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

On the combinatorics of iterated stochastic integrals

This paper derives several identities for the iterated integrals of a general semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Their form and derivations are combinatorial. The formulae simplify for continuous or finite-variation semimartingales, especially for counting processes. The results are motivated by chaotic representation of martingales, and a simple such application is given

Various identities for iterated integrals of a semimartingale

This paper derives several identities for iterated integrals of a semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Some, like for counting or finite-variation processes, are apparently new. Others, like two of the three formulae for continuous semimartingales, are generalizations of well-known formulae

Chaotic expansion of powers and martingale representation

This paper extends a recent martingale representation result of [N-S] for a L´evy process to filtrations generated by a rather large class of semimartingales. As in [N-S], we assume the underlying processes have moments of all orders, but here we allow angle brackets to be stochastic. Following their approach, including a chaotic expansion, and incorporating an idea of strong orthogonalization from [D], we show that the stable subspace generated by Teugels martingales is dense in the space of square-integrable martingales, yielding the representation. While discontinuities are of primary interest here, the special case of a (possibly infinite-dimensional) Brownian filtration is an easy consequence

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

On the combinatorics of iterated stochastic integrals

This paper derives several identities for the iterated integrals of a general semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Their form and derivations are combinatorial. The formulae simplify for continuous or finite-variation semimartingales, especially for counting processes. The results are motivated by chaotic representation of martingales, and a simple such application is given.Semimartingale; iterated integrals; power jump processes; Ito's formula; stochastic exponential; chaotic representation

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

Chaotic expansion of powers and martingale representation (v1.2)

This paper extends a recent martingale representation result of [N-S] for a L\'{e}vy process to filtrations generated by a rather large class of semimartingales. As in [N-S], we assume the underlying processes have moments of all orders, but here we allow angle brackets to be stochastic. Following their approach, including a chaotic expansion, and incorporating an idea of strong orthogonalization from [D], we show that the stable subspace generated by Teugels martingales is dense in the space of square-integrable martingales, yielding the representation. While discontinuities are of primary interest here, the special case of a (possibly infinite-dimensional) Brownian filtration is an easy consequence.Martingale Representation, chaotic expansion, power brackets, Teugels martingales, Hilbert space, strong orthogonalization