14) Derivatives Pricing with Marked Point Processes Using Tick-by-Tick Data
Forthcoming: Quantitative Finance
Abstract:
I propose to model stock price tick-by-tick data via a non-explosive marked point process. The arrival of trades is driven by a counting process in which the waiting-time between trades possesses a Mittag-Leffler survival function and price revisions have an infinitely divisible distribution. I show that the partial-integro-differential equation satisfied by the value of European-style derivatives contains a non-local operator in time-to-maturity known as the Caputo fractional derivative. Numerical examples are provided for a marked point process with conditionally Gaussian and with conditionally CGMY price innovations. Furthermore, the infinitesimal generator of the marked point process I derive to price derivatives coincides with that of a Lévy process of either finite or infinite activity.
13) Assessing the Performance of Different Volatility Estimators: A Monte Carlo Analysis (with Dimitris Karyampas)
Forthcoming: Applied Mathematical Finance
Abstract:
We test the performance of different volatility estimators that have recently been proposed in the literature and which have been designed to deal with problems arising when ultra high frequency data are employed: microstructure noise and price discontinuities. Our goal is to provide an extensive simulation analysis for different levels of noise and frequency of jumps to compare the performance of the proposed volatility estimators. We conclude that the MLE-F, a two-step parametric volatility estimator proposed by Cartea and Karyampas (2010), outperforms most of the well known high-frequency volatility estimators when different assumptions about the path properties of stock dynamics are used.
12) How Much Should We Pay for Interconnecting Electricity Markets? A Real Options Approach (with Carlos González-Pedraz)
Energy Economics, Volume 34, Issue 1, January 2012, p 14–30
Abstract:
An interconnector is an asset that gives the owner the option to transmit electricity between two locations. In financial terms, the value of an interconnector is the same as a strip of real options written on the spread between power prices in two markets. We model the spread based on a: seasonal trend, mean-reverting Gaussian process, and mean-reverting jump process. We express the value of these real options in closed-form. We apply our valuation tool to five pairs of European neighboring markets to value a hypothetical one-year lease of the interconnector. We show valuations for different assumptions about the seasonal component of the spread, and different liquidity caps which proxy for the depth of the interconnected power markets. We derive no-arbitrage lower bounds for the value of the interconnector in terms of electricity futures contracts. We find that, depending on the depth of the market, the jumps in the spread can account for between 1% and 40% of the total value of the interconnector. The two markets where an interconnector would be most (resp. least) valuable are Germany and the Netherlands (resp. France and Germany). Finally, we provide rules of thumb to interpret the different interconnector values.
11) Volatility and Covariation of Financial Assets: A High-Frequency Analysis (with Dimitris Karyampas)
Journal of Banking and Finance 35(12), December 2011, p 3319-3334.
Abstract:
Using high frequency data for the price dynamics of equities we measure the impact that market microstructure noise has on estimates of the: (i) volatility of returns; and (ii) variance-covariance matrix of n assets. We propose a Kalman-.filter-based methodology that allows us to deconstruct price series into the true efficient price and the microstructure noise. This approach allows us to employ volatility estimators that achieve very low Root Mean Squared Errors (RMSEs) compared to other estimators that have been proposed to deal with market microstructure noise at high frequencies. Furthermore, this price series decomposition allows us to estimate the variance covariance matrix of n assets in a more efficient way than the methods so far proposed in the literature. We illustrate our results by calculating how microstructure noise affects portfolio decisions and calculations of the equity beta in a CAPM setting.
10) How Duration Between Trades of Underlying Securities Affects Option Prices (with Thilo Meyer-Brandis)
Review of Finance, Volume 14, Issue 4, October 2010, p 749-785
Abstract:
We propose a model for stock price dynamics that explicitly incorporates random waiting times between trades, also known as duration, and show how option prices can be calculated using this model. We use ultra-high-frequency data for blue-chip companies to motivate a particular choice of waiting-time distribution and then calibrate risk-neutral parameters from options data. We also show that the convexity commonly observed in implied volatilities may be explained by the presence of duration between trades. Furthermore, we find that, ceteris paribus, implied volatility decreases in the presence of longer durations, a result consistent with the findings of Engle (2000) and Dufour and Engle (2000) which demonstrates the relationship between levels of activity and volatility for stock prices.
9) Option Pricing with Lévy-Stable Processes Generated by Lévy-Stable Integrated Variance (with Sam Howison)
Quantitative Finance Vol. 9, No. 4, June 2009, pp 397–409
Abstract:
In this paper we show how to calculate European-style option prices when the log-stock and stock returns processes follow a symmetric Lévy-Stable process. We extend our results to price European-style options when the log-stock process follows a negatively skewed Lévy-Stable process.
8) Modelling Electricity Prices with Forward Looking Capacity Constraints (with M.G. Figueroa and H. Geman)
Applied Mathematical Finance, Volume 16, Issue 2, 2009, p 103-122.
Abstract:
We present a spot price model for wholesale electricity prices which incorporates forward looking information that is available to all market players. We focus on information that measures the extent to which the capacity of the England and Wales generation park will be constrained over the next 52 weeks. We propose a measure of `tight market conditions', based on capacity constraints, which identifies the weeks of the year when price spikes are more likely to occur. We show that the incorporation of this type of forward looking information, not uncommon in the electricity markets, improves the modelling of spikes (timing and magnitude) and the different speeds of mean reversion.
