PRICING ASIAN OPTIONS FOR JUMP DIFFUSION

We construct a sequence of functions that uniformly converge (on compact sets) to the price of an Asian option, which is written on a stock whose dynamics follow a jump diffusion. The convergence is exponentially fast. We show that each element in this sequence is the unique classical solution of a parabolic partial differential equation (not an integro‐differential equation). As a result we obtain a fast numerical approximation scheme whose accuracy versus speed characteristics can be controlled. We analyze the performance of our numerical algorithm on several examples.     

PROPERTIES OF OPTION PRICES IN MODELS WITH JUMPS

We study convexity and monotonicity properties of option prices in a model with jumps using the fact that these prices satisfy certain parabolic integro–differential equations. Conditions are provided under which preservation of convexity holds, i.e., under which the value, calculated under a chosen martingale measure, of an option with a convex contract function is convex as a function of the underlying stock price. The preservation of convexity is then used to derive monotonicity properties of the option value with respect to the different parameters of the model, such as the volatility, the jump size, and the jump intensity.     

Error analysis of finite difference and Markov chain approximations for option pricing

Mijatovi? and Pistorius proposed an efficient Markov chain approximation method for pricing European and barrier options in general one‐dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are nonsmooth, are rarely available. In this paper, we solve this problem for general one‐dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with nonsmooth payoffs. In particular, we show that for call‐/put‐type payoffs, convergence is second order, while for digital‐type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well‐known smoothing techniques that can restore second‐order convergence for digital‐type payoffs and explain oscillations observed in the convergence for options with nonsmooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.     

On Jumps and ARCH Effects in Natural Resource Prices: An Application to Pacific Northwest Stumpage Prices

Continuous-time models of natural resource prices usually preclude the possibility of large changes (jumps) resulting from unexpected events. To test for the presence of jumps and/or ARCH effects, we combine bounds and the Monte Carlo test technique to obtain finite-sample, level-exact p -values. We apply this methodology to stumpage prices from the Pacific Northwest and find evidence of jumps and ARCH effects. To assess the impact of neglecting jumps on the decision to harvest old-growth timber, we develop an autonomous, infinite-horizon stopping model for which we provide a new method of resolution. Our numerical results show the importance of modeling jumps explicitly.     

Neue Sterblichkeitstafeln der “Varma” für minderwertige Leben

Abstract

Die in der Versicherung minderwertiger Leben angewendeten hypothetischen Sterblichkeitstafeln zerfallen ihrem Klassifizierungsgrunde nach in zwei Gruppen: die Klassifizierung gründet sich 1:0 auf die Art der Minderwertigkeitsursache, 2:0 auf den geschatzten Grad der Minderwertigkeit.     

Insurance premiums under demand constraints

Abstract

An alternative approach to the variance principle of premium determination is explored. The approach rationalises the principle in terms of an economic theory and formalises the notion that loadings in addition to the ‘fair’ premium are related to competition and expenses.     

Correlation estimation using components of Japanese candlesticks

Using the wick’s difference from the classical Japanese candlestick representation of daily open, high, low, close prices brings efficiency when estimating the correlation in a bivariate Brownian motion. An interpretation of the correlation estimator given in [Rogers, L.C.G. and Zhou, F., Estimating correlation from high, low, opening and closing prices. Ann. Appl. Probab., 2008, 18(2), 813–823] in the light of wicks’ difference allows us to suggest modifications, which lead to an increased efficiency and robustness over the baseline model. An empirical study of four major financial markets confirms the advantages of the modified estimator.     

Existence of a calibrated regime switching local volatility model

By Gyöngy's theorem, a local and stochastic volatility model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility (LV) function is equal to the ratio of the Dupire LV function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a stochastic differential equation (SDE) nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented in Guyon and Henry‐Labordère [Risk Magazine, 25, 92–97], provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no global existence result is available for the SDE nonlinear in the sense of McKean. When the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level, we prove existence of a solution to the associated Fokker–Planck equation under the condition that the range of the squared stochastic volatility factor is not too large. We then deduce existence to the calibrated model by extending the results in Figalli [Journal of Functional Analysis, 254(1), 109–153].     

MINIMUM GUARANTEED PAYMENTS AND COSTLY CANCELLATION RIGHTS: A STOPPING GAME PERSPECTIVE

We consider the valuation and optimal exercise policy of a δ‐penalty minimum guaranteed payment option in the case where the value of the underlying dividend‐paying asset follows a linear diffusion. We characterize both the value and optimal exercise policy of the considered game option explicitly and demonstrate that increased volatility increases the value of the option and postpones exercise by expanding the continuation region where exercising is suboptimal. An interesting and natural implication of this finding is that the value of the embedded cancellation rights of the issuer increase as volatility increases.     

On pricing barrier control in a regime-switching regulated market

We study a pricing barrier control problem in a regime-switching regulated market. In doing so, we analyze a class of one-dimensional reflected regime-switching diffusion processes. Such diffusion models arise as the key approximating processes in a regulated financial market system with the presence of regime changes. Our main goal is to determine optimal pricing barriers as solutions of long-run average mean–variance optimization problems. More precisely, the optimal barrier, if exists, will be to maximize the long-run average expected return (i.e. steady-state mean) subject to a selected level of long-run average risk (i.e. steady-state variance).