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We find Feynman–Kac type representation theorems for generalized diffusions. To do this we need to establish existence, uniqueness and regularity results for equations with measure-valued coefficients.

We compare two methods for superreplication of options with convex pay-off functions. One method entails the overestimation of an unknown covariance matrix in the sense of quadratic forms. With this method the value of the superreplicating portfolio is given as the solution of a linear Black–Scholes BS-type equation. In the second method, the choice of quadratic form is made pointwise. This leads to a fully non-linear equation, the so-called Black–Scholes–Barenblatt (BSB) equation, for the value of the superreplicating portfolio. In general, this value is smaller for the second method than for the first method. We derive estimates for the difference between the initial values of the superreplicating strategies obtained using the two methods.

Given a centred distribution, can one find a time-homogeneous martingale diffusion starting at zero which has the given law at time 1? We answer the question affirmatively if generalized diffusions are allowed.

Standard arbitrage theory shows that the absence of arbitrage is equivalent to the existence of an equivalent local martingale measure. In most models used for option pricing, including, for exampl...

We study the Black–Scholes equation in stochastic volatility models. In particular, we show that the option price is the unique classical solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility. If the boundary is attainable, then this boundary behaviour serves as a boundary condition and guarantees uniqueness in appropriate function spaces. On the other hand, if the boundary is non-attainable, then the boundary behaviour is not needed to guarantee uniqueness, but is nevertheless very useful for instance from a numerical perspective.

We study the optimal liquidation strategy for a call spread in the case when an investor, who does not hedge, believes in a volatility that differs from the implied volatility. The liquidation problem is formulated as an optimal stopping problem, which we solve explicitly. We also provide a sensitivity analysis with respect to the model parameters.

Boundary values and finite difference methods for the single factor term structure equation

We investigate which jump-diffusion models are convexity preserving. The study of convexity preserving models is motivated by monotonicity results for such models in the volatility and in the jump parameters. We give a necessary condition for convexity to be preserved in several-dimensional jump-diffusion models. This necessary condition is then used to show that, within a large class of possible models, the only convexity preserving models are the ones with linear coefficients., Comment: 14 pages

There are many references showing that a classical solution to the Black-Scholes equation is a stochastic solution. However, it is the converse of this theorem which is most relevant in applications and the converse is also more mathematically interesting. In the present article we establish such a converse. We find a Feynman-Kac type theorem showing that the stochastic representation yields a classical solution to the corresponding Black-Scholes equation with appropriate boundary conditions under very general conditions on the coefficients. We also obtain additional regularity results in the one-dimensional case. Stochastic formulas for option prices are often easy to formulate and to implement in for instance Monte Carlo algorithms. It is natural from the point of view of applications to let zero be an absorbing barrier for processes describing the risky assets. In Section 3 we discuss such a representation in the case of time- and level-dependent volatilities. However, it is often advantageous to instead solve the corresponding Black-Scholes equation and thus be able to use results from the theory of partial differential equations. In the literature one often omits specifying the boundary conditions on the lateral part of the parabolic boundary. This causes no problem when the risky assets are modelled with geometric Brownian motion, since assets in this model reach zero with probability zero. However, for many models, such as the constant elasticity of variance models, the values of the assets can vanish with positive probability and boundary conditions need to be specified. Furthermore, in numerical applications one is often helped by knowing the boundary behavior of the solution even if these conditions are mathematically redundant to specify. In fact, there are many references showing that a classical solution to the Black-Scholes equation is a stochastic solution, compare Theorem 2.5. How- ever, it is the converse of this theorem which is most relevant in applications as described above and the converse is also more mathematically subtle. In the present article we establish such a converse. We find a a Feynman-Kac type theorem showing that the stochastic representation yields a classical so- lution to the corresponding Black-Scholes equation with appropriate bound- ary conditions, compare Theorem 5.5. We also obtain additional regularity results in the one-dimensional case.

We show that the American put option price is log-concave as a function of the log-price of the underlying asset. Thus the elasticity of the price decreases with increasing stock value. We also consider related contracts of American type, and we provide an example showing that not all American option prices are log-concave in the stock log-price.

We prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary., 10 pages

There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

In the present paper we find necessary and sufficient conditions on the coefficients of a parabolic equation for convexity to be preserved. A parabolic equation is said to preserve convexity if given a convex initial condition, any solution of moderate growth remains a convex function of the spatial variables for each fixed time.

