics or form a view of the market, the algorithmic trader's profits are higher and more certain. Even though the trader executes a strategy based on a directional view, the sources of profits are both from making the spread as well as capital appreciation of inventories. Higher volatility of prices considerably impairs the trader's ability to learn from price innovations, but this adverse effect can be circumvented by learning from a collection of assets that co-move.

Optimal Accelerated Share Repurchases [ PDF ] with Damir Kinzebulatov and Dmitri H. Rubisov

An accelerated share repurchase (ASR) allows a firm to repurchase a significant portion of its shares immediately, while shifting the burden of reducing the impact and uncertainty in the trade to an intermediary. The intermediary must then purchase the shares from the market over several days, weeks, or as much as several months. Some contracts allow the intermediary to specify when the repurchase ends, at which point the corporation and the intermediary exchange the difference between the arrival price and the TWAP over the trading period plus a spread. Hence, the intermediary effectively has an American option embedded within an optimal execution problem. As a result, the firm receives a discounted spread relative to the no early exercise case. In this work, we address the intermediary's optimal execution and exit strategy taking into account the impact that trading has on the market. We demonstrate that it is optimal to exercise when the TWAP exceeds \zeta(t) S_t where S_t is the fundamental price of the asset and \zeta(t) is deterministic. Moreover, we develop a dimensional reduction of the stochastic control and stopping problem and implement an efficient numerical scheme to compute the optimal trading and exit strategies.

Algorithmic Trading with Model Uncertainty [ PDF ] with Álvaro Cartea and Ryan Donnelly, SIAM Financial Mathematics, Forthcoming

Because algorithmic traders acknowledge that their models are incorrectly specified we allow for ambiguity in their choices to make their models robust to misspecification. We show how to include misspecification to: (i) the arrival rate of market orders (MOs), (ii) the fill probability of limit orders, and (iii) the dynamics of the midprice of the asset they trade. In the context of market making, we demonstrate that market makers (MMs) adjust their quotes to reduce inventory risk and adverse selection costs. Moreover, robust market making increases the strategy's Sharpe ratio and allows the MM to fine tune the tradeoff between the mean and the standard deviation of profits. Our framework adopts a robust optimal control approach and we provide existence and uniqueness results for the robust optimal strategies as well as a verification theorem. The behavior of the ambiguity averse MM generalizes that of a risk averse MM, and coincide in only one circumstance.

A real options model to evaluate the effect of environmental policies on the oil sands rate of expansion [ PDF ] with Laleh Kobari and Yuri Lawryshyn, Energy Economics, Volume 45, September 2014, Pages 155-165

Canadian oil sands hold the third largest recognized oil deposit in the world. While the rapidly expanding oil sands industry in western Canada has driven economic growth, the extraction of the oil comes at a significant environmental cost. It is believed that the government policies have failed to keep up with the rapid oil sands expansion, creating serious challenges in managing the environmental impacts. This paper presents a practical, yet financially sound, real options model to evaluate the rate of oil sands expansion, under different environmental cost scenarios resulting from governmental policies, while accounting for oil price uncertainty and managerial flexibilities. Our model considers a multi-plant/multi-agent setting, in which labor costs increase for all agents and impact their optimal strategies, as new plants come online. Our results show that a stricter environmental cost scenario delays investment, but leads to a higher rate of expansion once investment begins. Once constructed, a plant is highly unlikely to shut down. Our model can be used by government policy makers, to gauge the impact of policy strategies on the oil sands expansion rate, and by oil companies, to evaluate expansion strategies based on assumptions regarding market and taxation costs.

Optimal Execution with a Price Limiter [ PDF ] with Damir Kinzebulatov, RISK, July 2014

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 Álvaro Cartea, Mathematical Finance, Vol. 25(3), 576-611

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, Quantitative Finance, 14(2) pg. 369-382

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 Álvaro Cartea and Jason Ricci, SIAM Journal of Financial Mathematics, 5.1 (2014): 415-444.

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.

The Generalized Shiryaev's Problem and Skorohod Embedding [ PDF ] with Alex Kreinin and Angel Valov,Theory Probab. Appl., 58(3), 493–502

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, Quantitive Finance, 14(2) pg. 259-270

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, Commodities, Energy and Environmental Finance, vol. 74,chap. Incorporating Managerial Information into Real Option Valuation, pp. 213–238.Springer (2015)

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 Álvaro Cartea, Applied Mathematical Finance, 20 (6) pg. 512-547

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. Euro Journal of Finance, 19 (7-8) pg. 625-644

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 Journal of Financial Mathematics (2) pp. 665-691 (2015)

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. Int. J. Theor. Appl. Finan. 16, 1350034 (2013)

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 LÂ´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 LÂ´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 explicitâ€“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

The variance gamma jump model is known to describe the volatility smile for shortdatedoptions accurately. However, implementation for exotic path-dependent optionscan prove difficult. Here, Claudio Albanese, Sebastian Jaimungal and Dmitri Rubisov usethe method of lines to develop an alternative approach, allowing prices to be calculatedin a more straightforward manner, either analytically or through numerical integration