Density functionals, with an option-pricing application

We present a method of estimating density-related functionals, without prior knowledge of the density’s functional form. The approach revolves around the specification of an explicit formula for a new class of distributions that encompasses many of the known cases in statistics, including the normal, gamma, inverse gamma, and mixtures thereof. The functionals are based on a couple of hypergeometric functions. Their parameters can be estimated, and the estimates then reveal both the functional form of the density and the parameters that determine centering, scaling, etc. The function to be estimated always leads to a valid density, by design, namely, one that is nonnegative everywhere and integrates to 1. Unlike fully nonparametric methods, our approach can be applied to small datasets. To illustrate our methodology, we apply it to finding risk-neutral densities associated with different types of financial options. We show how our approach fits the data uniformly very well. We also find that our estimated densities’ functional forms vary over the dataset, so that existing parametric methods will not do uniformly well

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Options and market making

Options and market making are recurring themes in Mathematical Finance. This thesis explores both topics with the ultimate goal of developing an options market making model for exchange-traded vanilla options. We start the derivation of closed-form optimal controls for an asset-agnostic market making model with multiple assets via an ergodic limit. We then investigate the intraday dynamics of options and its connection with spot volatility to gain insights on the high-frequency option price dynamics and on volatility and Greeks estimation. Finally, we develop a market making model for exchange-traded vanilla options that encompasses relevant features that we observe empirically. Closed-form solutions for the options market making model can be obtained via small time-to-horizon asymptotics. The optimal spreads in the small time-to-horizon regime allow us to empirically study options spreads and trading activity.Open Acces

Optimal algorithmic trading and market microstructure

The efficient frontier is a core concept in Modern Portfolio Theory. Based on this idea, we will construct optimal trading curves for different types of portfolios. These curves correspond to the algorithmic trading strategies that minimize the expected transaction costs, i.e. the joint effect of market impact and market risk. We will study five portfolio trading strategies. For the first three (single-asset, general multi-asseet and balanced portfolios) we will assume that the underlyings follow a Gaussian diffusion, whereas for the last two portfolios we will suppose that there exists a combination of assets such that the corresponding portfolio follows a mean-reverting dynamics. The optimal trading curves can be computed by solving an N-dimensional optimization problem, where N is the (pre-determined) number of trading times. We will solve the recursive algorithm using the "shooting method", a numerical technique for differential equations. This method has the advantage that its corresponding equation is always one-dimensional regardless of the number of trading times N. This novel approach could be appealing for high-frequency traders and electronic brokers.quantitative finance; optimal trading; algorithmic trading; systematic trading; market microstructure

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

"The Contributions of Professors Fischer Black, Robert Merton, and Myron Scholes to the Financial Services Industry"

This paper is written as a tribute to Professors Robert Merton and Myron Scholes, winners of the 1997 Nobel Prize in economics, as well as to their collaborator, the late Professor Fischer Black. We first provide a brief and very selective review of their seminal work in contingent claims pricing. We then provide an overview of some of the recent research on stock price dynamics as it relates to contingent claim pricing. The continuing intensity of this research, some 25 years after the publication of the original Black-Scholes paper, must surely be regarded as the ultimate tribute to their work. We discuss jump-diffusion and stochastic volatility models, subordinated models, fractal models, and generalized binomial tree models, for stock price dynamics and option pricing. We also address questions as to whether derivatives trading poses a systemic risk in the context of models in which stock price movements are endogenized, and give our views on the "LTCM crisis" and liquidity risk.

The impact of transactions costs in the UK stock market: Evidence and implications

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.There has been an increasing interest in the finance literature regarding the impact of transactions costs on US equity markets. The US empirical evidence indicates that transactions costs influence both trading volume (Atkins and Dyl (1997)) and asset returns (Amihud and Mendelson (1986)). Additionally, the theoretical finance literature also indicates that transactions costs affect equilibrium asset returns (Fisher (1994)). In this thesis we assess the impact of transactions costs on the UK equity markets, from four aspects. Firstly, we provide empirical support to the hypothesis that transactions costs affect the "holding period" of an asset in the portfolio of an investor. Secondly, we provide robust results showing that transactions costs affect equilibrium asset returns. Thirdly, we explain the variability of transactions costs with the use of information asymmetry, proxied by the variance of analysts' forecasts, in the spirit of Kim and Verrecchia (1994, 2001). Finally, we find that stock price and trading volume reaction to changes in the FTSE 100 list can be explained by liquidity effects, as proxied by the bid-ask spread. We provide overwhelming evidence, suggesting that transactions costs are important in UK equity markets.Economics and Finance Department, Brunel Universit

We develop a sequential trade model of Iceberg order execution in a limit order book. The Iceberg-trader has the freedom to expose his trading intentions or (partially) shield the true order size against other market participants. Order exposure can cause drastic market reactions (“market impact”) in the end leading to higher transaction costs. On the other hand the Iceberg trader faces a loss-in-priority when he hides his intentions, as most electronic limit order books penalize the usage of hidden liquidity. Thus the Iceberg-trader is faced with the problem to find the right trade-off. Our model provides optimal exposure strategies for Iceberg traders in limit order book markets. In particular, we provide a range of analytical statements that are in line with recent empirical findings on the determinants of trader’s exposure strategies. In this framework, we also study the market impact also market impact of limit orders. We provide optimal exposure profiles for a range of hightech stocks from the US S&P500 and how they scale with the state-of-the-book. We finally test the Iceberg’s performance against the limit orders and find that Iceberg orders can significantly enhance trade performance by up to 60%.

Hidden Liquidity, Iceberg Orders, Limit Order Book, Market Impact of Limit Orders, Optimal Exposure, Trading Strategies, Iceberg versus Limit Order, Pre-trade transparency, Agency-Trading.