Recent Advances in Financial Engineering 2014

The following sections are included:

• PREFACE
• TMU Finance Workshop 2014
• Program
• CONTENTS

Our goal in this paper is to look at stochastic financial models and especially on those properties of the involved distributions which are expressed in terms of the moments. The questions discussed and the results presented reveal the role which the moments play in the analysis of distributions. Interesting conclusions can be derived in both cases when available are finitely many moments and when we know all moments.

Among the results included in the paper are sharp lower and/or upper bounds for option prices in terms of a finite number of moments. However the main attention is paid to the determinacy of distributions by their moments. While any light tailed distribution is uniquely determined by its moments, the uniqueness may fail for heavy tailed distributions. And, here is the point. Heavy tailed distributions are essentially involved in stock market modelling, and most of them are non-unique in terms of the moments. This is why the phenomenon non-uniqueness of distributions deserves a special attention.

We treat distributions on the positive half-line used to describe, e.g., stock prices and option prices, and distributions on the whole real line describing log-returns. For reader's convenience we give a brief and unified general picture of existing results about uniqueness and non-uniqueness of distribution in terms of their moments. Some of the results are classical and well-known. In several cases we provide here new arguments along with presenting new recent results on the moment determinacy of random variables and their non-linear transformations and also of stochastic processes which are solutions to SDEs.

Utility indifference pricing is an effective method for investors to construct a strategy in an incomplete market. In fact, if an investor can trade a random endowment under the criteria shown by utility indifference pricing, they can devise financial contracts that are optimized according to their preferences. However, because it does not have the direct implication of equilibrium, the value of the random endowment given by indifference pricing is not necessarily the same as the market price. In this study, we attempt to derive the equilibrium of random endowment under the framework of indifference pricing. However, letting the utility function be of exponential type means that any trade involving random endowment will not appear in equilibrium. Thus, we show that non-zero trade in equilibrium appears by introducing uncertainty in a model, which is one of the sources of market incompleteness.

Shortcomings of continuous and static microstructure models are noted with motivation provided by data from the NASDAQ. The influence of order imbalance on microstructure dynamics is incorporated in to a model which allows the agent to adjust their strategy based on an easily observable quantity. The predictive power of order imbalance allows the agent to decide when they should trade more agressively to take advantage of beneficial price movements, and when they should trade more conservatively to protect against adverse selection effects. High imbalance results in a stronger inclination to place limit buy orders with the opposite effect on limit sell orders.

We formulate a short-selling strategy of a stock and seek the optimal timing of short covering in the presence of a random recall and a loan fee rate in an illiquid stock loan market. The aim is to study how the optimal trading strategy of the short-seller is influenced by the relevant features of the stock loan market. We consider a regime-switching stock price model that captures the transition in between the bull and the bear markets. The solution to the optimal stopping problem is obtained in closed-form based on the techniques in Guo and Zhang (2005). We provide the numerical example to illustrate of importance of a regime-dependent stopping rule for the short-seller's problem.

The paper examines how changes in reversibility of investment affect a firm's investment timing and quantity strategies, in the presence of manager's private information. We find that even under asymmetric information, higher reversibility of investment decreases the investment trigger. More importantly, the quantity under asymmetric information is no longer independent of the degree of reversibility of investment, but increases with it.

This study proposes a joint pricing model for stocks and bonds in a noarbitrage framework. A stock price representation is obtained in a manner consistent with the quadratic Gaussian term structure model, in which the short rate is the quadratic form of the state variables. In this study, specifying the dividend as a function using the quadratic form of the state variables leads to a stock price representation that is exponential-quadratic in the state variables. We prove that the coefficients determining the stock price have to satisfy some matrix equations, including an algebraic Riccati equation. Moreover, we specify the sufficient condition in which the matrix equations do have a unique solution. In our empirical analysis using Japanese data, we obtain estimates with a good fit to the actual data. Furthermore, we estimate the risk premiums for stocks and bonds and analyze how the BOJ's unconventional monetary policy has affected these risk premiums.

In pricing an option on multiple assets, volatilities of individual assets and the correlations among different assets are necessary inputs. In practice, implied volatility surface for individual underlying asset can be used to calibrate the marginal distribution, but information on correlations is generally difficult to obtain. By regarding the correlation between two assets as an ambiguous parameter, we obtain the worst case price to option seller. The corresponding pricing problem is formulated as a stochastic optimal control problem in which the uncertain correlation takes the role as a control function. Consequently, the Black-Scholes equation is replaced by an HJB equation for deriving of the option price bounds. We solve this problem using the framework of stochastic volatility asymptotics and explain how volatility surfaces of individual assets can be robustly pulled together to estimate the price bounds of an option on multiple assets. Empirical study with foreign exchange option data illustrates the practical use of the proposed approach.

In a non-recourse collateralized loan agreement, the lender's recovery is only limited to the market value of the asset as a collateral. In practice, a loan lender can protect himself against any loss by introducing either a margin requirement or a callable feature in the loan contract. In this respect, it is natural to ask: (1) which one of the two mentioned features should be more preferable to the loan manager; (2) how the borrower reacts towards either one of these two features. In the present work, we address these issues by solving explicitly the respective valuation problems of collateralized lending with margin requirement and callable feature under the Black-Scholes model. For the callable feature, we also provide systematic discussions on (1) how to identify whether the smooth-fit condition holds or fails at the optimal stopping boundaries of the associated Dynkin game, and (2) how to solve it when the smooth-fit condition fails at one or both boundaries. We finally utilize these explicit solutions to conduct detailed comparative analysis.

We study the control band policy arising in the context of cash balance management. A policy is specified by four parameters (d, D, U, u). The controller pushes the process up to D as soon as it goes below d and pushes down to U as soon as it goes above u, while he does not intervene whenever it is within the set (d, u). We focus on the case when the underlying process is a spectrally one-sided Lévy process and obtain the expected fixed and proportional controlling costs as well as the holding costs under the band policy.

The Sharpe ratio is not monotone: an investment strategy which yields a higher return may have a lower Sharpe ratio. This may lead to suboptimal investment decisions.

In this paper we present a modification of the Sharpe ratio, which is monotone with respect to second-order stochastic dominance. We study its properties and obtain a representation which allows to compute it in an efficient way.