Asians and cash dividends: Exploiting symmetries in pricing theory

In this article we present new results for the pricing of arithmetic Asian options within a Black-Scholes context. To derive these results we make extensive use of the local scale invariance that exists in the theory of contingent claim pricing. This allows us to derive, in a natural way, a simple PDE for the price of arithmetic Asians options. In the case of European average strike options, a proper choice of numeraire reduces the dimension of this PDE to one, leading to a PDE similar to the one derived by Rogers and Shi. We solve this PDE, finding a Laplace-transform representation for the price of average strike options, both seasoned and unseasoned. This extends the results of Geman and Yor, who discussed the case of average price options. Next we use symmetry arguments to show that prices of average strike and average price options can be expressed in terms of each other. Finally we show, again using symmetries, that plain vanilla options on stocks paying known cash dividends are closely related to arithmetic Asians, so that all the new techniques can be directly applied to this case.Comment: 19 pages, no figure

Tradable Schemes

In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very elegant way, due to the fact that in our tradable based formulation there appear no drift terms in the PDE. We construct a mixed scheme based on this idea and apply it to price various types of arithmetic Asian options, as well as plain vanilla options (both european and american style) on stocks paying known cash dividends. We find prices which are accurate to $\sim 0.1$ in about 10ms on a Pentium 233MHz computer and to $\sim 0.001$ in a second. The scheme can also be used for market conform pricing, by fitting it to observed option prices.Comment: 13 pages, 5 tables, LaTeX 2

Scale invariance and contingent claim pricing II: Path-dependent contingent claims

This article is the second one in a series on the use of scaling invariance in finance. In the first paper, we introduced a new formalism for the pricing of derivative securities, which focusses on tradable objects only, and which completely avoids the use of martingale techniques. In this article we show the use of the formalism in the context of path-dependent options. We derive compact and intuitive formulae for the prices of a whole range of well known options such as arithmetic and geometric average options, barriers, rebates and lookback options. Some of these have not appeared in the literature before. For example, we find rather elegant formulae for double barrier options with moving barriers, continuous dividends and all possible configurations of the barriers. The strength of the formalism reveals itself in the ease with which these prices can be derived. This allowed us to pinpoint some mistakes regarding geometric mean options, which frequently appear in the literature. Furthermore, symmetries such as put-call transformations appear in a natural way within the framework.contingent claim pricing, scale-invariance, homogeneity, partial differential equation

Tradable Schemes

In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very elegant way, due to the fact that in our tradable based formulation there appear no drift terms in the PDE. We construct a mixed scheme based on this idea and apply it to price various types of arithmetic Asian options, as well as plain vanilla options (both european and american style) on stocks paying known cash dividends. We find prices which are accurate to ~0.1% in about 10ms on a Pentium 233MHz computer and to ~0.001% in a second. The scheme can also be used for market conform pricing, by fitting it to observed option prices.contingent claim pricing, numeric methods, asian options, cash dividend, partial differential equation

Asians and cash dividends: Exploiting symmetries in pricing theory

In this article we present new results for the pricing of arithmetic Asian options within a Black-Scholes context. To derive these results we make extensive use of the local scale invariance that exists in the theory of contingent claim pricing. This allows us to derive, in a natural way, a simple PDE for the price of arithmetic Asians options. In the case of European average strike options, a proper choice of numeraire reduces the dimension of this PDE to one, leading to a PDE similar to the one derived by Rogers and Shi. We solve this PDE, finding a Laplace-transform representation for the price of average strike options, both seasoned and unseasoned. This extends the results of Geman and Yor, who discussed the case of average price options. Next we use symmetry arguments to show that prices of average strike and average price options can be expressed in terms of each other. Finally we show, again using symmetries, that plain vanilla options on stocks paying known cash dividends are closely related to arithmetic Asians, so that all the new techniques can be directly applied to this case.contingent claim pricing, scale invariance, asian options, partial differential equation

Symmetries in Jump-Diffusion Models with Applications in Option Pricing and Credit Risk

It is a well known fact that local scale invariance plays a fundamental role in the theory of derivative pricing. Specific applications of this principle have been used quite often under the name of `change of numeraire', but in recent work it was shown that when invoked as a fundamental first principle, it provides a powerful alternative method for the derivation of prices and hedges of derivative securities, when prices of the underlying tradables are driven by Wiener processes. In this article we extend this work to the pricing problem in markets driven not only by Wiener processes but also by Poisson processes, i.e. jump-diffusion models. It is shown that in this case too, the focus on symmetry aspects of the problem leads to important simplifications of, and a deeper insight into the problem. Among the applications of the theory we consider the pricing of stock options in the presence of jumps, and Levy-processes. Next we show how the same theory, by restricting the number of jumps, can be used to model credit risk, leading to a `market model' of credit risk. Both the traditional Duffie- Singleton and Jarrow-Turnbull models can be described within this framework, but also more general models, which incorporate default correlation in a consistent way. As an application of this theory we look at the pricing of a credit default swap (CDS) and a first-to- default basket option.option pricing, jump diffusion, local scale invariance, homogeneity, partial differential difference equations

Estimating Cardinalities with Deep Sketches

We introduce Deep Sketches, which are compact models of databases that allow us to estimate the result sizes of SQL queries. Deep Sketches are powered by a new deep learning approach to cardinality estimation that can capture correlations between columns, even across tables. Our demonstration allows users to define such sketches on the TPC-H and IMDb datasets, monitor the training process, and run ad-hoc queries against trained sketches. We also estimate query cardinalities with HyPer and PostgreSQL to visualize the gains over traditional cardinality estimators.Comment: To appear in SIGMOD'1