Budhi Arta Surya

Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels

We revisit the previous work of Leland [J Finance 49:1213–1252, 1994], Leland and Toft [J Finance 51:987–1019, 1996] and Hilberink and Rogers [Finance Stoch 6:237–263, 2002] on optimal capital structure and show that the issue of determining an optimal endogenous bankruptcy level can be dealt with analytically and numerically when the underlying source of randomness is replaced by that of a general spectrally negative Lévy process. By working with the latter class of processes we bring to light a new phenomenon, namely that, depending on the nature of the small jumps, the optimal bankruptcy level may be determined by a principle of continuous fit as opposed to the usual smooth fit. Moreover, we are able to prove the optimality of the bankruptcy level according to the appropriate choice of fit.

Optimal Capital Structure with Scale Effects under Spectrally Negative Levy Models

The optimal capital structure model with endogenous bankruptcy was first studied by Leland (1994) and Leland & Toft (1996), and was later extended to the spectrally negative Levy model by Hilberink Rogers (2002) and Kyprianou Surya (2007). This paper incorporates scale effects by allowing the values of bankruptcy costs and tax benefits to be dependent on the firms asset value. By using the fluctuation identities for the spectrally negative Levy process, we obtain a candidate bankruptcy level as well as a sufficient condition for optimality. The optimality holds in particular when, monotonically in the asset value, the value of tax benefits is increasing, the loss amount at bankruptcy is increasing, and its proportion relative to the asset value is decreasing. The solution admits a semi-explicit form in terms of the scale function. A series of numerical studies are given to analyze the impacts of scale effects on the bankruptcy strategy and the optimal capital structure.

Two-dimensional Hull-White model for stochastic volatility and its nonlinear filtering estimation

Abstract This paper presents an extension of the Hull-White model for stochastic volatility. It considers a two-dimensional case where returns of two assets are correlated. The main objective is to estimate the volatility of each asset online given the observation of the returns of the asset prices, taking account the correlation between the asset prices. We propose recursive filtering equations derived from the results of Jazwinski [8] and Maybeck [12] for the estimation.

Optimal double stopping of a Brownian bridge

We study optimal double stopping problems driven by a Brownian bridge. The objective is to maximize the expected spread between the payoffs achieved at the two stopping times. We study several cases where the solutions can be solved explicitly by strategies of a threshold type.

Generalized Phase-Type Distribution and Competing Risks for Markov Mixtures Process

Phase-type distribution has been an important probabilistic tool in the analysis of complex stochastic system evolution. It was introduced by Neuts in 1975. The model describes the lifetime distribution of a finite-state absorbing Markov chains, and has found many applications in wide range of areas. It was brought to survival analysis by Aalen in 1995. However, the model has lacks of ability in modeling heterogeneity and inclusion of past information which is due to the Markov property of the underlying process that forms the model. We attempt to generalize the distribution by replacing the underlying by Markov mixtures process. Markov mixtures process was used to model jobs mobility by Blumen et al. in 1955. It was known as the mover-stayer model describing low-productivity workers tendency to move out of their jobs by a Markov chains, while those with high-productivity tend to stay in the job. Frydman later extended the model to a mixtures of finite-state Markov chains moving at different speeds on the same state space. In general the mixtures process does not have Markov property. We revisit the mixtures model for mixtures of multi absorbing states Markov chains, and propose generalization of the phase-type distribution under competing risks. The new distribution has two main appealing features: it has the ability to model heterogeneity and to include past information of the underlying process, and it comes in a closed form. Built upon the new distribution, we propose conditional forward intensity which can be used to determine rate of occurrence of future events (caused by certain type) based on available information. Numerical study suggests that the new distribution and its forward intensity offer significant improvements over the existing model.

