Bi-stable chemical systems are the basic building blocks for intracellular memory and cell fate decision circuits. These circuits are built from molecules, which are present at low copy numbers and are slowly diffusing in complex intracellular geometries. The stochastic reaction-diffusion kinetics of a double-negative feedback system and a MAPK phosphorylation-dephosphorylation system is analysed with Monte-Carlo simulations of the reaction-diffusion master equation. The results show the geometry of intracellular reaction compartments to be important both for the duration and the locality of biochemical memory. Rules for when the systems lose global hysteresis by spontaneous separation into spatial domains in opposite phases are formulated in terms of geometrical constraints, diffusion rates and attractor escape times. The analysis is facilitated by a new efficient algorithm for exact sampling of the Markov process corresponding to the reaction-diffusion master equation.

/pdf/spontaneous-separation-of-bi-stable-biochemical-systems-into-59v4vsiwm8.pdf

Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases

https://ch4170.files.wordpress.com/2011/08/multiscalemodeling.pdf

A review of multiscale modeling of metal-catalyzed reactions: Mechanism development for complexity and emergent behavior

Free And Moving Boundary Problems

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Handbook Of Integral Equations

We introduce a new analytical approach to price American options. Using an explicit and intuitive proxy for the exercise rule, we derive tractable pricing formulas using a short-maturity asymptotic expansion. Depending on model parameters, this method can accurately price options with time-to-maturity up to several years. The main advantage of our approach over existing methods lies in its straightforward extension to models with stochastic volatility and stochastic interest rates. We exploit this advantage by providing an analysis of the impact of volatility mean-reversion, volatility of volatility, and correlations on the American put price.

/pdf/pricing-american-options-under-stochastic-volatility-and-59hdq88eh2.pdf

Pricing American Options under Stochastic Volatility and Stochastic Interest Rates

An installment option is a derivative financial security where the price is paid in installments instead of as a lump sum at the time of purchase. The valuation of these options involves a free boundary problem in that at each installment date, the holder of the derivative has the option of continuing to pay the premiums or allowing the contract to lapse, and the decision will depend upon whether the present value of the expected pay-off is greater or less than the present value of the remaining premiums. Using a model installment option where the premiums are paid continuously rather than on discrete dates, an integral equation is derived for the position of this free boundary by applying a partial Laplace transform to the underlying partial differential equation for the value of the security. Asymptotic analysis of this integral equation allows us to deduce the behavior of the free boundary close to expiry.

Laplace transforms and installment options

Laplace transform methods are used to study the valuation of American call and put options with constant dividend yield, and to derive integral equations giving the location of the optimal exercise boundary. In each case studied, the main result of this paper is a nonlinear Fredholm-type integral equation for the location of the free boundary. The equations differ depending on whether the dividend yield is less than or exceeds the risk-free rate. These integral equations contain a transform variable, so the solution of the equations would involve finding the free boundary that satisfies the equations for all values of this transform variable. Expressions are also given for the transform of the value of the option in terms of this free boundary.

Laplace transforms and American options

We address the pricing of American straddle options. We use
partial Laplace transform techniques due to Evans et al. (1950) to derive a pair of integral equations giving the locations of the optimal exercise boundaries for an American straddle option with a constant dividend yield.

https://downloads.hindawi.com/journals/jam/2002/428081.pdf

Laplace transforms and the American straddle

We use an asymptotic expansion to study the behavior of the American put option close to expiry for the case where the dividend yield is less than or equal to the risk-free interest rate. Series solutions are obtained for the location of the free boundary and the price of the option in that limit. Options are derivative financial securities whose value is based on the value of some other underlying security. Equity options are options on an individual stock, or an index of such stocks such as the Dow Jones Industrial Average (DJIA) or S&P500, and such options are traded both on exchanges and over-the-counter. Although more exotic pay-os are possible, exchange-traded options tend to be either vanilla calls or puts. If the price of the underlying stock is S and E is the exercise price of the option, which is specified in the option contract, a vanilla call pays an amount max(S E,0) at expiry while a vanilla put pays max(E S,0). In addition to being classified by their pay-os, options are also classified by when they can be exercised: European options can be exercised only at expiry while American options can be exercised at any time at or before expiry, while the less-common Bermudan options can be exercised on a finite number of pre-specified dates. Because they can be exercised only at expiry, it is straightforward to price European options using the Black-Scholes option pricing formula (6): essentially, one calculates the probability distribution for the stock price at expiry and uses that to price the option. American options are considerably harder to price because the possibility of early exercise leads to a free boundary problem, with this free boundary separating the region where it is optimal to hold from the region where exercise is optimal. In theory, exercise should take place only on this free boundary, known as the optimal exercise boundary. This sort of free boundary problem is common in diusion problems such as melting and solidification problems and is referred to as a Stefan problem. To date, no closed form solution is known for American options, except for one or two very special cases such as the American call without dividends when exercise is never optimal meaning that the option has the same value as a European.

/pdf/the-american-put-option-close-to-expiry-4caeqbj2kt.pdf

The american put option close to expiry

We use an asymptotic expansion to study the behavior of installment options close to expiry. Installment options are contracts where the price is paid over the life of the option rather than as a lump sum at the time of purchase, and where the contract can be allowed to lapse at any time. Series solutions are obtained for the location of the free boundary and the price of the option.

https://downloads.hindawi.com/archive/2006/060824.pdf

Ghada Alobaidi

Papers

Laplace transforms and installment options

Laplace transforms and American options

Laplace transforms and the American straddle

The american put option close to expiry

Installment options close to expiry