We study the term structure equation for single-factor models that predict nonnegative short rates. In particular, we show that the price of a bond or a bond option is the unique classical solution to a parabolic differential equation with a certain boundary behavior for vanishing values of the short rate. If the boundary is attainable then this boundary behavior serves as a boundary condition and guarantees uniqueness of solutions. On the other hand, if the boundary is nonattainable then the boundary behavior is not needed to guarantee uniqueness but it is nevertheless very useful, for instance, from a numerical perspective., Comment: Published in at http://dx.doi.org/10.1214/10-AAP698 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Pairs trading is a common strategy used by hedge funds. When the spread between two highly correlated assets is observed to deviate from historical observations, a long position is taken in the underpriced asset, and a short position in the overpriced one. If the spread narrows, both positions are closed, thus generating a profit. We study when to optimally liquidate a pairs trading strategy when the difference between the two assets is modeled by an Ornstein–Uhlenbeck process. We also provide a sensitivity analysis in the model parameters.

Given a simply connected domain ω in the complex plane with boundary of Minkowski dimension α, we show that the area with respect to the Poincare metric divided by the area with respect to the quasihyperbolic metric of the super level sets of the distance function to ∂ω is asymptotically bounded by α2 as the distance decreases to zero. These bounds are established using a symmetrization argument, showing that the Poincare area of a subdomain of ω can be estimated in terms of a conformal modulus of its complement. We find a generalization of this result to locally simply connected domains and also show that the curvature of the quasi-hyperbolic metric tends to -α on the average, as we approach ∂ω. This behavior of the curvature leads us to conjecture that the bound α2 can be replaced by α.

This paper studies eigenvalue bounds and isoperimetric inequalities for Rieman-nian spaces with cone type singularities along a codimension-2 subcomplex. These “cone-manifolds” include orientable orbifolds, and singular geometric structures on 3-manifolds studied by W. Thurston and others.We first give a precise definition of “cone-manifold” and prove some basic results on the geometry of these spaces. We then generalise results of S.-Y. Cheng on upper bounds of eigenvalues of the Laplacian for disks in manifolds with Ricci curvature bounded from below to cone-manifolds, and characterise the case of equality in these estimates.We also establish a version of the Lévy-Gromov isoperimetric inequality for cone-manifolds. This is used to find lower bounds for eigenvalues of domains in cone-manifolds and to establish the Lichnerowicz inequality for cone-manifolds. These results enable us to characterise cone-manifolds with Ricci curvature bounded from below of maximal diameter.

A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we derive existence and uniqueness results for the Black--Scholes equation, and we provide convexity theory for option pricing and derive related ordering results with respect to volatility. We show that American options are convexity preserving, whereas European options preserve concavity for general payoffs and convexity only for bounded contracts., Comment: Published in at http://dx.doi.org/10.1214/08-AAP579 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study the convexity and model parameter monotonicity properties for prices of bonds and bond options when the short rate is modeled by a diffusion process. We provide sharp conditions on the model parameters under which the convexity of the price in the short rate is guaranteed. Under these conditions, the price is decreasing in the drift and increasing in the volatility of the short rate. We also study the convexity properties of the logarithm of the price and find simple conditions on the coefficients that guarantee that the price is log-convex or log-concave.

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., 14 pages

We use a notion of stochastic time, here called volatility time, to show convexity of option prices in the underlying asset if the contract function is convex as well as continuity and monotonicity of the option price in the volatility. The volatility time is obtained as the almost surely unique stopping time solution to a random ordinary differential equation related to volatility. This enables us to write price processes, or processes modeled by local martingales, as Brownian motions with respect to volatility time. The results are shown under very weak assumptions and are of independent interest in the study of stochastic differential equations. Options on several underlying assets are also studied and we prove that if the volatility matrix is independent of time, then the option prices decay with time if the contract function is convex. However, the option prices are no longer necessarily convex in the underlying assets and the option prices do not necessarily decay with time, if a time-dependent volatility is allowed.

We study pricing equations in jump-to-default models, and we provide conditions under which the option price is the unique classical solution, with a special focus on boundary conditions. In particular, we find precise conditions ensuring that the option price at the default boundary coincides with the recovery payment. We also study spatial convexity of the option price, and we explore the connection between preservation of convexity and parameter monotonicity.

