354 research outputs found
No arbitrage without semimartingales
We show that with suitable restrictions on allowable trading strategies, one
has no arbitrage in settings where the traditional theory would admit arbitrage
possibilities. In particular, price processes that are not semimartingales are
possible in our setting, for example, fractional Brownian motion.Comment: Published in at http://dx.doi.org/10.1214/08-AAP554 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Option pricing models without probability: a rough paths approach
We describe the pricing and hedging of financial options without the use of
probability using rough paths. By encoding the volatility of assets in an
enhancement of the price trajectory, we give a pathwise presentation of the
replication of European options. The continuity properties of rough-paths allow
us to generalise the so-called fundamental theorem of derivative trading,
showing that a small misspecification of the model will yield only a small
excess profit or loss of the replication strategy. Our hedging strategy is an
enhanced version of classical delta hedging where we use volatility swaps to
hedge the second order terms arising in rough-path integrals, resulting in
improved robustness
Weak and strong no-arbitrage conditions for continuous financial markets
We propose a unified analysis of a whole spectrum of no-arbitrage conditions for finan- cial market models based on continuous semimartingales. In particular, we focus on no-arbitrage conditions weaker than the classical notions of No Arbitrage opportunity (NA) and No Free Lunch with Vanishing Risk (NFLVR). We provide a complete characterization of the considered no-arbitrage conditions, linking their validity to the characteristics of the discounted asset price process and to the existence and the properties of (weak) martingale deflators, and review classical as well as recent results
Asymptotic Maturity Behavior of the Term Structure
Pricing and hedging of long-term interest rate sensitive products require to extrapolate the term structure beyond observable maturities. For the resulting limiting term structure we show two results by postulating no arbitrage in a bond market with infinitely increasing maturities: long zero-bond yields and long forward rates (i) are monotonically increasing and (ii) equal their minimal future value. Both results constrain the asymptotic maturity behavior of stochastic yield curves. They are fairly general and extend beyond semimartingale modeling. Hence our framework embeds arbitrage-free term structure models and imposes restrictions on their specification.bond markets, yield curve, long forward rates, no arbitrage, asymptotic maturity
Implicit transaction costs and the fundamental theorems of asset pricing
This paper studies arbitrage pricing theory in financial markets with
implicit transaction costs. We extend the existing theory to include the more
realistic possibility that the price at which the investors trade is dependent
on the traded volume. The investors in the market always buy at the ask and
sell at the bid price. Implicit transaction costs are composed of two terms,
one is able to capture the bid-ask spread, and the second the price impact.
Moreover, a new definition of a self-financing portfolio is obtained. The
self-financing condition suggests that continuous trading is possible, but is
restricted to predictable trading strategies having c\'adl\'ag
(right-continuous with left limits) and c\'agl\'ad (left-continuous with right
limits) paths of bounded quadratic variation and of finitely many jumps. That
is, c\'adl\'ag and c\'agl\'ad predictable trading strategies of infinite
variation, with finitely many jumps and of finite quadratic variation are
allowed in our setting. Restricting ourselves to c\'agl\'ad predictable trading
strategies, we show that the existence of an equivalent probability measure is
equivalent to the absence of arbitrage opportunities, so that the first
fundamental theorem of asset pricing (FFTAP) holds. It is also shown that the
use of continuous and bounded variation trading strategies can improve the
efficiency of hedging in a market with implicit transaction costs. To better
understand how to apply the theory proposed we provide an example of an
implicit transaction cost economy that is linear and non-linear in the order
size.Comment: International Journal of Theoretical and Applied Finance, 20(04) 201
Variance optimal hedging for continuous time additive processes and applications
For a large class of vanilla contingent claims, we establish an explicit
F\"ollmer-Schweizer decomposition when the underlying is an exponential of an
additive process. This allows to provide an efficient algorithm for solving the
mean variance hedging problem. Applications to models derived from the
electricity market are performed
On stochastic calculus related to financial assets without semimartingales
This paper does not suppose a priori that the evolution of the price of a
financial asset is a semimartingale. Since possible strategies of investors are
self-financing, previous prices are forced to be finite quadratic variation
processes. The non-arbitrage property is not excluded if the class
of admissible strategies is restricted. The classical notion of
martingale is replaced with the notion of -martingale. A calculus
related to -martingales with some examples is developed. Some
applications to no-arbitrage, viability, hedging and the maximization of the
utility of an insider are expanded. We finally revisit some no arbitrage
conditions of Bender-Sottinen-Valkeila type
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