International Journal of Theoretical and Applied Finance

This paper addresses the problem of approximating the price of options on discrete and continuous arithmetic averages of the underlying, i.e. discretely and continuously monitored Asian options, in local volatility models. A “path-integral”-type expression for option prices is obtained using a Brownian bridge representation for the transition density between consecutive sampling times and a Laplace asymptotic formula. In the limit where the sampling time window approaches zero, the option price is found to be approximated by a constrained variational problem on paths in time-price space. We refer to the optimizing path as the most-likely path (MLP). An approximation for the implied normal volatility follows accordingly. The small-time asymptotics and the existence of the MLP are also rigorously recovered using large deviation theory.

We develop a general framework for applying the Kelly criterion to the stock market. By supplying an arbitrary probability distribution modeling the future price movement of a set of stocks, the Kelly fraction for investing each stock can be calculated by inverting a matrix involving only first and second moments. The framework works for one or a portfolio of stocks and the Kelly fractions can be efficiently calculated. For a simple model of geometric Brownian motion of a single stock we show that our calculated Kelly fraction agrees with existing results. We demonstrate that the Kelly fractions can be calculated easily for other types of probabilities such as the Gaussian distribution and correlated multivariate assets.

Correlation plays an important role in pricing multi-asset options. In this work we incorporate stochastic correlation into pricing quanto options which is one special and important type of multi-asset option. Motivated by the market observations that the correlations between financial quantities behave like a stochastic process, instead of using a constant correlation, we allow the asset price process and the exchange rate process to be stochastically correlated with a parameter which is driven either by an Ornstein–Uhlenbeck process or a bounded Jacobi process. We derive an exact quanto option pricing formula in the stochastic correlation model of the Ornstein–Uhlenbeck process and a highly accurate approximated pricing formula in the stochastic correlation model of the bounded Jacobi process, where correlation risk has been hedged. The comparison between prices using our pricing formula and the Monte-Carlo method are provided.

Traditional approaches to Arbitrage Pricing Theory (APT) propose a factor model, but empirical applications of APT are, nowadays, based on seemingly unrelated regression. I drop the factor model and assume only that the market is ergodic. This enables me to apply the theory of Hilbert spaces in a natural way. The expected return on any asset can always be approximated by an affine-linear function of its betas and we are able to estimate the relative number of assets that violate the APT equation by taking the expected returns and betas in the market into account. I present a simple sufficient condition for the APT equation in its inexact form. Further, I show that the APT equation holds true in its exact form if and only if an equilibrium market is exhaustive, which means that it must be possible to replicate the betas and idiosyncratic risk of each asset by some strategy that diversifies away all approximation errors in the market.

We investigate the pricing of temperature derivatives under weather forecasts modeled by initially enlarged filtrations. For this purpose, we introduce a mean-reverting temperature model with seasonality and derive expressions for the so-called forward temperature. Although our analysis focuses on cumulative average temperature (CAT) futures, the presented derivation techniques can likewise be applied to other weather derivatives such as heating degree day (HDD) or cooling degree day (CDD) futures, for instance. We also treat option pricing and utility maximizing portfolio selection in temperature markets under additional information on future weather behavior. We finally prove an anticipative sufficient stochastic minimum principle in an enlarged filtration setup and apply the result to minimal variance hedging of temperature derivatives under weather forecasts. In this context, we derive explicit minimal variance hedging portfolios for different weather-related claims.

This paper focuses on optimal asset allocation with stochastic interest rates in regime-switching models. A class of stochastic optimal control problems with Markovian regime-switching is formulated for which a verification theorem is provided. The theory is applied to solve two portfolio optimization problems (a portfolio of stock and savings account and a portfolio of mixed stock, bond and savings account) while a regime-switching Vasicek model is assumed for the interest rate. Closed-form solutions are obtained for a regime-switching power utility function. Numerical results are provided to illustrate the impact of regime-switching on the optimal investment decisions.

In this work, we deal with market frictions which are given by fixed transaction costs independent of the volume of the trade. The main question that we study is the minimization of shortfall risk in the Black–Scholes (BS) model under constraints on the initial capital. This problem does not have an analytical solution and so numerical schemes come into the picture. The Cox–Ross–Rubinstein (CRR) binomial models are an efficient tool for approximating the BS model. In this paper, we study in detail the CRR models with fixed transaction costs. In particular, we construct an augmented state-action space forming a Markov decision process (MDP) and provide a proof for the existence of optimal control/policy. We further suggest a dynamic programming algorithm for calculating the optimal hedging strategy and its corresponding shortfall risk. In the absence of transaction costs, there is an analytical solution in both CRR and BS models, and so we use them for testing our algorithm and its convergence. Moreover, we point out various insights provided by our numerical results, for example, regarding the change in the investor’s activity in the presence of friction.