Double Exponential Jump Diffusion Process Research Articles

An Algorithm for the Pricing and Timing of the Option to make a Two-Stage Investment with Credit Guarantees

We develop a jump-diffusion model for a guarantee-investment combination financing mode (G-I mode) that is recently popular in financial practice. We assume that a borrower has exclusively an option to invest in a project in two stages. The project’s cash flow follows a double exponential jump-diffusion process and it is increased by a growth factor once the second-stage investment is exercised. The first-stage investment cost is financed by a bank loan with the guarantee provided by an insurer, who promises to provide the second-stage investment cost as well as take the lender’s all default losses. In return for the guarantee and investment, the borrower pays a guarantee fee upon first investment and grants a fraction of equity upon second investment to the insurer. In sharp contrast to prior papers on guarantee, the guarantee costs are contracted prior to investment. We provide closed-form solutions and produce a numerical algorithm for the timing and pricing of the two investment options.

An algorithm for the pricing and timing of the option to make a two-stage investment with loan guarantees

We develop a jump-diffution model for a guarantee-investment combination financing mode (G-I mode) that is recently popular in financial practice. We assume that a borrower has exclusively an option to invest in a project in two stages. The project's cash flow follows a double exponential jump-diffusion process and it is increased by a growth factor once the second-stage investment is exercised. The first-stage investment cost is financed by a bank loan with the guarantee provided by an insurer, who promises to provide the second-stage investment cost as well as take the lender's all default losses. In return for the guarantee and investment, the borrower pays a guarantee fee upon first investment and grants a fraction of equity upon second investment to the insurer. In sharp contrast to prior papers on guarantee, the guarantee costs are contracted prior to investment. We provide closed-form solutions and produce a numerical algorithm for the timing and pricing of the two investment options.

LONG MEMORY VERSION OF STOCHASTIC VOLATILITY JUMP-DIFFUSION MODEL WITH STOCHASTIC INTENSITY

It is widely accepted that certain financial data exhibit long range dependence. We consider a fractional stochastic volatility jump diffusion model in which the stock price follows a double exponential jump diffusion process with volatility described by a long memory stochastic process and intensity rate expressed by an ordinary Cox, Ingersoll, and Ross (CIR) process. By calibrating the model with real data, we examine the performance of the model and also, we illustrate the role of long range dependence property by comparing our presented model with the Heston model.

On the probability of default in a market with price clustering and jump risk

In this paper, we consider the probability of default for financial variables under a tractable stochastic model which can capture both the price clustering phenomenon and the jump risk. We assume that the logarithm of the price dynamics is driven by a so-called sticky double exponential jump diffusion process. In particular, the price clustering is incorporated by time changing the jump diffusion process, and the jump risk is described through a Poisson stochastic integral related to the compound Poisson process with double exponential jumps. Under such a model, we study the default events in the structural credit risk framework. The explicit solutions of Laplace transforms associated with default times are obtained. Based on the derived results, we numerically analyze the effects of the price clustering and the jump risk on the distributions of default times and on the bond’s credit spreads.

Pricing and hedging barrier options under a Markov-modulated double exponential jump diffusion-CIR model

A semi-closed-form valuation model is presented for barrier options whose underlying asset follows a mean-reverting and regime-switching double exponential jump diffusion process, and the interest rate is modulated by a mean-reverting square root model. The proposed model captures the impact of regime-switching uncertainty on barrier option prices and their hedge parameters in long and short business cycles. The model provides richer economic insight and is more appropriate for valuing barrier options in commodity markets as well as in equity and foreign-exchange markets, when an economy faces regime-switching uncertainty.

Real options under a double exponential jump-diffusion model with regime switching and partial information

We consider the irreversible investment in a project which generates a cash flow following a double exponential jump-diffusion process and its expected return is governed by a continuous-time two-state Markov chain. If the expected return is observable, we present explicit expressions for the pricing and timing of the option to invest. With partial information, i.e. if the expected return is unobservable, we provide an explicit project value and an integral-differential equation for the pricing and timing of the option. We provide a method to measure the information value, i.e. the difference between the option values under the two different cases. We present numerical solutions by finite difference methods. By numerical analysis, we find that: (i) the higher the jump intensity, the later the option to invest is exercised, but its effect on the option value is ambiguous; (ii) the option value increases with the belief in a boom economy; (iii) if investors are more uncertain about the economic environment, information is more valuable; (iv) the more likely the transition from boom to recession, the lower the value of the option; (v) the bigger the dispersion of the expected return, the higher the information value; (vi) a higher cash flow volatility induces a lower information value.

Occupation times of Lévy-driven Ornstein–Uhlenbeck processes with two-sided exponential jumps and applications

For an Ornstein–Uhlenbeck process driven by a double exponential jump diffusion process, we obtain formulas for the joint Laplace transform of it and its occupation times. The approach used is new and can be extended to investigate the occupation times of an Ornstein–Uhlenbeck process driven by a more general Lévy process. Besides, some applications to price occupation–time options are presented.

