International Journal of Theoretical and Applied Finance

In this work, we develop an efficient methodology for analyzing risk in the wealth balance (hedging error) distribution arising from a mean square optimal dynamic hedge on a European call option, where the underlying stock price process is modeled on a multinomial lattice. By exploiting structure in mean square optimal hedging problems, we show that moments of the resulting wealth balance may be computed directly and efficiently on the stock lattice through the backward iteration of a matrix. Based on this moment information, convex optimization techniques are then used to estimate the Value-at-Risk of the hedge. This methodology is applied to a numerical example where the Value-at-Risk is estimated for a hedged European call option on a stock modeled on a trinomial lattice.

An informationally efficient price keeps investors as a group in the state of maximum uncertainty about the next price change. The Entropy Pricing Theory (EPT) captures this intuition and suggests that, in informationally efficient markets, perfectly uncertain market beliefs must prevail. When the entropy functional is used to index collective market uncertainty, then the entropy-maximizing consensus beliefs must prevail. The EPT resolves the ambiguity of arbitrage-free valuation in incomplete markets. The EPT produces a new bond option model that is similar to Black–Scholes' with the lognormal distribution replaced by a beta distribution. Unlike alternative models, the beta model is valid for arbitrary term structure dynamics and for arbitrary credit risk of the underlying bonds. Option replication and hedging under the beta model accounts for random changes in the underlying bond price, price volatility and short-term interest rates.

In this work we introduce an adaptive method of portfolio optimization. The basic idea is to describe essential movements of the stock price using a hidden Markov model and to calculate the optimal portfolio using a recursive algorithm. The portfolio optimization is adaptive in the sense that the standard EM-algorithm fits the model to historical data, which improves the portfolio performance.

In the general framework that is offered by the market model, each LIBOR interest rate is a lognormal martingale under its own probability measure. The advantage is that the approach is consistent with the way cap, floor and swaption volatilities are quoted. The joint distribution of several LIBOR or swap rates under a common probability measure is somehow more complicated because it requires the specification of a drift term structure and the specification of a correlation term structure. In this paper, the correlation between the LIBORs is represented by a function of the LIBOR maturities. The form of this function is inspired by the stochastic string theory that was recently introduced in finance for the modelling of yield curves. The function is fitted to the volatilities of the LIBOR and swap rates so that it is consistent with market observations and does not rely on statistical analysis of any historical data.

We introduce a general class of analytically tractable models for the dynamics of an asset price based on the assumption that the asset-price density is given by the mixture of known basic densities. We consider the lognormal-mixture model as a fundamental example, deriving explicit dynamics, closed form formulas for option prices and analytical approximations for the implied volatility function. We then introduce the asset-price model that is obtained by shifting the previous lognormal-mixture dynamics and investigate its analytical tractability. We finally consider a specific example of calibration to real market option data.