P - T

A differential equation which has a function of multiple variables when all variables with the exception of interest fixed during the differentiation. It is essentially a differential equation which expresses one or more quantities in terms of partial derivatives.

Because of the difficulty in solving PDEs, numerous methods exist to solve them, including Bäcklund Transformations, Green's Function, Integral Transformations, or numerical methods such as finite difference methods.

> Detailed description of PDEs can be found here

A probability distribution.

Similar to standard Monte Carlo simulation methods, with the exception of 1 aspect. Monte Carlo methods utilize psuedo-random numbers to generate the output, whereas, Quasi - Monte Carlo utilizes quasi-random number sequences, also referred to as low discrepancy sequences in order to generate the output.

To be familiar with Quasi - Monte Carlo simulation, we must first be aware of the purposes put forward by the Monte Carlo method and it's primary uses. For our purposes, we focus on simulating the outcome of - for example a European option by using random numbers to portray the outcome of our variables (namely volatility, interest rate, etc.). If we attempt to calculate the European option price by using only the estimated variable factors, the result is likely to be flawed due to a number of reasons discussed in the Monte Carlo section.

The problems with Monte Carlo simulation are several-fold, and to overcome this, analysts can employ the use of low discrepancy sequences opposed to psuedo random number sequences to generate a substantially more accurate simulation with high numbers of iterations.

We make a comparison of the pricing methods in our spreadsheets section.

Radon-Nikodym (Derivative)

A somewhat abstract term, random walk in finance is commonly used to describe the movement of stock prices. Stock prices are said to follow a random walk which can be represented by what is called Brownian motion, a science term commonly referred to when describing atomic movements. Stock prices take on similar movements in that they are 'random', and this essentially forms the foundation of option pricing models through the modeling of stock price movements.

The Greek letter Rho can represent a number of things within mathematics, but in the context of derivatives, it commonly refers to the sensitivity of the option value to interest rates. Rho of 10.5 means that for every 1% increase in interest rates, there is a corresponding 10.5% increase in the option value.

The Rho for a call and put option on a dividend paying stock are respectively given as:

Call:

Put:

Where S is the asset price, D is the dividend yield, T is the time to maturity and N(x) is the cumulative normal distribution function of d.

In calculus, the Riemann Integral is the standard integral used.

Defined as the degree of asymmetry of a given distribution. Skewness comes in two forms; negative skewness, which is when it has a long negative tail, and also positive skewness, where there is a long positive tail.

A common term used to describe the deviation from the mean of a population sample.

Consider a portfolio of 20 stocks with an average return over 1 year of 20% and a standard deviation of 10%. This is useful as a representation of how much the stock returns "deviate" from the mean. The SD of 10% does not mean that the expected returns of the stocks is going to be 20% +/- 10%, but if you consider a normal distribution, a SD of +/- 1 deviations mean that 68% the stock returns lie between 10% and 30%.

If we consider a function f(x) which has continuous derivatives up to the (n+1)th order, then the function can be expanded to the following:

The term is given as:

Which can be simplified to:

Where:

This is illustrated as a Taylor series as x converges bringing = 0 as n tends towards infinity. The above is said to be a Taylor Series of the function f(x) about a.

Taylor series can be applied to a finance context, as finite Taylor approximations can be applied to calculating values of option greeks as well as application of the Taylor series to the Black-Scholes PDE.

If we look at the Taylor expansion for a vanilla call option:

Where C is the call option value, S is the stock price, r is the risk free rate, y is the dividend yield, is the volatility of the stock and T is the Time to maturity.

What does the above PDE actually show? If you haven't seen it already, it represents the greeks. Look at the first term:

What is that? None other than the delta of the call option. Looking along the line of polynomials, we see that all the greeks are represented.

From this, it seems clear that Taylor expansions are vital to derivatives option pricing.

One of the greeks of option pricing, it is considered to be the change in the option value with respect to the change in time. For standard European options under the Black-Scholes framework, the Theta can be derived as:

For a European call option with dividends:

Where D is the dividend yield, T is the time to maturity, and N(x) is the cumulative normal distribution for the terms and which are defined within the Black-Scholes formula. For stocks which do not pay any dividends, the formula simplifies to:

Similarly for a European put option with dividends:

And without dividends:

There will be a more in depth look at the theta function in the options section shortly.