If I remember rightly, cos⁡π2=1\cos\frac{\pi}{2} =1

An account of some of the less rigorous utterances of applied mathematicians

A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 2: Bermudan options.

We discuss the `continuity correction' that should be applied to connect the prices of discretely sampled American put options (i.e. Bermudan options) and their continuously-sampled equivalents. Using a matched asymptotic expansions approach we compute the correction and relate it to that discussed by Broadie, Glasserman & Kou (Mathematical Finance 7, 325 (1997)) for barrier options. In the Bermudan case, the continuity correction is an order of magnitude smaller than in the corresponding barrier problem. We also show that the optimal exercise boundary in the discrete case is slightly higher than in the continuously sampled case

Risk-Neutral Pricing of Financial Instruments in Emission Markets: A Structural Approach

We present a novel approach to the pricing of financial instruments in emission markets, for example, the EU ETS. The proposed structural model is positioned between existing complex full equilibrium models and pure reduced form models. Using an exogenously specified demand for a polluting good it gives a causal explanation for the accumulation of CO2 emissions and takes into account the feedback effect from the cost of carbon to the rate at which the market emits CO2. We derive a forward-backward stochastic differential equation for the price process of the allowance certificate and solve the associated semilinear partial differential equation numerically. We also show that derivatives written on the allowance certificate satisfy a linear partial differential equation. The model is extended to emission markets with multiple compliance periods and we analyse the impact different intertemporal connecting mechanisms, such as borrowing, banking and withdrawal, have on the allowance price.Comment: Section 5 in this version is new and contains an asymptotic analysis of the problem under consideratio

A free boundary problem arising in a model for shallow water entry at small deadrise angles

A free boundary problem arising in a model for inviscid, incompressible shallow water entry at small deadrise angles is derived and analysed. The relationship between this novel free boundary problem and the well-known viscous squeeze film problem is described. An inverse method is used to construct explicit solutions for certain body profiles and to find criteria under which the splash sheet can `split'. A variational inequality formulation, conservation of certain generalized moments and the Schwarz function formulation are introduced

A non-arbitrage liquidity model with observable parameters for derivatives

We develop a parameterised model for liquidity effects arising from the trading in an asset. Liquidity is defined via a combination of a trader's individual transaction cost and a price slippage impact, which is felt by all market participants. The chosen definition allows liquidity to be observable in a centralised order-book of an asset as is usually provided in most non-specialist exchanges. The discrete-time version of the model is based on the CRR binomial tree and in the appropriate continuous-time limits we derive various nonlinear partial differential equations. Both versions can be directly applied to the pricing and hedging of options; the nonlinear nature of liquidity leads to natural bid-ask spreads that are based on the liquidity of the market for the underlying and the existence of (super-)replication strategies. We test and calibrate our model set-up empirically with high-frequency data of German blue chips and discuss further extensions to the model, including stochastic liquidity

Option pricing with Lévy-Stable processes generated by Lévy-Stable integrated variance.

We show how to calculate European-style option prices when the log-stock price process follows a Lévy-Stable process with index parameter 1≤α≤2 and skewness parameter -1≤β≤1. Key to our result is to model integrated variance as an increasing Lévy-Stable process with continuous paths in ΤLévy-Stable processes; Stable Paretian hypothesis; Stochastic volatility; α-stable processes; Option pricing; Time-changed Brownian motion;

The motion of a viscous filament in a porous medium or Hele-Shaw cell: a physical realisation of the Cauchy-Riemann Equations

We consider the motion of a thin filament of viscous fluid in a Hele-Shaw cell. The appropriate thin film analysis and use of Lagrangian variables leads to the Cauchy-Riemann system in a surprisingly direct way. We illustrate the inherent ill-posedness of these equations in various contexts

On the pricing and hedging of volatility derivatives

We consider the pricing of a range of volatility derivatives, including volatility and variance swaps and swaptions. Under risk-neutral valuation we provide closed-form formulae for volatility-average and variance swaps for a variety of diffusion and jump-diffusion models for volatility. We describe a general partial differential equation framework for derivatives that have an extra dependence on an average of the volatility. We give approximate solutions of this equation for volatility products written on assets for which the volatility process fluctuates on a time-scale that is fast compared with the lifetime of the contracts, analysing both the ``outer'' region and, by matched asymptotic expansions, the ``inner'' boundary layer near expiry

Global existence, singular solutions, and ill-posedness for the Muskat problem

The Muskat, or Muskat--Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell under applied pressure gradients or fluid injection/extraction. In contrast to the Hele-Shaw problem (the one-phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem. For the stable, forward Muskat problem, in which the higher viscosity fluid expands into the lower viscosity fluid, we show global in time existence for initial data that is a small perturbation of a flat interface. The initial data in this result may contain weak (e.g., curvature) singularities. For the unstable, backward problem, in which the higher viscosity fluid contracts, we construct singular solutions that start off with smooth initial data, but develop a point of infinite curvature at finite time