Financial Derivatives and Partial Differential Equation

Financial Derivatives and Partial Differential Equation
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Almgren (2002) focuses on the application of partial differential equations to determine the value of financial derivatives. An asset provides claim to future cash flows, while a derivative is a financial asset or security whose values are derived from or depends on the value of another asset. The author begins by addressing how the price fluctuation of stocks, which is one form of asset, can be evaluated using statistics, and he extends this concept to valuing derivatives such as options using a partial differential equation. Statistics is also useful to determine the expected size of the price motions or fluctuations. This is a summary of the application of partial differential equation on financial derivatives, specifically, assets and derivatives, European option valuation and the American and exotic options.
Assets and derivatives
Differential equations have been used in finance to value derivatives and one of the most widely used approaches using these equations is the Black-Scholes option pricing model. The model consists of three equations where the value of an option depends on factors such as the stock price, the option’s time to expiration, the price fluctuations, strike price and the standard, the risk-free rate. This approach is similar to Louis Bachelier‘s 1900 model where he valued options using the expected value of the payoffs and a probabilistic model. From the two models researchers have focused on how to use partial differential equations to determine the expected payoffs considering price fluctuations (Almgren, 2002).
The intrinsic value of a call option is the difference between the stock price/or underlying price (S) and the strike price (exercise price), (K) and the Black Scholes adds other variables and uses partial differentiation equations. The author also focuses on the application of the Itô’s Lemmain the Black Scholes model in European options, where calculates the derivative of a time-dependent function of a stochastic process (Almgren, 2002). The author also considers the change in the between two portfolios as Δ (V- π) where are the portfolios where V is the value of the option changes (Almgren, 2002)
European option valuation
* For the simplest derivative is a forward or futures contract where the future expiration date T, for a specified strike price K where the market place price S.
* The holder makes a profit/ loss S-K
For European option
* Market at price S (t)
* Payoff function Λ (S)
* Expiration t = T.
* For t < T, the option value V depends on the underlying price S and on time t, such that price as V(S, t)
* Λ (S) is a payoff function that determines the value of an option at expiration t
* V(S, T)= Λ (S)