Real Options under Choquet-Brownian Ambiguity

Real options models characterized by the presence of ambiguity have been recently proposed. But based on recursive multiple-priors approaches to solve ambiguity, these seminal models reduce individual preferences to extreme pessimism by considering only the worst case scenario. In contrast, by relying on dynamically consistent Choquet-Brownian motions to model the dynamics of ambiguous expected cash flows, we show that a much broader spectrum of attitudes towards ambiguity may be accounted for. In the case of a perpetual real option to invest, ambiguity aversion delays the moment of exercise of the option, while the opposite holds true for an ambiguity lover.Real Options; Ambiguity; Irreversible investment; Optimal stopping; Knightian uncertainty; Choquet-Brownian motions

A unified pricing of variable annuity guarantees under the optimal stochastic control framework

In this paper, we review pricing of variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework, and review the existing numerical methods. For numerical valuation of these contracts, we develop a direct integration method based on Gauss-Hermite quadrature with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is very efficient when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We also present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a benchmark for practitioners and researchers developing pricing of variable annuity guarantees.Comment: Keywords: variable annuity, guaranteed living and death benefits, guaranteed minimum accumulation benefit, optimal stochastic control, direct integration metho

Stochastic modeling of private equity: an equilibrium based approach to fund valuation

In this paper, we present a new approach to measure the returns of private equity investments based on a stochastic model of the dynamics of a private equity fund. Our stochastic model of a private equity fund consists of two independent stages: the stochastic model of the capital drawdowns and the stochastic model of the capital distributions over a fund's lifetime. Capital distributions are assumed to follow lognormal distributions in our approach. A mean-reverting square-root process is applied to model the rate at which capital is drawn over time. Applying equilibrium intertemporal asset pricing consideration, we are able to derive closed-form solutions for the market value and time-weighted model returns of a private equity fund. --Private Equity Funds,Stochastic Modeling,Mean-Reverting Square-Root Process,Incomplete Markets

An analysis of spending behaviour under liquidity constraints with an application to financial hedging

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Stock Price Dynamics and Option Valuations under Volatility Feedback Effect

According to the volatility feedback effect, an unexpected increase in squared volatility leads to an immediate decline in the price-dividend ratio. In this paper, we consider the properties of stock price dynamics and option valuations under the volatility feedback effect by modeling the joint dynamics of stock price, dividends, and volatility in continuous time. Most importantly, our model predicts the negative effect of an increase in squared return volatility on the value of deep-in-the-money call options and, furthermore, attempts to explain the volatility puzzle. We theoretically demonstrate a mechanism by which the market price of diffusion return risk, or an equity risk-premium, affects option prices and empirically illustrate how to identify that mechanism using forward-looking information on option contracts. Our theoretical and empirical results support the relevance of the volatility feedback effect. Overall, the results indicate that the prevailing practice of ignoring the time-varying dividend yield in option pricing can lead to oversimplification of the stock market dynamics.Comment: 23 pages, 7 figures, 2 table

Real Option Theory and Application to the Fishery Industry. A survey of the literature

WP 08/14; This paper would be a review of the literature of the main and innovative methodologies of evaluation of real investments: the real option approach (ROA). In particular, the aim of this work is to define an optimal methodology and to select the main drivers that permit to make a more accurate evaluation of the investments in the fishery market. ROA methodology comes from the need to overtake the traditional theory of the net present value (NPV) and from the need for the management of a fishery enterprise to adapt to the future market conditions and to the competitive behavior in the changes of the fishery techniques. ROA was born from the theory of Dixit & Pyndick (1994) that started to use the models of the financial option theories in order to evaluate investments in other sectors like oil, energy, ICT, manufacturing. From a theoretical point of view, indeed, real investments are characterized by “irreversibility” and “possibility of delay” since a manager can defer, expand, abandon an initial project in different years of its own operational life. In this context, despite of the financial option models ROA has a real investment as underlying asset. If the enterprise decides to invest in a real investment it means that the enterprise exercises an option and this decision is irreversible. In the context of the Ritmare project, we would use the same methodological approach by using the evaluation of the investments in the fishery market. Our first step is to provide a review of the main papers that focus on ROA in the fishery with some empirical applications. Finally, we also try to underline the main drivers or variables of the literature that permits to use the ROA and to present a possible scheme of work to apply to the fishery market, by using data at regional or municipal leve

