We shall study backward stochastic differential equations and we will present a new approach for the existence of the solution. This type of equation appears very often in the valuation of financial derivatives in complete markets. Therefore, the ide
ntification of the solution as the unique element in a certain Banach space where a suitably chosen functional attains its minimum becomes interesting for numerical computations.
A New Trinomial Recombination Tree Algorithm and Its Applications
An investor with constant absolute risk aversion trades a risky asset with general It^o-dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading-order optimal trading policy and the associated w
elfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.
The goal of this paper is to prove a result conjectured in Follmer and Schachermayer [FS07], even in slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth cond
ition on the mean-variance tradeoff process of S. We show that S then allows asymptotic exponential arbitrage with exponentially decaying failure probability, which is a strong and quantitative form of long-term arbitrage. In contrast to Follmer and Schachermayer [FS07], our result does not assume that S is a diffusion, nor does it need any ergodicity assumption.
We study a stochastic game where one player tries to find a strategy such that the state process reaches a target of controlled-loss-type, no matter which action is chosen by the other player. We provide, in a general setup, a relaxed geometric dynam
ic programming principle for this problem and derive, for the case of a controlled SDE, the corresponding dynamic programming equation in the sense of viscosity solutions. As an example, we consider a problem of partial hedging under Knightian uncertainty.
A stochastic model for pure-jump diffusion (the compound renewal process) can be used as a zero-order approximation and as a phenomenological description of tick-by-tick price fluctuations. This leads to an exact and explicit general formula for the
martingale price of a European call option. A complete derivation of this result is presented by means of elementary probabilistic tools.
During the last decade Levy processes with jumps have received increasing popularity for modelling market behaviour for both derviative pricing and risk management purposes. Chan et al. (2009) introduced the use of empirical likelihood methods to est
imate the parameters of various diffusion processes via their characteristic functions which are readily avaiable in most cases. Return series from the market are used for estimation. In addition to the return series, there are many derivatives actively traded in the market whose prices also contain information about parameters of the underlying process. This observation motivates us, in this paper, to combine the return series and the associated derivative prices observed at the market so as to provide a more reflective estimation with respect to the market movement and achieve a gain of effciency. The usual asymptotic properties, including consistency and asymptotic normality, are established under suitable regularity conditions. Simulation and case studies are performed to demonstrate the feasibility and effectiveness of the proposed method.
This work focuses on the indifference pricing of American call option underlying a non-traded stock, which may be partially hedgeable by another traded stock. Under the exponential forward measure, the indifference price is formulated as a stochastic
singular control problem. The value function is characterized as the unique solution of a partial differential equation in a Sobolev space. Together with some regularities and estimates of the value function, the existence of the optimal strategy is also obtained. The applications of the characterization result includes a derivation of a dual representation and the indifference pricing on employee stock option. As a byproduct, a generalized Itos formula is obtained for functions in a Sobolev space.
This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model subject to inter-temporal default risk, and provides a semigroup approximation for the utility indifference price. The key tool is the splitt
ing method, whose convergence is proved based on the Barles-Souganidis monotone scheme, and the convergence rate is derived based on Krylovs shaking the coefficients technique. We apply our methodology to study the counterparty risk of derivatives in incomplete markets.
Density expansions for hypoelliptic diffusions $(X^1,...,X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,...,X_T^l)$, at time $T>0$, with $l leq d$. Global conditions are found which replace the w
ell-known not-in-cutlocus condition known from heat-kernel asymptotics. Our small noise expansion allows for a second order exponential factor. As application, new light is shed on the Takanobu--Watanabe expansion of Brownian motion and Levys stochastic area. Further applications include tail and implied volatility asymptotics in some stochastic volatility models, discussed in a compagnion paper.