Optimal Portfolio for Terminal Wealth Under Inflation

In this study, the scheme of Dynamic Portfolio consisted of three assets (Stock, Bond and Money account) were generated for investor who want to maximize the discounted expected utility for terminal wealth along finite time horizon in complete market in which inflation rate, interest rate and stock price modeled as stochastic process. As stock prices had evolved with Constant Elasticity of Variance (CEV) process, interest rate and inflation rate had undergone Ornstein-Uhlenbeck (OU) processes. The continuous time framework to solve the Stochastic Differential Equation (SDE) then to be adapted. In this paper, our aim is to maximize discounted utility for terminal wealth in the end of investment period. The above problem is known as stochastic optimal control problem. The wealth process, price level and interest rate had been modelled as state variable and asset portion had been appointed as control function. As Hamilton Jacobi Bellman (HJB) equation had been derived, we were assuming the corresponding Partial Differential Equation was separable. Hence, the closed form solution of optimal trading strategy can be obtained. Simulation was conducted to the real Indonesian financial time series data. Along with Merton’s seminal work and Brennan and Xia (2002), our result of risky asset trading strategies were not presence as feedback form of wealth process. We’ve found that the terminal wealth as long as the asset proportion in the portfolio is depend on the investor’s risk averse degree as well as asset characteristics.

Optimal portfolio in discrete-time under HARA utility function

In this study we are going to discuss about optimal dynamic portfolio strategy given the new information of the market to the investor. The objective is to find the optimal strategy that maximizes the expected total hyperbolic absolute risk aversion (HARA)-utility of investor weight portfolio over finite life time. There are two assets that take place in to the dynamic portfolio model, risky asset and risk-free bond with constant interest rate. The underlying stock price is obtained under binomial process of Markov chain approximation of diffusion process. The stochastic dynamic programming is used as the approach to solve the problem. In contrast to the continuous-time counterpart, the optimal trading strategies are found to be time-dependent in recursive manners. Sufficient conditions for short selling are given in terms of physical and martingale probabilities of the stock price.

Maintaining Hedonistic Lifestyle from Trading: Optimal Portfolio Strategy for Consumption with Binomial Trees

Motivated by observation that seminal work of Samuelson obtaining optimal investment on risky asset which was independent of time and wealth process, in this paper we were replacing the source of uncertainty in the risky asset by binomial process in order to improve the model. Risky asset is determined to move in two possible directions of up and down in the next time instant with certain probability. Working under this model, we manage to obtain explicit solution for the optimal investment and consumption decisions expressed as time-dependent functions of wealth process in feedback form, something more appropriate in intertemporal decision - that our decision today is affecting our condition tomorrow. We specify prohibition of short selling on each asset and we exemplify the results by numerical examples altogether with Monte Carlo simulation.

Finite Maturity Optimal Stopping of Levy Processes with Running Cost, Stopping Cost and Terminal Gain

Motivated by problems in mathematical finance and insurance, this paper discusses optimal stopping problem in general setting. It considers discounted running cost and stopping cost in addition to terminal gain in the objective function, subject to be optimized over finite-time period. The underlying source of uncertainty is modeled by Levy processes. We derive early exercise premium representation for the value function based on a partial integro-differential free-boundary problem associated with the optimal stopping problem. The representation gives rise to a nonlinear integral equation for the optimal stopping boundary. The integral equation generalizes that of found in Kim (1990), Myneni (1992), Carr et al. (1990), Jacka (1991), Pham (1997), and Peskir (2005). The boundary can be characterized as a unique solution of the integral equation within the class of continuous decreasing function of time to maturity. We show that the continuity of the boundary holds when the stopping cost function is either time-independent or decreasing in time. Uniqueness of such solution holds when the running cost and stopping cost functions satisfy a differential inequality. By reformulating the free-boundary problem as a linear complementarity, the problem is solved iteratively by adapting the implicit-explicit method of Cont and Voltchkova (2005) and the Brennan-Schwartz (1977) algorithm that was implemented in Jaillet et al. (1990) and Almendral (2005) for the pricing of American put option. We give an example in optimal capital structure. We also verify numerically the recent results in Kyprianou and Surya (2007) that the smooth pasting condition may not hold for general Levy processes.

Optimal investment and consumption strategies for small investor using Bellman's principle of optimality

This paper discusses optimal investment and consumption strategies in discrete-time setting for a small utility-maximizing investor in a finite-time horizon. The investor is interested in maximizing his/her final utility of wealth with respect to his/her investment and consumption strategies. Within discrete-time framework, we solve the problem using Bellmans principle of optimality. To illustrate the problem, we give some numerical examples based on lattice modelling of stock price movement and make use of MAPLE programming language.