Hedge Portfolios in Markets with Price Discontinuities

We consider a market consisting of multiple assets under jump-diffusion dynamics with European style options written on these assets. It is well-known that such markets are incomplete in the Harrison and Pliska sense. We derive a pricing relation by adopting a Radon-Nikodym derivative based on the exponential martingale of a correlated Brownian motion process and a multivariate compound Poisson process. The parameters in the Radon-Nikodym derivative define a family of equivalent martingale measures in the model, and we derive the corresponding integro-partial differential equation for the option price. We also derive the pricing relation by setting up a hedge portfolio containing an appropriate number of options to "complete" the market. The market prices of jump-risks are priced in the hedge portfolio and we relate these to the choice of the parameters in the Radon-Nikodym derivative used in the alternative derivation of the integro-partial differential equation.incomplete markets; equivalent martingale measure; compound Poisson processes; Radon-Nikodym derivative; multi-asset options; integro-partial differential equation

A Modern View on Merton's Jump-Diffusion Model

Merton has provided a formula for the price of a European call option on a single stock where the stock price process contains a continuous Poisson jump component, in addition to a continuous log-normally distributed component. In Merton's analysis, the jump-risk is not priced. Thus the distribution of the jump-arrivals and the jump-sizes do not change under the change of measure. We go onto introduce a Radon-Nikodym derivative process that induces the change of measure from the market measure to an equivalent martingale measure. The choice of parameters in the Radon-Nikodym derivative allows us to price the option under different financial-economic scenarios. We introduce a hedging argument that eliminates the jump-risk in some sort of averaged sense, and derive an integro-partial differential equation of the option price that is related to the one obtained by Merton.financial derivatives; compound Poisson processes; equivalent martingale measure; hedging portfolio

Exchange Options Under Jump-Diffusion Dynamics

Margrabe provides a pricing formula for an exchange option where the distributions of both stock prices are log-normal with correlated Wiener components. Merton has provided a formula for the price of a European call option on a single stock where the stock price process contains a continuous Poisson jump component, in addition to a continuous log-normally distributed component. We use Merton’s analysis to extend Margrabe’s results to the case of exchange options where both stock price processes also contain compound Poisson jump components. A Radon-Nikod´ym derivative process that induces the change of measure from the market measure to an equivalent martingale measure is introduced. The choice of parameters in the Radon-Nikod´ym derivative allows us to price the option under different financial-economic scenarios. We also consider American style exchange options and provide a probabilistic intepretation of the early exercise premium.American options; exchange options; compound Poisson processes; equivalent martingale measure

An Analysis of American Options under Heston Stochastic Volatility and Jump-Diffusion Dynamics

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square root process as used by Heston (1993), and by a Poisson jump process as introduced by Merton (1976). Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalises in an intuitive way the structure of the solution to the corresponding European option pricing problem in the case of a call option and constant interest rates obtained by Scott (1997).American options; stochastic volatility; jump-diffusion processes; Volterra integral equations; free boundary problem; method of lines

Interview with Gerald Cheang

Gerald (Jerry) was not born in the Salinas Chinatown but he was able provide information pertaining to his childhood growing up in Stockton, California, which also had a huge Chinese population and where Jerry grew up. He talks about his education in American Schools as well as Chinese School and the importance of going to school and getting an education. With his education that he received he was able to go to college and become a dentist still to this day. Jerry gives us a look at what his parents did for a living during WWII and how he contributed to the family during this time as well as post war. Jerry moved to Salinas in the mid 60s, due to not wanting to raise his children in the big city of San Francisco. The interview also focuses on his contributions and involvements with the Revitalization Project and also with the Asian Cultural Encounter. He wants to bring awareness to the public and hopes for the public to get involved in as well as CSUMB, because without their contributions nothing could get accomplished such as the oral history museum and the garden.https://digitalcommons.csumb.edu/ohcma_chinatown/1022/thumbnail.jp

A systems analysis of the chemosensitivity of breast cancer cells to the polyamine analogue PG-11047

<p>Abstract</p> <p>Background</p> <p>Polyamines regulate important cellular functions and polyamine dysregulation frequently occurs in cancer. The objective of this study was to use a systems approach to study the relative effects of PG-11047, a polyamine analogue, across breast cancer cells derived from different patients and to identify genetic markers associated with differential cytotoxicity.</p> <p>Methods</p> <p>A panel of 48 breast cell lines that mirror many transcriptional and genomic features present in primary human breast tumours were used to study the antiproliferative activity of PG-11047. Sensitive cell lines were further examined for cell cycle distribution and apoptotic response. Cell line responses, quantified by the GI₅₀(dose required for 50% relative growth inhibition) were correlated with the omic profiles of the cell lines to identify markers that predict response and cellular functions associated with drug sensitivity.</p> <p>Results</p> <p>The concentrations of PG-11047 needed to inhibit growth of members of the panel of breast cell lines varied over a wide range, with basal-like cell lines being inhibited at lower concentrations than the luminal cell lines. Sensitive cell lines showed a significant decrease in S phase fraction at doses that produced little apoptosis. Correlation of the GI₅₀values with the omic profiles of the cell lines identified genomic, transcriptional and proteomic variables associated with response.</p> <p>Conclusions</p> <p>A 13-gene transcriptional marker set was developed as a predictor of response to PG-11047 that warrants clinical evaluation. Analyses of the pathways, networks and genes associated with response to PG-11047 suggest that response may be influenced by interferon signalling and differential inhibition of aspects of motility and epithelial to mesenchymal transition.</p> <p>See the related commentary by Benes and Settleman: <url>http://www.biomedcentral.com/1741-7015/7/78</url></p

Approximation with neural networks activated by ramp sigmoids

AbstractAccurate and parsimonious approximations for indicator functions of d-dimensional balls and related functions are given using level sets associated with the thresholding of a linear combination of ramp sigmoid activation functions. In neural network terminology, we are using a single-hidden-layer perceptron network implementing the ramp sigmoid activation function to approximate the indicator of a ball. In order to have a relative accuracy ϵ, we use T=c(d2/ϵ2) ramp sigmoids, a result comparable to that of Cheang and Barron (2000) [4], where unit step activation functions are used instead. The result is then applied to functions that have variation Vf with respect to a class of ellipsoids. Two-hidden-layer feedforward neural nets with ramp sigmoid activation functions are used to approximate such functions. The approximation error is shown to be bounded by a constant times Vf/T112+Vfd/T214, where T1 is the number of nodes in the outer layer and T2 is the number of nodes in the inner layer of the approximation fT1,T2

A Better Approximation for Balls

AbstractUnexpectedly accurate and parsimonious approximations for balls in Rd and related functions are given using half-spaces. Instead of a polytope (an intersection of half-spaces) which would require exponentially many half-spaces (of order (1ε)d) to have a relative accuracy ε, we use T=c(d2/ε2) pairs of indicators of half-spaces and threshold a linear combination of them. In neural network terminology, we are using a single hidden layer perceptron approximation to the indicator of a ball. A special role in the analysis is played by probabilistic methods and approximation of Gaussian functions. The result is then applied to functions that have variation Vf with respect to a class of ellipsoids. Two hidden layer feedforward sigmoidal neural nets are used to approximate such functions. The approximation error is shown to be bounded by a constant times Vf/T1/21+Vfd/T1/42, where T1 is the number of nodes in the outer layer and T2 is the number of nodes in the inner layer of the approximation fT1, T2