Jump-diffusion Market Research Articles

Deep learning for quadratic hedging in incomplete jump market

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based on a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feed-forward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black–Scholes model serves as a benchmark for the algorithm’s performance. The results that indicate the algorithm’s good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle.

Robust optimal asset–liability management with delay and ambiguity aversion in a jump-diffusion market

In this paper, we consider a robust optimal asset–liability management problem with delay for an ambiguity-averse investor (AAI), who does not hold a firm belief towards not only the drift parameters but also the jump intensity in the wealth process. The objective of AAI is to maximise the robust value involving the expected utility of a weighted sum of terminal wealth and a penalisation of model uncertainty. In addition, the wealth process is modelled by a stochastic delay equation with jump via introducing the performance-related capital inflow/outflow. Applying the technique of robust optimal theory and Hamilton–Jacobi–Bellman equation, we derive the optimal results and the proof of verification theorem for the general utility functions and liability processes. In the case of the exponential utility function and a positive liability process, we prove the existence and uniqueness of the robust optimal investment strategy and find it naturally satisfies the constraint of no-short selling. Meanwhile, we adopt a special method to solve the corresponding Riccati equation and obtain the value function in explicit form. Furthermore, the results under the complete memory are discussed in detail. Finally, some numerical examples are provided to illustrate the influence of model parameters on the optimal strategy as well as the economic interpretation behind it.

Pricing of spread and exchange options in a rough jump–diffusion market

Asset dynamics with rough volatility recently received a great deal of attention in finance because they are consistent with empirical observations. This article provides a detailed analysis of the impact of roughness on prices of spread and exchange options. We consider a bivariate extension of the rough Heston model with jumps and derive the joint characteristic functions of asset log-returns under the risk neutral measure and under a measure using a risky asset as numeraire. These characteristic functions are expressed in terms of solutions of fractional differential equations (FDE’s). To infer these FDE’s, we rewrite the rough model as an infinite dimensional Markov process and propose a finite dimensional approximation. Next, we show that characteristic functions of log-returns admit a representation in terms of forward differential equations. FDE’s are obtained by passing to the limit. Spread and exchange options are valued with a two or a one dimensional discrete Fourier Transform. The numerical illustration reveals that considering a rough instead of Brownian volatility does not systematically increase exchange option prices.

American Option Pricing and Filtering with a Hidden Regime-Switching Jump Diffusion

The valuation of an American-style contingent claim is discussed in a hidden Markov regime-switching jump-diffusion market, where the evolution of a hidden economic state process over time is described by a continuous-time, finite-state, hidden Markov chain. Filtering theory is applied to introduce a filtered market where the valuation problem is discussed. A probabilistic approach to American option pricing is considered, where a decomposition formula for the price of an American put option is given as the sum of its European counterpart and an early exercise premium. Then the valuation of a perpetual American put option is considered. A (semi-)analytical approximation to the perpetual American put price is obtained. Numerical results for the perpetual American put prices and critical values are provided to illustrate the approximation and to examine the impacts of probability beliefs on hidden economic regimes and jumps on the put prices and critical values.

Mean-Variance Asset-Liability Management in a Non-Markovian Regime-Switching Jump-Diffusion Market with Random Horizon

This paper investigates the problem of mean-variance asset-liability management (ALM) in a market where the dynamics of assets are non-Markovian regime-switching models driven by a Brownian motion, a Poisson random measure and a continuous time finite-state Markov chain. It is assumed that the time horizon is uncertain relying not only on asset prices and liability values, but also on other factors. The insurer aims to minimize the variance of the terminal surplus given an expected terminal surplus subject to the risk of paying out random liabilities of an extensive Cramer-Lundberg model. By further developing the solvability of a linear backward stochastic differential equation (BSDE) with two types of jumps, namely jumps modelled by a Poisson random measure and by basic martingales related to a Markov chain, closed-form expressions for both the efficient strategy and the efficient frontier are obtained and represented by unique solutions to several relevant BSDEs.

Mean-variance problem for an insurer with dependent risks and stochastic interest rate in a jump-diffusion market

In this paper, we study an optimal investment and reinsurance problem in which the interest rate is driven by the Vasicek process and two dependent classes of insurance business are correlated through a common shock component. The goal of insurer is to find an optimal investment-reinsurance strategy to minimize the variance of terminal net wealth for a given expected terminal net wealth. By using the linear-quadratic optimal control theory and the corresponding Hamilton-Jacobi-Bellman (HJB) equation, the closed-form expressions for the value function and optimal strategies are obtained under the special case of perfect correlation between the bond and stock processes. We present that the solution of the HJB equation is no longer a classical solution, but a viscosity solution due to the non-negativity constraint of the reinsurance strategy. Furthermore, the efficient strategies and efficient frontier are derived explicitly. Finally, we explore some numerical examples to show the influence of model parameters on the optimal investment and reinsurance strategies.

