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

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

A Numerical Approach to Pricing Exchange Options under Stochastic Volatility and Jump-Diffusion Dynamics

We consider a method of lines (MOL) approach to determine prices of European and American exchange options when underlying asset prices are modeled with stochastic volatility and jump-diffusion dynamics. As with any other numerical scheme for partial differential equations (PDEs); the MOL becomes increasingly complex when higher dimensions are involved; so we first simplify the problem by transforming the exchange option into a call option written on the ratio of the yield processes of the two assets. This is achieved by taking the second asset yield process as the numéraire. Under the equivalent martingale measure induced by this change of numéraire; we derive the exchange option pricing integro-partial differential equations (IPDEs) and investigate the early exercise boundary of the American exchange option. We then discuss a numerical solution of the IPDEs using the MOL; its implementation using computing software and possible alternative boundary conditions at the far limits of the computational domain. Our analytical and numerical investigation shows that the near-maturity behavior of the early exercise boundary of the American exchange option is significantly influenced by the dividend yields and the presence of jumps in the underlying asset prices. Furthermore; with the numerical results generated by the MOL; we are able to show that key jump and stochastic volatility parameters significantly affect the early exercise boundary and exchange option prices. Our numerical analysis also verifies that the MOL performs more efficiently; than other finite difference methods or simulation approaches for American options; since the MOL integrates the computation of option prices; greeks and the early exercise boundary; and does so with the least error

Representation of exchange option prices under stochastic volatility jump-diffusion dynamics

In this article, we provide representations of European and American exchange option prices under stochastic volatility jump-diffusion (SVJD) dynamics following models by Merton [Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 1976, 3(1-2), 125–144], Heston [A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud., 1993, 6(2), 327–343], and Bates [Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Rev. Financ. Stud., 1996, 9(1), 69–107]. A Radon–Nikodým derivative process is also introduced to facilitate the shift from the objective market measure to other equivalent probability measures, including the equivalent martingale measure. Under the equivalent martingale measure, we derive the integro-partial differential equation that characterizes the exchange option prices. We also derive representations of the European exchange option price using the change-of-numéraire technique proposed by Geman et al. [Changes of numéraire, changes of probability measure and option pricing. J. Appl. Probab., 1995, 32(2), 443–458] and the Fourier inversion formula derived by Caldana and Fusai [A general closed-form spread option pricing formula. J. Bank. Finance, 2013, 37, 4893–4906], and show that these two representations are comparable. Lastly, we show that the American exchange option price can be decomposed into the price of the European exchange option and an early exercise premium

Correction: Exchange Option under Jump-diffusion Dynamics

In this note, we provide the correct formula for the price of the European exchange option given in Cheang, G. H. L., & Chiarella, C. (2011. Exchange options under jump-diffusion dynamics. Applied Mathematical Finance, 18, 245–276) in a bi-dimensional jump diffusion model