Some Financial Mathematics

The one-period rate of return of an asset at time t. where p t = the asset price at time t. Note: Also if there was continuous compounding during the period t – 1 to t at a rate r t then the value of the asset would be: r t = ln (1 + R t ) is called the log return.

Note: If we have an initial capital of A 0, an nominal interest rate of r with continuous compounding then the value of the capital at time T is:

The k-period rate of return of an asset at time t. again Also

Thus the k-period rate of return of an asset at time t. If there was continuous compounding during the period t – k to t at a rate r t (k) then the value of the asset would be at time t:

Thus the k-period continuous compounding rate of return of an asset at time t is: Taking t = k, let p 0 = the value that an asset is purchased at time t = 0. Then the value of the asset of time t is: If are independent identically distributed mean 0, then is a random walk.

Some distributional properties of R t and r t

Ref: Analysis of Financial Time Series Ruey S. Tsay

Conditional Heteroscedastic Models Models for asset price volatility

Volatility is an important factor in the trading of options (calls & puts) A European call option is an option to buy an asset at a fixed price (strike price) on a given date (expiration date) A European put option is an option to sell an asset at a fixed price (strike price) on a given date (expiration date) If you can exercise the option prior to the expiration date it is an American option.

Black-Scholes pricing formula c t = the cost of the call option, P t = the current price, K = the strike price, l = time to expiration r = the risk-free interest rate  t = the conditional standard deviation of the log return of the specified asset  (x) = the cumulative distribution function for the standard normal distribution

Conditional Heteroscedastic Models for log returns {r t } Let P t-1 denote the information available at time t – 1 (i.e. all linear functions of { …, r t-3, r t-2, r t-1 }) Let  t = E [r t | P t-1 ] and u t = r t –  t. Assume an “ARMA(p,q)” model, i.e. P t-1 ] = E[(r t –  t ) 2 | P t-1 ]

The conditional heteroscedastic models are concerned with the evolution of P t-1 ] = E[(r t –  t ) 2 | P t-1 ] = var[ u t | P t-1 ]

The ARCH(m) model Auto-regressive conditional heteroscedastic model where {z t } are independent identically distributed (iid) variables with mean zero variance 1.

The GARCH(m,s) model Generalized ARCH model where { z t } are independent identically distributed (iid) variables with mean zero variance 1.

Excel files illustrating ARCH and GARCH models CH models