7) A Multivariate Commodity Analysis and Applications to Risk Management (with Reik Boerger, Ruediger Kiesel and Gero Schindlmayr)
Journal of Futures Markets, Volume 29, Issue 3, March 2009, p 197-217.
Abstract:
The understanding of joint asset return distributions is an important ingredient for managing risks of portfolios. While this is a well-discussed issue in fixed income and equity markets, it is a challenge for energy commodities. In this paper we are concerned with describing the joint return distribution of energy related commodities futures, namely power, oil, gas, coal and carbon. The objective of the paper is threefold. First, we conduct a careful analysis of empirical returns and show how the class of multivariate generalized hyperbolic distributions performs in this context. Second, we present how risk measures can be computed for commodity portfolios based on generalized hyperbolic assumptions. And finally, we discuss the implications of our findings for risk management analyzing the exposure of power plants which represent typical energy portfolios. Our main findings are that risk estimates based on a normal distribution in the context of energy commodities can be statistically improved using generalized hyperbolic distributions. Those distributions are flexible enough to incorporate many characteristics of commodity returns and yield more accurate risk estimates. Our analysis of the market suggests that carbon allowances can be a helpful tool for controlling the risk exposure of a typical energy portfolio representing a power plant.
6) Spot Price Modeling and the Valuation of Electricity Forward Contracts: The Role of Demand and Capacity (with Pablo Villaplana Conde)
Journal of Banking and Finance 32, Issue 12, (2008), pp. 2502-2519.
Abstract:
We propose a model where wholesale electricity prices are explained by two state variables: demand and capacity. We derive analytical expressions to price forward contracts and to calculate the forward premium. We apply our model to the PJM, England and Wales, and Nord Pool markets. Our empirical findings indicate that volatility of demand is seasonal and that the market price of demand risk is also seasonal and positive, both of which exert an upward (seasonal) pressure on the price of forward contracts. We assume that both volatility of capacity and the market price of capacity risk are constant and find that, depending on the market and period understudy, it could either exert an upward or downward pressure on forward prices. In all markets we find that the forward premium exhibits a seasonal pattern. During the months of high volatility of demand, forward contracts trade at a premium. During months of low volatility of demand, forwards can either trade at a relatively small premium or, even in some cases, at a discount, i.e. they exhibit a negative forward premium.
5) Pricing Forward Contracts in Power Markets by the Certainty Equivalence Principle: explaining the sign of the market risk premium (with Fred Espen Benth and Ruediger Kiesel)
Journal of Banking and Finance 32, Issue 10, (2008), pp. 2006-2021.
Abstract:
In this paper we provide a framework that explains how the market risk premium, defined as the difference between forward prices and spot forecasts, depends on the risk preferences of market players and the interaction between buyers and sellers. In commodities markets this premium is an important indicator of the behavior of buyers and sellers and their views on the market spanning between short-term and long-term horizons. We show that under certain assumptions it is possible to derive explicit solutions that link levels of risk aversion and market power with market prices of risk and the market risk premium. We apply our model to the German electricity market and show that the market risk premium exhibits a term structure which can be explained by the combination of two factors. Firstly, the levels of risk aversion of buyers and sellers, and secondly, how the market power of producers, relative to that of buyers, affects forward prices with different delivery periods.
4) UK Gas Markets: the Market Price of Risk and Applications to Multiple Interruptible Supply Contracts (with Thomas Williams)
Energy Economics, Volume 30, Issue 3, pages 829-846, May 2008.
Abstract:
We employ the Schwartz and Smith model to explore the dynamics of the UK gas markets. We discuss in detail the short-term and long-term market prices of risk borne by the market players and how deviations from expected cyclical storage affect the short-term market price of risk. Finally, we illustrate an application of the model by pricing interruptible supply contracts that are currently traded in the UK.
3) On the fluid limit of the continuous-time random walk with general Lévy jump distribution functions (with Diego del-Castillo-Negrete)
Physical Review E, 76, 2007.
Abstract:
The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions $psi(t)$ and general jump distribution functions $eta(x)$. There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times, $psi ~sim t^{-(1+eta)}$, and algebraic decaying jump distributions, $eta ~sim x^{-(1+alpha)}$, corresponding to Lévy stable processes, the fluid limit leads to the fractional diffusion equation of order $alpha$ in space and order $eta$ in time. Here we consider the CTRW for the most general Lévy stochastic processes in the Lévy-Khintchine representation for the jump distribution function and obtain an integro-differential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated Lévy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. In particular, it is observed that the dynamics exhibits a transition from a super-diffusive regime to a sub-diffusive regime.
2) Fractional Diffusion Models of Option Prices in Markets with Jumps (with Diego del-Castillo-Negrete)
Physica A, 374, pages 749–763, 2007.
Abstract:
Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a Lévy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular Lévy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derived.
1) Pricing in Electricity Markets: a Mean Reverting Jump Diffusion Model with Seasonality (with Marcelo G. Figueroa)
Applied Mathematical Finance, Volume 12, No 4, 313-335, December 2005.
Abstract:
In this paper we present a mean-reverting jump diffusion model for the electricity spot price. We obtain a closed-form solution for forward contracts and calibrate it to market data from England and Wales. Finally, based on the calibrated forward curve we present months, quarters, and seasons-ahead forward surfaces.