Given a simply connected domain Ω in the complex plane with boundary of Minkowski dimension α, we show that the area with respect to the Poincare metric divided by the area with respect to the quasi-hyperbolic metric of the super level sets of the distance function to ∂Ω is asymptotically bounded by α2 as the distance decreases to zero. Provided that the distance function has no local maximum points arbitrarily close to the boundary, we improve this estimate to α. These bounds are established using a symmetrization argument, showing that the Poincare area of a subdomain of Ω can be estimated in terms of a conformal modulus of its complement. This modulus is then estimated using test functions constructed from the lengths of the level curves of the distance function. We find a generalization of these results to locally simply connected domains and also show that the curvature of the quasi-hyperbolic metric tends to —α on the average, as we approach ∂Ω.

We study Dupire's equation for local volatility models with bubbles, i.e. for models in which the discounted underlying asset follows a strict local martingale. If option prices are given by risk-neutral valuation, then the discounted option price process is a true martingale, and we show that the Dupire equation for call options contains extra terms compared to the usual equation. However, the Dupire equation for put options takes the usual form. Moreover, uniqueness of solutions to the Dupire equation is lost in general, and we show how to single out the option price among all possible solutions. The Dupire equation for models in which the discounted derivative price process is merely a local martingale is also studied.

There is no canonical way of defining a product of distributions. In the present paper we compare two different methods of defining a product of distributions. These methods are based on the sequential and functional-analytic approaches to distributions. Introduction. In many applications of distribution theory a product of distributions occurs. There is, however, no canonical way to define such a product. In this article we shall discuss two different methods of defining a product of two distributions. The sequential approach leads to one way of defining a product, and, using Fourier analysis, we obtain a second product based on the functional-analytic definition of distributions. We call this product the product in the Fourier sense; it is essentially defined as in [1]. We shall compare the two definitions and prove that if that product exists in the Fourier sense, it also exists in the sequential sense, and the products are equal. This article is based on an undergraduate thesis published at the Department of Mathematics at Uppsala University in September 1981. Professor Christer 0. Kiselman gave me invaluable help both with my undergraduate thesis and the present paper. I take this opportunity to thank him. The sequential method. First let us study a product based on the sequential approach. To do this we will need so-called 8-sequences: Let an, n > 1, be positive real numbers such that an--* 0 as n xo. Let dn E C0&(B(0, an)) where B(O, an) {X E Rm: lxl < an}. Assume that

Let M n , n ≥ 3 {M^n},n \geq 3 , be an oriented minimally immersed complete hypersurface in Euclidean space. We show that for n = 3 , 4 , 5 , or 6 n = 3,4,5,{\text { or }}6 , the index of M n {M^n} is finite if and only if the total scalar curvature of M n {M^n} is finite, provided that the volume growth of M n {M^n} is bounded by a constant times r n {r^n} , where r r is the Euclidean distance function. We also note that this result does not hold for n ≥ 8 n \geq 8 . Moreover, we show that the index of M n {M^n} is bounded by a constant multiple of the total scalar curvature for all n ≥ 3 n \geq 3 , without any assumptions on the volume growth of M n {M^n} .

The multi-dimensional Black–Scholes equation is solved numerically for a European call basket option using a priori–a posteriori error estimates. The equation is discretized by a finite difference method on a Cartesian grid. The grid is adjusted dynamically in space and time to satisfy a bound on the global error. The discretization errors in each time step are estimated and weighted by the solution of the adjoint problem. Bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations. Comparisons are made with Monte Carlo and quasi-Monte Carlo methods in one dimension, and the performance of the method is illustrated by examples in one, two, and three dimensions.

On etudie des estimations de valeur propre des recouvrements de Riemann ramifies des varietes compactes. On demontre que si φ:M n →N n est un recouvrement de Riemann ramifie, et {μ i } i=0 ∞ et {λ i } i=0 ∞ sont les valeurs propres de l'operateur de Laplace-Beltrami sur M et N, alors Σ i=0 ∞ exp(-μit)≤kΣ i=0 ∞ exp(−λit) pour tout t positif, ou k est le nombre de couches du recouvrement