Option Pricing Under Dividend Barrier Strategies

In this paper, we investigate the optimal dividend problem under Parisian ruin with affine penalty payments at Parisian ruin time. The underlying risk process is assumed to be a spectrally negative Levy risk process. With the help of the dynamic programming principle, we prove that the value function associated to our optimal control problem is the smallest solution with certain characteristics to the corresponding Hamilton–Jacobi–Bellman (HJB) equation. In addition, the form of the performance function under barrier dividend strategy is expressed in terms of various extended scale functions. Then we identify a condition under which the performance function under certain barrier strategy is also a solution to the HJB equation, which in turn illustrates the optimalilty of such barrier dividend strategy among all admissible strategies. Various numerical examples are also given when the underlying risk process is compound Poisson process, Brownian motion with drift and jump-diffusion process.

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the corresponding price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such "X-factor" we associate a market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents "noise". The noise term is modelled by an independent Brownian bridge that spans the interval from the present to the time at which the value of the factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk-neutral measure, conditional on the information provided by the market filtration. In the case where the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price process. Dividend growth is taken into account by introducing appropriate structure on the market factors. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black–Scholes–Merton type. We consider the case where the rate at which information is revealed to the market is constant, and the case where the information rate varies in time. Option pricing formulae are obtained for both cases. The information-based framework generates a natural explanation for the origin of stochastic volatility in financial markets, without the need for specifying on an ad hoc basis the dynamics of the volatility.

For spectrally negative Lévy risk processes we consider a general version of de Finetti's optimal dividend problem in which the ruin time is replaced with a general drawdown time from the running maximum in its value function. We identify a condition under which a barrier dividend strategy is optimal among all admissible strategies if the underlying process does not belong to a small class of compound Poisson processes with drift, for which the take-the-money-and-run dividend strategy is optimal. It generalizes the previous results on dividend optimization from ruin time based to drawdown time based. The associated drawdown functions are discussed in detail for examples of spectrally negative Lévy processes.

Using physical probabilistic measure of price process and the principle of fair premium , we deal with pricing formula of option on Foreign currency option under the assumption that foreign option price process driven by fractional Brownian motion process ,we obtain the pricing formula of foreign option.

Option markets, empirical price data, and theoretical arguments all indicate that asset prices in actively traded markets are driven by Lévy flights and/or tempered Lévy flights, not by Brownian motion. So here we model asset prices in the real world by (1) where dZL is a Lévy or tempered Lévy flight. Derivatives based on such assets cannot be made risk free by dynamic hedging, so these derivatives cannot be priced using the standard Black–Scholes–Merton (BSM) arbitrage-free pricing criterion. Therefore, we develop a pricing theory based on a more general pricing criterion: that the expected return of all diversifiable portfolios is at the risk-free rate. We show that even though derivatives based on Lévy or tempered Lévy-driven assets cannot be hedged to make risk-free portfolios, they can be hedged to make diversifiable portfolios. This allows us to conclude that these option prices are given by (2) where the expected value uses the “real-world” probability measure, but with the asset price X(t) replaced by the alternative “pricing process” (3) which grows at the risk-free rate. We analyze these models to obtain explicit asymptotic formulas for European option prices. This analysis shows that as the time to expiry decreases, eventually all these Lévy and tempered Lévy-based models reduce to the same canonical model, and that European option prices and implied volatilities are given by similarity solutions under this canonical model. These similarity solutions are then examined to assess the mishedging that arises from Brownian-based models in a Lévy world.

The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.

In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index H > 1 / 2 , (ii) is proven to be satisfied by a rough volatility model with H < 1 / 2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist.

We consider the dual of the generalized Erlang(n) risk model with a barrier dividend strategy. We derive integro‐differential equations with boundary conditions satisfied by the expectation of the sum of discounted dividends until ruin and the moment‐generating function of the discounted dividend payments until ruin, respectively. The results are illustrated by several examples.

In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.