Essays in Equilibrium Finance

The first chapter, Open-Loop Equilibria and Perfect Competition in Option Exercise Games , a joint work with Professor Kerry Back, is concerned with the optimal exercise and valuation of growth options within a partial equilibrium setting. A finite number of firms invest irreversibly into production capacities. It is well known (see e.g. Dixit and Pindyck (1994)) that irreversibility creates an option-like feature and it is usually not optimal for a monopolistic firm to exercise its investment option when its option is 'at the money'. But what if there is more than one firm competing for market shares? Competing firms invest earlier in equilibrium. With the number of firms going to infinity, the value of the option of waiting to invest approaches zero. In the limit, investment is undertaken as soon as its net present value reaches zero. Our contribution is to provide a rigorous proof for the statement that the strategies in Grenadier (2002) – properly reinterpreted - form an open-loop equilibrium. As open loop strategies lack subgame perfectness, we further show that perfect competition forms a subgame perfect equilibrium already for two firms. In general equilibrium, the central planner's problem is analogous to the monopolistic problem in partial equilibrium. The planner maximizes utility over all admissible investment paths just as a firm maximizes profits. So it is natural to hypothesize that there also exists an 'option premium of waiting to invest' for a welfare maximizing central planner. But how would a delay reconcile with perfect competition and zero profits for firms? This question is addressed in the second chapter within a stylized general equilibrium model with irreversible investment. While it is true that with a single consumption good there are no relative prices within a particular instant of time, i.e. there are no intra-period prices, prices can still be related on their intertemporal dimension. It are precisely the dynamics of intertemporal prices, i.e. interest rates and future prices, that reconcile investment delays with zero profits in the context of the model. Longer term interest rates and futures on wages contain the expected growth-effect of optimally exercised growth options, rendering current investment opportunities unprofitable whenever a delay is efficient. In this sense, the term-structure of future prices reflects the option premium of waiting and leads to optimal delay in investment. Interestingly, this mechanism is similar to what Keynes termed the 'speculative motive' for money demand and liquidity preference. Thinking about Keynes' theory and the speculative motive in particular, naturally leads to questions linked to liquidity preference, such as “What exactly is a liquidity trap? The third and last chapter attempts to make one first step towards this direction by approaching a more elementary question. It asks: Why do people exchange real goods against a piece of paper that neither provides intrinsic utility nor (unlike in Keynes times) constitutes a claim on a real good such as gold? Why is money a safe asset whose value people (can) rely upon? In the model presented in Chapter 3 money is 'safe': Fiat money has strictly positive value in the unique trembling hand equilibrium. This holds as each bank note is both: a witness for the existence of some agent in the economy with debt, backed by collateral, and the only matter that allows the debtor to settle her debt. Debtors fear to lose the collateral and compete with each other for not defaulting. Hence they compete for money. This creates money demand and thereby ensures positive money value. As not only a single but all debtors in the economy demand money, idiosyncratic shocks to solvency wash out. This makes fiat money a safe asset

Essays in Quantitative Finance

This thesis contributes to the quantitative finance literature and consists of four research papers.Paper 1. This paper constructs a hybrid commodity interest rate market model with a stochastic local volatility function that allows the model to simultaneously fit the implied volatility of commodity and interest rate options. Because liquid market prices are only available for options on commodity futures (not forwards), a convexity correction formula is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given cross-correlation structure between forward interest rates and commodity prices. When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), whereas in the case of futures, a (rapidly converging) iterative fitting procedure is presented. The cross-correlation fitting is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the “orthonormal Procrustes” problem. The calibration approach is demonstrated on market data for oil futures.Paper 2. This paper describes an efficient American Monte Carlo approach for pricing Bermudan swaptions in the LIBOR market model using the Stochastic Grid Bundling Method (SGBM) which is a regression-based Monte Carlo method in which the continuation value is projected onto a space in which the distribution is known. We demonstrate an algorithm to obtain accurate and tight lower–upper bound values without the need for the nested Monte Carlo simulations that are generally required for regression-based methods.Paper 3. The credit valuation adjustment (CVA) for over-the-counter derivatives are computed using the portfolio’s exposure over its lifetime. Usually, future exposure is approximated by Monte Carlo simulations. For derivatives that lack an analytical approximation for their mark-to-market (MtM) value, such as Bermudan swaptions, the standard practice is to use the regression functions from the least squares Monte Carlo method to approximate their simulated MtMs. However, such approximations have significant bias and noise, resulting in an inaccurate CVA charge. This paper extend the SGBM to efficiently compute expected exposure, potential future exposure, and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates how different measures, such as spot and terminal measures, can simultaneously be employed in the SGBM framework to significantly reduce the variance and bias.Paper 4. This paper presents an algorithm for simulation of options on Lévy driven assets. The simulation is performed on the inverse transition matrix of a discretised partial differential equation. We demonstrate how one can obtain accurate option prices and deltas on the variance gamma (VG) and CGMY model through finite element-based Monte Carlo simulations

Real option approach to investments in electricity generating capacity

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2000.Includes bibliographical references (leaves 101-105).by Jean-Baptiste Fayet.S.M