Modelling of tradeable securities with dividends

We propose a generalized framework for the modeling of tradeable securities with dividends which are not necessarily cash dividends at fixed times or continuously paid dividends. In our setup the dividend processes are only required to be semi-martingales. We give a definition of self-financing replication which incorporates dividend processes, and we show how this allows us to translate standard results for the pricing and hedging of derivatives on assets without dividends to the case of assets with dividends. We then apply this framework to analyze and compare the different assumptions that have been made in earlier dividend models. We also study the case where we have uncertain dividend dates, and we look at securities which are not equity-based such as futures and credit default swaps, since our weaker assumptions on the dividend process allow us to consider these other applications as well

Numeraire Invariance and application to Option Pricing and Hedging

Numeraire invariance is a well-known technique in option pricing and hedging theory. It takes a convenient asset as the numeraire, as if it were the medium of exchange, and expresses all other asset and option prices in units of this numeraire. Since the price of the numeraire relative to itself is identically 1 at all times, this reduces pricing and hedging to a market with zero-interest rates. A somewhat controversial implication is that the modelling focus should be more on the asset price ratios rather than on the asset price processes themselves. The idea of numeraire invariance is already implicit in Merton (1973), and since then many authors have contributed to its development. After a brief survey of its origins, we state and prove the numeraire invariance principle for general semimartingale price processes, following essentially Duffie [3]. We then present its application to unique pricing in arbitrage-free models and discuss nondegeneracy and unique hedging

Exchange Options Under Jump-Diffusion Dynamics

Margrabe provides a pricing formula for an exchange option where the distributions of both stock prices are log-normal with correlated Wiener components. Merton has provided a formula for the price of a European call option on a single stock where the stock price process contains a continuous Poisson jump component, in addition to a continuous log-normally distributed component. We use Merton’s analysis to extend Margrabe’s results to the case of exchange options where both stock price processes also contain compound Poisson jump components. A Radon-Nikod´ym derivative process that induces the change of measure from the market measure to an equivalent martingale measure is introduced. The choice of parameters in the Radon-Nikod´ym derivative allows us to price the option under different financial-economic scenarios. We also consider American style exchange options and provide a probabilistic intepretation of the early exercise premium.American options; exchange options; compound Poisson processes; equivalent martingale measure

Exchange Options

The contract is described and market examples given. Essential theoretical developments are introduced and cited chronologically. The principles and techniques of hedging and unique pricing are illustrated for the two simplest nontrivial examples: the classical Black-Scholes/Merton/Margrabe exchange option model brought somewhat uptodate from its form three decades ago, and a lesser exponential Poisson analogue to illustrate jumps. Beyond these, a simplified Markovian SDE/PDE line is sketched in an arbitrage-free semimartingale setting. Focus is maintained on construction of a hedge using Itˆo’s formula and on unique pricing, now for general homogenous payoff functions. Clarity is primed as the multivariate log-Gaussian and exponential Poisson cases are worked out. Numeraire invariance is emphasized as the primary means to reduce dimensionality by one to the projective space where the SDE dynamics are specified and the PDEs solved (or expectations explicitly calculated). Predictable representation of a homogenous payoff with deltas (hedge ratios) as partial derivatives or partial differences of the option price function is highlighted. Equivalent martingale measures are utilized to show unique pricing with bounded deltas (and in the nondegenerate case unique hedging) and to exhibit the PDE or closed-form solutions as numeraire-deflated conditional expectations in the usual way. Homogeneity, change of numeraire, and extension to dividends are discussed

Multifrequency Jump-Diffusions: An Equilibrium Approach

This paper proposes that equilibrium valuation is a powerful method to generate endogenous jumps in asset prices, which provides a structural alternative to traditional reduced-form specifications with exogenous discontinuities. We specify an economy with continuous consumption and dividend paths, in which endogenous price jumps originate from the market impact of regime-switches in the drifts and volatilities of fundamentals. We parsimoniously incorporate shocks of heterogeneous durations in consumption and dividends while keeping constant the number of parameters. Equilibrium valuation creates an endogenous relation between a shock's persistence and the magnitude of the induced price jump. As the number of frequencies driving fundamentals goes to infinity, the price process converges to a novel stochastic process, which we call a multifractal jump-diffusion.

Numeraire Invariance and application to Option Pricing and Hedging

This is a short version of the paper of Exchange Options (2007), concentrating on the principle of numeraire invariance. It emphasizes application to unique pricing in arbitrage-free model, the derivation of hedge ratios and the PDE when price ratios are diffusions, explicit representations in the multivariate Poisson model, and the role played by homogeneity.Numeraire invariance, hedging, self-financing trading strategy, predictable representation, unique pricing, arbitrage-free, martingale, homogeneous payoff, Markovian, It\^o's formula, SDE, PDE, geometric Brownian motion, exponential Poisson process

Put-call parities and the value of early exercise for put options on a performance index

Futures Markets;finance

Closed formula for options with discrete dividends and its derivatives

We present a closed pricing formula for European options under the BlackScholes model and formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and by expressing the spatial derivatives as expectations under special measures, as in Carr, together with an unusual change of measure technique that relies on the replacement of the initial condition. The closed formulas are attained for the case where no dividend payment policy is considered. Despite its small practical relevance, a digital dividend policy case is also considered which yields approximation formulas. The results are readily extensible to time dependent volatility models but no so for local-vol type models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis using the formulas here obtained. The closed formulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods. --equity option,discrete dividend,hedging,analytic formula