Government guarantees are often priced by appealing to an analogy with option pricing techniques pioneered by Black and Scholes (1973) and Merton (1973). Of course, in reality, the government does not purchase options in the open market. Instead, barring a Ponzi game opportunity, the government must adjust tax revenue or spending in response to shocks in the value of the asset being insured for the government to maintain its intertemporal budget constraint. Nonetheless, this article derives an exact mathematical relationship between fiscal policy and option pricing. Explicitly deriving the option pricing equivalence from first principles proves that the analogy is, in fact, correct and suitable for use by public finance economists and government officials to estimate the cost of government guarantees. Starting from first principles also has the benefit of showing that an additional mathematical term, which multiplies the option price, is necessary to capture the pay-as-you-go nature of most perpetual governme nt guarantees-- a term that is missing in the previous work. The author develops the analysis in the context of a highly relevant example: investing the Social Security trust fund in equities. Advocates of this policy have argued that the higher expected returns would help prefund future benefits and, therefore, would require less of a tax increase on future workers. Several government agencies have scored hypothetical legislation and have concluded likewise. The results herein show just the opposite. By focusing on expected returns, trust fund investment creates an instant windfall for current workers and is mathematically equivalent to a tax increase on all future workers, relative to a baseline policy of maintaining the current payroll tax rate. National saving is reduced. The author computes President Clinton's proposal to invest in equities to be equivalent to increasing the future payroll tax by 0.8 percent in perpetuity. A policy recommendation to invest about 40 percent of the trust fund in stocks analyzed recently by the Social Security Administration is equ ivalent to a 21 percent increase in the future payroll tax. INTRODUCTION As Robert Merton noted in his Nobel address, the use of option pricing technology has extended well beyond financial operations and has been used in numerous valuation problems, including government guarantees. The list is extensive and includes deposit insurance, student loan guarantees, loans to business, the government right to change quotas, and many other guarantees. [1] Of course, in reality, the government does not purchase options for the purpose of hedging. Instead, barring a Ponzi game opportunity, the government must adjust tax revenue or spending in response to shocks in the value of the asset being insured for the government to maintain its intertemporal budget constraint. Government policy then becomes state contingent and the government's intertemporal budget constraint must be taken explicitly into account. Although most analyses of guarantees rationalize using options pricing based on an analogy, this article starts with fiscal policy instruments and proves that the analogy is, in fact, true--with a modification. The author derives an exact mathematical equivalence between state-contingent fiscal policies and option pricing. Deriving implicit option prices from the government's intertemporal budget constraint not only brings an even greater sense of reality to the case for using arbitrage pricing technology to value government guarantees, but it is also important because many government guarantees are not prefunded. The option pricing formulae must be augmented with a multiplicative term that recognizes the pay-as-you-go nature of most perpetual guarantees. As a relevant example, this article considers the proposal to alter the Social Security trust fund's portfolio from exclusively holding government debt to holding equity as well. …

This chapter introduces the use of a time-homogenous Markov chain for the valuation of options. In the computational finance literature, three categories of numerical techniques have been explored extensively for the valuation of contingent claims. The first category involves the use of a lattice structure to approximate the price movement of the underlying asset under the risk neutral probability measure and then computes the price of a contingent claim as a discounted expected payoff. The lattice approach essentially discretizes both time and state in a particular way. The second category is the finite difference/element approach. This technique numerically solves the partial differential equation that the value function of a contingent claim must obey under the no-arbitrage condition. The third category is the Monte Carlo method, which simulates the system under the risk-neutral probability measure so that the expectation of a contingent payoff can be approximated by the sample average.

In this paper, we propose a discrete-time counterpart of the classical compound Poisson and several applications of a bivariate extension of the Dickson-Waters discretisation. We expose numerous examples, including forth-looking scenario of ruin and surplus models We consider the integro-differential equation m′(u) = λ+δ c1 m(u) − λ c1 ∫ u 0 to expose application of a compound-Poisson risk models in barrier strategies invoking dividend surplus. Keywords— Bivariate risk process, Discretisation, Dividend barrier strategy, Copulas, Capital allocation, Proportional reinsurance

The time evolution of prices of different financial quantities is often represented as a partial differential equation (PDE) with independent variables being time and prices of some other, often underlying, assets. Let $$V(S_t, t)$$ be the price of an option at time t when the share price of the underlying stock is $$S_t$$ . See Appendix A.1 for background on mathematical finance that is used in what follows. Under the Black–Scholes set-up, we have a risk-less asset bond $$B_t$$ and a risky asset stock $$S_t$$ . They evolve as where r is the interest rate, $$\mu $$ is the drift, $$\sigma $$ is the volatility and W is a standard Brownian motion(BM). We apply Ito’s formula to the option price to get Consider the discounted option price $$B_t^{-1}V(S_t,t)$$ . By the Fundamental Theorem of Arbitrage Pricing (see Appendix), the discounted option price must be a martingale under the risk-neutral measure . Also, under the risk-neutral measure $$\mu =r$$ . We have, from the above, For a martingale, the coefficient of the dt term has to be zero, otherwise there is a systematic drift.

In this article, we explain main concepts of prospect theory and cumulative prospect theory within the rational dynamic asset pricing framework. We derive option pricing formulas when asset returns are altered by a generalized prospect theory value function or a modified Prelec’s weighting probability function. We introduce new parametric classes for prospect theory value functions and probability weighting functions consistent with rational dynamic pricing theory. After studying the behavioral finance notion of “greed and fear” from the perspective of rational dynamic asset pricing theory, we derive the corresponding option pricing formulas when asset returns follow continuous diffusions or discrete binomial trees. We define a mixed subordinated variance gamma process to model asset return and derive the corresponding option pricing formula. Finally, we apply the proposed probability weighting functions to study the greedy or fearful disposition of option traders when asset returns follow a mixed subordinated variance gamma process. The results indicate availability bias and diminishing sensitivity of option traders.

Pricing VIX Futures: Evidence from Integrated Physical and Risk-Neutral Probability Measures