First-Passage Time Model Driven by Lévy Process for Pricing CoCos

Contingent convertible bonds (CoCos) are typical form of contingent capital that converts into equity of issuing firm or writes down if a prespecified trigger occurs. This paper proposes a general Lévy framework for pricing CoCos. The Lévy framework indicates that the difficulty in giving closed-form expression for CoCos price is the possible introduction of the Lévy process whose first-passage time problem has not been solved. According to characteristics of new Lévy measure after the measure transform, three specific Lévy models driven by drifted Brownian motion, spectrally negative Lévy process, and double exponential jump diffusion process are proposed to give the solution keeping the form of the driving process unchanged under the measure transform. These three Lévy models provide closed-form expressions for CoCos price while the latter two possess them up to Laplace transform, whose pricing results are given by combining with numerical Fourier inversion and Laplace inversion. Numerical results show that negative jumps have large influence on CoCos pricing and the Black-Scholes model would overestimate CoCos price by simply compressing jumps information into volatility while the other two models would give more accurate CoCos price by taking jump risk into consideration.

A solvable singular control problem driven by a jump diffusion process with applications

ABSTRACTWe consider a class of singular control problems driven by a double exponential jump diffusion process, which come from the reversible investment problem. In some interesting cases (e.g., the running cost function is given by the so-called Cobb-Douglas production function), we give the explicit solutions to the singular control problem by using the connection between singular control and optimal switching. We solve a collection of consistent optimal switching problems and yield the explicit solution for the singular control problem. We then give an application to a particular inventory control problem in a single random period.

Investment and financing for SMEs with a partial guarantee and jump risk

We consider a small- and medium-sized enterprise (SME) with a funding gap intending to invest in a project, of which the cash flow follows a double exponential jump-diffusion process. In contrast to traditional corporate finance theory, we assume the SME is unable to get a loan directly from a bank and hence it enters into a partial guarantee agreement with an insurer and a lender. Utilizing a real options approach, we develop an investment and financing model with a partial guarantee. We explicitly derive the pricing and timing of the option to invest. We find that if the funding gap rises, the option value decreases but its investment threshold first declines and then increases. The larger the guarantee level, the lower the option value and the later the investment. The optimal coupon rate decreases with project risk and a growth of the guarantee level can effectively reduce agency conflicts.

Occupation times of refracted double exponential jump diffusion processes

Motivated by our previous research, we investigate the problem of computing the occupation times of a refracted double exponential jump diffusion process. For such a process, we derive formulas for the Laplace transform of its occupation times.

Valuing equity-linked death benefits with a threshold expense strategy

We investigate equity-linked investment products with a threshold expense strategy, under which an insurance company will collect expenses continuously from the policyholder’s account only when the account value is lower than a pre-specified level. The logarithmic value of the policyholder’s account, before deducting any fees, is described by a jump diffusion process which is independent of the time-to-death random variable. The distribution of the time-to-death random variable is approximated by a combination of exponential distributions, which are dense in the space of density functions on [0,∞). We characterize the Laplace transform of the distribution of a general refracted jump diffusion process through some integro-differential equations. Besides, the distribution of a refracted double exponential jump diffusion process at an independent exponential random variable is derived, from which closed-form formulas to evaluate the total expenses and the fair fee rates are obtained. Finally, we illustrate our results by some numerical examples.

Pricing vulnerable European options under a two-sided jump model via Laplace transforms

In this paper, we propose a model for pricing the vulnerable European options when the value of the firm's asset of the counterparty follows a double exponential jump-diffusion process and correlates with the underlying asset value of the option. The two-sided jump process can model sudden down and up changes in the firm's asset value, which provides more economic insights on vulnerable options pricing. Incorporating the early default and payoff conditions, and based on the explicit formula of the joint Laplace transforms of the first passage times for the two-sided jump process and a correlated Bownian motion derived by us, we present a simple analytical pricing formula for vulnerable options via two-dimensional Laplace transforms, one for the space and one for the time, and the formula can be implemented by numerical Laplace inversion. The numerical results show that our pricing formula is correct and efficient.

