International Journal of Theoretical and Applied Finance

For a large class of ℝ₊ valued, continuous local martingales (M_tt ≥ 0), with M₀ = 1 and M_∞ = 0, the put quantity: Π_M (K,t) = E ((K - M_t)⁺) turns out to be the distribution function in both variables K and t, for K ≤ 1 and t ≥ 0, of a probability γ_M on [0,1] × [0, ∞[. In this paper, the first in a series of three, we discuss in detail the case where , for (B_t, t ≥ 0) a standard Brownian motion.

We study a model of a financial market in which two risky assets are paying dividends with rates changing their initial values to other constant ones when certain events occur. Such events are associated with the first times at which the value processes of issuing firms, modeled by geometric Brownian motions, fall to some prescribed levels. The asset price dynamics are described by exponential diffusion processes with random drift rates and independent driving Brownian motions. We derive closed form expressions for rational values of European contingent claims, under full and partial information.

We investigate triangular arbitrage within the spot foreign exchange market using high-frequency executable prices. We show that triangular arbitrage opportunities do exist, but that most have short durations and small magnitudes. We find intra-day variations in the number and length of arbitrage opportunities, with larger numbers of opportunities with shorter mean durations occurring during more liquid hours. We demonstrate further that the number of arbitrage opportunities has decreased in recent years, implying a corresponding increase in pricing efficiency. Using trading simulations, we show that a trader would need to beat other market participants to an unfeasibly large proportion of arbitrage prices to profit from triangular arbitrage over a prolonged period of time. Our results suggest that the foreign exchange market is internally self-consistent and provide a limited verification of market efficiency.

We present a fast and accurate FFT-based method of computing the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a Lévy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying Lévy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short.

In this paper, we analyze the importance of curvature term structure movements on forecasts of interest rates. An extension of the exponential three-factor Diebold and Li (2006) model is proposed, where a fourth factor captures a second type of curvature. The new factor increases model ability to generate volatility and to capture nonlinearities in the yield curve, leading to a significant improvement of forecasting ability. The model is tested against the original Diebold and Li model and some other benchmarks. Based on a forecasting experiment with Brazilian fixed income data, it obtains significantly lower bias and root mean square errors for most examined maturities, and under three different forecasting horizons. Robustness tests based on two sub-sample analyses partially confirm the favorable results.

We develop the complete 6-dimensional classical symmetry group of the partial differential equation (PDE) that governs the fair price of a simple Asian option within a simple market model. The symmetries we expose include the 5-dimensional symmetry group partially noted by Rogers and Shi, and communicated implicitly by the change of numéraire arguments of Večeř (in which symmetries reduce the original 2 + 1 dimensional simple Asian option PDE to a 1 + 1 dimensional PDE). Going beyond this previous work, we expose a new 1-dimensional space of symmetries of the Asian PDE that cannot reasonably be found by inspection. We demonstrate that the new symmetry could be used to formulate a new, "nonlinear" derivative security that has a 1 + 1 dimensional PDE formulation. We indicate that this nonlinear security has a closed-form pricing formula similar to that of the Black–Scholes equation for a particular market dependent payoff, and show that hedging the short position in this particular exotic option is stable for all market parameters. We also demonstrate the patently Lie-algebraic method for obtaining the already well-known "Rogers–Shi–Večeř" reduction.

Motivated by the interplay between structural and reduced form credit models, we propose to model the firm value process as a time-changed Brownian motion that may include jumps and stochastic volatility effects, and to study the first passage problem for such processes. We are lead to consider modifying the standard first passage problem for stochastic processes to capitalize on this time change structure and find that the distribution functions of such "first passage times of the second kind" are efficiently computable in a wide range of useful examples. Thus this new notion of first passage can be used to define the time of default in generalized structural credit models. Formulas for defaultable bonds and credit default swaps are given that are both efficiently computable and lead to realistic spread curves. Finally, we show that by treating joint firm value processes as dependent time changes of independent Brownian motions, one can obtain multifirm credit models with rich and plausible dynamics and enjoying the possibility of efficient valuation of portfolio credit derivatives.