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Automated market makers (AMMs) are a new prototype of trading venues which are revolutionising the way market participants interact. At present, the majority of AMMs are constant function market makers (CFMMs) where a deterministic trading function determines how markets are cleared. A distinctive characteristic of CFMMs is that execution costs are given by a closed-form function of price, liquidity, and transaction size. This gives rise to a new class of trading problems. We focus on constant product market makers and show how to optimally trade a large position in an asset and how to execute statistical arbitrages based on market signals. We employ stochastic optimal control tools to devise two strategies. One strategy is based on the dynamics of prices in competing venues and assumes constant liquidity in the AMM. The other strategy assumes that AMM prices are efficient and liquidity is stochastic. We use Uniswap v3 data to study price, liquidity, and trading cost dynamics, and to motivate the models. Finally, we perform consecutive runs of in-sample estimation of model parameters and out-of-sample liquidation and arbitrage strategies to showcase the performance of the strategies., Paper in [SSRN 4144743]

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 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

We show how to calculate European-style option prices when the log-stock price process follows a Lévy-Stable process with index parameter 1≤α≤2 and skewness parameter -1≤β≤1. Key to our result is to model integrated variance as an increasing Lévy-Stable process with continuous paths in Τ

We employ the Schwartz and Smith (2000) 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

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

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

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

In this paper we present a mean-reverting jump diffusion model for the electricity spot price and derive the corresponding forward in closed-form. Based on historical spot data and forward data from England and Wales we calibrate the model and present months, quarters, and seasons–ahead forward surfaces

In this paper we derive analytic expressions for the value of European Put and Call options when the stock process follows an exponential Lévy-Stable process. It is shown that the generalised Black-Scholes operator for the Lévy-Stable case can be obtained as an asymptotic approximation of a process where the random variable follows a Damped- Lévy process. Finally, it is also shown that option prices under the Lévy-Stable case generate the volatility smile encountered in the financial markets when the Black-Scholes framework is employed The first author acknowledges financial support from JP Morgan