International Journal of Theoretical and Applied Finance

In this paper we consider a nonlinear filtering approach to the estimation of asset price volatility. We are particularly interested in models which are suitable for high frequency data. In order to describe some of the typical features of high frequency data we consider marked point process models for the asset price dynamics. Both jump-intensity and jump-size distribution of this marked point process depend on a hidden state variable which is closely related to asset price volatility. In our setup volatility estimation can therefore be viewed as a nonlinear filtering problem with marked point process observations. We develop efficient recursive methods to compute approximations to the conditional distribution of this state variable using the so-called reference probability approach to nonlinear filtering.

We consider a general discrete-time dynamic financial market with three assets: a riskless bond, a security and a derivative. The market is incomplete (a priori) and at equilibrium. We assume also that the agents of the economy have short-sales constraints on the stock and that the payoff at the expiry of the derivative asset is a monotone function of the underlying security price. The derivative price process is not identified ex ante. This leads the agents to act as if there were no market for this asset at the intermediary dates. Using some nice properties of the pricing probabilities, which are admissible at the equilibrium, we prove that it suffices to consider the subset of the risk-neutral probabilities that overestimate the low values of the security and underestimate its high values with respect to the true probability. This approach greatly reduces the interval of admissible prices for the derivative asset with respect to no-arbitrage, as showed numerically.

We consider the case of an agent with recursive preferences (Stochastic Differential Utility or SDU) who cannot observe the random drift of the stock price process. This partial information problem can be transformed into a problem with full information, in which the drift is replaced by its expected value conditional on the information given by the stock price history. We are then in a position to use recent results for maximizing SDUs under full information and their connection to Forward-Backward Stochastic Differential Equations. They enable us to study qualitative differences between the cases with full and partial information. We also analyze some special cases in which we obtain semi-explicit results and compute the SDU numerically.

We consider the mean-variance hedging when an investor observes just the stock prices. We explain how the theory developed in Gouriéroux, Laurent and Pham (1998) and Rheinländer and Schweizer (1997) can be extended to this framework. We then focus to a diffusion model when drift of stock prices are not observed directly but only through a measurement process. By using filtering techniques, we obtain explicit formulae for optimal mean-variance hedging strategies and for the associated minimal risk. Closed-form expressions are provided in the case of a Bayesian investor and when the stock drift is modelled as a linear Gaussian process.

This paper is an extension of [13] where we have studied a financial market with informed and non-informed agents. The informed agent has an initial insider information. We analyze the completeness of the market and the family of risk-neutral probabilities. Some cases of arbitrage opportunities are done.

We present an approach for the pricing of illiquid bonds (and bond derivatives) in an arbitrage-free way and which is consistent with the observed prices of liquid bonds. The basic model is a multifactor term structure model with abstract latent factors. The approach is based on stochastic filtering techniques, leading to a continuous update of the distribution of the latent factors on the basis of the information coming from the observations. This allows our model to continuously "track" the real market.

We consider an investor with initial wealth X₀<1, who wishes to maximize the probability of achieving a goal, X_T = 1, when the stock's drift — modeled as a linear mean-reverting diffusion — is not observed directly but only via the measurement process. Adopting a martingale approach, a generalized Cameron–Martin (1945) formula then enables explicit computation of the value of the problem as well as the wealth process. The dynamic optimal allocation can then be determined using Clark's formula.

We consider a new point of view to portfolio selection in presence of transaction costs which allows us to map the problem into one without costs. The proposed approach connects all the quantities of interest to the average lag between two trades and the probabilities of reaching a price fluctuation of a given size. This active portfolio managing in presence of costs is based on quantities directly measurable in real markets and it leads to analytic results in the Wiener case.

Closed form formulae for European barrier options are well known from the literature. This is not the case for American barrier options, for which no closed form formulae have been published. One has therefore had to resort to numerical methods. Lattice models like a binomial or a trinomial tree, for valuation of barrier options are known to converge extremely slowly, compared to plain vanilla options. Methods for improving the algorithms have been described by several authors. However, these are still numerical methods that are quite computer intensive. In this paper we show how some American barrier options can be valued analytically in a very simple way. This speeds up the valuation dramatically as well as give new insight into barrier option valuation.

We consider an option c which is contingent on an underlying that is not a traded asset. This situation typically arises in the context of real options. We investigate the situation when there is a "surrogate" traded asset S whose price process is highly correlated with that of . An illustration would be the cases where S and model two different brands of crude oil. The main result of the paper shows that in this case one cannot draw any non-trivial conclusions on the price of the option by only using no-arbitrage arguments.

In a second step we try to isolate hedging strategies on the traded asset S which minimize the variance of the hedging error. We show in particular, that the naive strategy of simply replacing by S fails to be optimal and we are able to quantify how far it is from being optimal.