Fast Pricing of Energy Derivatives with Mean-Reverting Jump-diffusion Processes

ABSTRACT Most energy and commodity markets exhibit mean-reversion and occasional distinctive price spikes, which result in demand for derivative products which protect the holder against high prices. To this end, in this paper we present a few fast and efficient methodologies for the exact simulation of the spot price dynamics modelled as the exponential of the sum of a Gaussian Ornstein-Uhlenbeck process and an independent pure jump process, where the latter one is driven by a compound Poisson process with (bilateral) exponentially distributed jumps. These methodologies are finally applied to the pricing of Asian options, gas and hydro storages and swing options under different combinations of jump-diffusion market models, and the apparent computational advantages of the proposed procedures are emphasized.

CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs

<p style='text-indent:20px;'>This paper analyzes Conditional Value-at-Risk (CVaR) based partial hedging and its applications on equity-linked life insurance contracts in a Jump-Diffusion market model with transaction costs. A nonlinear partial differential equation (PDE) that an option value process inclusive of transaction costs should satisfy is provided. In particular, the closed-form expression of a European call option price is given. Meanwhile, the CVaR-based partial hedging strategy for a call option is derived explicitly. Both the CVaR hedging price and the weights of the hedging portfolio are based on an adjusted volatility. We obtain estimated values of expected total hedging errors and total transaction costs by a simulation method. Furthermore,our results are implemented to derive target clients’ survival probabilities and age of equity-linked life insurance contracts.</p>

A financial market with singular drift and no arbitrage

We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow and Protter (J Bank Finan 29:2803–2820, 2005) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay theta > 0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as theta > 0. This implies that there is no arbitrage in the market in that case. However, when theta goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al. (Math Finan Econom 10(3):223–262, 2016) and the references therein.

Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market

We apply dynamic programming principle to discuss two optimal investment problems by using zero-sum and nonzero-sum stochastic game approaches in a continuous-time Markov regime-switching environment within the frame work of behavioral finance. We represent different states of an economy and, consequently, investors’ floating levels of psychological reactions by a D-state Markov chain. The first application is a zero-sum game between an investor and the market, and the second one formulates a nonzero-sum stochastic differential portfolio game as the sensitivity of two investors’ terminal gains. We derive regime-switching Hamilton–Jacobi–Bellman–Isaacs equations and obtain explicit optimal portfolio strategies with Feynman–Kac representations of value functions. We illustrate our results in a two-state special case and observe the impact of regime switches by comparative results.

Time-consistent mean–variance asset-liability management in a regime-switching jump-diffusion market

This paper investigates the time-consistent optimal control of a mean–variance asset-liability management problem in a regime-switching jump-diffusion market. The investor (a company) is investing in the market with one risk-less bond and one risky stock while subject to an uncontrollable liability. The risky stock and the liability processes are discontinuous with correlated jumps and modulated by a continuous-time observable Markov chain that represents the market state. This hybrid model can capture both long-term and short-term effects on the investing process resulting from the market movements and unexpected events. Under a game-theoretic framework, we derive the regime-switching jump-diffusion version of the extended Hamilton–Jacobi–Bellman (HJB) system as well as the verification theorem, based on which we obtain the closed-form equilibrium control and equilibrium value function in terms of five systems of ordinary differential equations by solving the extended HJB equation. Finally, a numerical analysis investigates the influence of changes in the model parameters on our solution, and it is discovered that regime switching and jump diffusion both have great effect on the investment and therefore should be considered in conjunction.

Pricing and hedging of general rating-sensitive claims in a jump-diffusion market model in the presence of stochastic factors

In this paper we solve a problem of finding a risk minimizing hedging strategy on a general Markovian market with ratings on which prices are influenced by additional factors and rating. The market is described by a system of SDEs driven by a Wiener process and a compensated Poisson random measure. Rating-sensitive claims are considered. We relate the problem of pricing and hedging the contracts described by a general cash-flow process to solving Cauchy/Dirichlet problems and subsequently to solving some linear system of equations. We illustrate our theory on two examples of different nature. The first is a general exponential Lévy model with stochastic volatility, and the second is a generalization of an exponential Lévy model with regime switching.