Investment and Financing for SMEs with a Partial Guarantee and Jump Risk

Many SMEs and young entrepreneurs experience financing constraints due to their low credibility and information asymmetry. To this end, a partial guarantee agreement is popular with Chinese entrepreneurs. In this paper, we consider an SME with a funding gap, who wants to invest in a project, of which the cash flow follows a double exponential jump-diffusion process. In contrast to traditional corporate finance theory, we assume the SME is unable to get loan directly from banks because of its high project risk, strong information asymmetry and low credibility of SMEs. To overcome such financing constraints, the SME turns to an insurer and enters into a partial guarantee agreement, where a bank lends cash to the SME with the insurer promising to undertake a fraction (guarantee level) of debt once the SME defaults. In return, the SME (borrower) allocates a fraction of equity and a fixed guarantee fee rate per unit time to the insurer. We assume the investment is irreversible and can be postponed. Utilizing a real options approach, we develop an investment and financing model with the partial guarantee. We explicitly derive the pricing and timing of the option to invest for the cash flow with both diffusion and jump risk. The two sources of project risk increase the value of the option and postpone investment. More importantly, we reveal that larger funding gaps or higher guarantee levels lead to a later investment and generally a slightly less value of the option to invest. Interestingly, raising the guarantee level can effectively reduce the borrower's moral hazard to increase equity values at the expense of the lender. Meanwhile, an increase in the guarantee level does not add disincentive for the SME to replenish its equity.

Financial Leverage and Cross Sectional Delta-Hedged Option Returns

This paper studies the relation between firm leverage and delta-hedged equity option return both theoretically and empirically. In a capital structure model with double-exponential jump diffusion process, the expected return of delta-hedged equity option portfolio is related to firm's structural characteristics such as leverage, debt covenants and volatility of the firm's asset. Under general conditions, these determinants are negatively related to the expected return of the option portfolio and the higher order terms of the determinants also play important roles. Empirically, we find that these structural variables can explain a large portion of the cross-sectional variation in the data and even subsume information in other determinants documented in the literature, such as idiosyncratic volatility and liquidity. The results from the double sorting portfolios support the theoretical implications. The empirical evidence also supports the nonlinear relation between the determinants and the delta-hedged equity option returns. These findings are robust across calls, puts and different moneyness levels.

Pricing currency options under double exponential jump diffusion in a Markov-modulated HJM economy

Extending the framework of Amin and Jarrow (J Int Money Financ 10:310–329, 1991) and Bo et al. (Insur Math Econ 46:461–469, 2010), this study provides a theoretical exploration of currency options pricing under the presence of interest-rate regime shifts and exchange-rate asymmetric jumps. Evidence of interest-rate regime shifts inferred from UK and US zero coupon bond yields provides support for the regime-switching specifications which we reflect upon the domestic and foreign forward rates. Results of statistical tests conducted on JPY/USD and EUR/USD FX rates provide further support the rationale behind using a double exponential jump diffusion process within a Markov modulated Heath–Jarrow–Morton economy. Our numerical results suggest that, the pricing performance of our model is closely comparable to the Bo-Wang-Yang model for at-the-money options, yet yields improvements in percentage root mean errors for in-the-money options.

Valuing Credit Default Swap under a double exponential jump diffusion model

This paper discusses the valuation of the Credit Default Swap based on a jump market, in which the asset price of a firm follows a double exponential jump diffusion process, the value of the debt is driven by a geometric Brownian motion, and the default barrier follows a continuous stochastic process. Using the Gaver-Stehfest algorithm and the non-arbitrage asset pricing theory, we give the default probability of the first passage time, and more, derive the price of the Credit Default Swap.

Pricing formulae for constant proportion debt obligation notes: The Laplace transform technique

In this paper we derive closed form pricing formulae for the constant proportion debt obligation (CPDO) by using the Laplace transform technique. First, we present the pricing equation as a combination of a pricing problem (conditional expectation) and a static part that depends only on time. Then, we indicate that the pricing problem is in fact a pricing of a barrier option written on the shortfall. Hence, we derive explicit solutions of such barrier option problems when the shortfall follows either a diffusion or a double exponential jump diffusion process. Finally, we illustrate and discuss the results using numerical applications.

Pricing discrete path-dependent options under a double exponential jump–diffusion model

We provide methodologies to price discretely monitored exotic options when the underlying evolves according to a double exponential jump diffusion process. We show that discrete barrier or lookback options can be approximately priced by their continuous counterparts’ pricing formulae with a simple continuity correction. The correction is justified theoretically via extending the corrected diffusion method of Siegmund (1985). We also discuss the jump effects on the performance of this continuity correction method. Numerical results show that this continuity correction performs very well especially when the proportion of jump volatility to total volatility is small. Therefore, our method is sufficiently of use for most of time.

Precautionary measures for credit risk management in jump models

Sustaining efficiency and stability by properly controlling the equity to asset ratio is one of the most important and difficult challenges in bank management. Due to unexpected and abrupt decline of asset values, a bank must closely monitor its net worth as well as market conditions, and one of its important concerns is when to raise more capital so as not to violate capital adequacy requirements. In this paper, we model the trade-off between avoiding costs of delay and premature capital raising, and solve the corresponding optimal stopping problem. In order to model defaults in a bank's loan/credit business portfolios, we represent its net worth by Lévy processes, and solve explicitly for the double exponential jump-diffusion process and for a general spectrally negative Lévy process.