Optimal Investment-Consumption-Insurance with Durable and Perishable Consumption Goods in a Jump Diffusion Market

We investigate an optimal investment-consumption and optimal level of insurance on durable consumption goods with a positive loading in a continuous-time economy. We assume that the economic agent invests in the financial market and in durable as well as perishable consumption goods to derive utilities from consumption over time in a jump-diffusion market. Assuming that the financial assets and durable consumption goods can be traded without transaction costs, we provide a semi-explicit solution for the optimal insurance coverage for durable goods and financial asset. With transaction costs for trading the durable good proportional to the total value of the durable good, we formulate the agent's optimization problem as a combined stochastic and impulse control problem, with an implicit intervention value function. We solve this problem numerically using stopping time iteration, and analyze the numerical results using illustrative examples.

A Note on the Hedging of Options by Malliavin Calculus in a Jump-Diffusion Market

We study a jump-diffusion market with two settings for the stock price process and expose the capability of Malliavin calculus in generation of the locally risk minimizing portfolio under weaker condition.

Optimal investment of variance-swaps in jump-diffusion market with regime-switching

We consider a general jump-diffusion market with regime-switching where the jump risk is modeled as a Markov-modulated Poisson random measure. In this incomplete market, we price the variance-swaps using a combination of the Esscher transform and change of measure on time-inhomogeneous Markov chains. We study the dynamic optimal investment problem of the variance-swaps and characterize the optimal feedback strategy. Moreover, a closed-form solution to the HJB PDE associated with the stochastic control problem is established and the verification theorem is proved. The numerical analysis based on a two-state Markov chain uncovers some robust features of the optimal investment strategy.

EQUILIBRIUM EQUITY PRICE WITH OPTIMAL DIVIDEND POLICY

This paper proposes an equilibrium model for evaluating equity with optimal dividend policy in a jump-diffusion market. In this model, a representative investor having power utility over an aggregate consumption process evaluates the equity as the expected value of the discounted dividends with his stochastic discount factor, while a firm paying the dividends from its own cash reserves manages to maximize the equity price. This situation is formulated as a singular stochastic control problem of jump-diffusion processes. We solve this problem and give the equilibrium equity price and the optimal dividend policy. Numerical examples show that the aggregate consumption process and the investor’s risk aversion have a significant impact on the equity price and the dividend policy. This model provides a structural explanation of equity risk premiums that is consistent with the standard theory of asset pricing.

Better than pre-committed optimal mean-variance policy in a jump diffusion market

Dynamic mean-variance investment model can not be solved by dynamic programming directly due to the nonseparable structure of variance minimization problem. Instead of adopting embedding scheme, Lagrangian duality approach or mean-variance hedging approach, we transfer the model into mean field mean-variance formulation and derive the explicit pre-committed optimal mean-variance policy in a jump diffusion market. Similar to multi-period setting, the pre-committed optimal mean-variance policy is not time consistent in efficiency. When the wealth level of the investor exceeds some pre-given level, following pre-committed optimal mean-variance policy leads to irrational investment behaviors. Thus, we propose a semi-self-financing revised policy, in which the investor is allowed to withdraw partial of his wealth out of the market. And show the revised policy has a better investment performance in the sense of achieving the same mean-variance pair as pre-committed policy and receiving a nonnegative free cash flow stream.

Equilibrium Equity Price with Optimal Dividend Policy Under Jump-Diffusion Processes

This paper proposes an equilibrium model for evaluating equity with optimal dividend policy in a jump-diffusion market. In this model, a representative investor having power utility over a total consumption process evaluates the equity as the expected value of the discounted dividends with his stochastic discount factor, while a firm paying the dividends from his own cash reserve manages to maximize the equity price. This situation is formulated as a singular stochastic control problem of jump-diffusion processes. We solve this problem and give the equilibrium equity price and the optimal dividend policy. Numerical examples show that the total consumption process and the investor's risk aversion have a significant impact on the equity price and the dividend policy and generate positive values of an expected excess return and the Sharpe ratio on equity compatible with a standard theory of asset pricing.

Optimal Asset-Liability Management for an Insurer Under Markov Regime Switching Jump-Diffusion Market

This paper considers an asset-liability management problem under a continuous time Markov regime-switching jump-diffusion market. We assume that the risky stock’s price is governed by a Markov regime-switching jump-diffusion process and the insurance claims follow a Markov regime-switching compound poisson process. Using the Markowitz mean-variance criterion, the objective is to minimize the variance of the insurer’s terminal wealth, given an expected terminal wealth. We get the optimal investment policy. At the same time, we also derive the mean-variance efficient frontier by using the Lagrange multiplier method and stochastic linear-quadratic control technique.

Errata corrige optimal portfolio and consumption with habit formation in a jump diffusion market

We correct (17), (19), (20), (21), (46) and (48) in [1].