Financial economics/Tutorials

Financial models

The Capital Asset Pricing Model

The rate of return, r,  from an equity asset is given by

$r=r_{f}\beta (r_{m}-r_{f})$

where

r_f  is the risk-free rate of return

r_m  is the equity market rate of return

(and r_m - r_f is known as the equity risk premium)

and β is the covariance of the asset's return with market's return divided by the variance of the market's return.

(for a proof of this theorem see David Blake Financial Market Analysis page 297 McGraw Hill 1990)

The Arbitrage Pricing Model

The rate of return on the ith asset in a portfolio of n assets, subject to the influences of factors j=1 to k is given by

${r_{i}}={r_{f}}+\sum _{j=1}^{k}{y_{j}}{\beta _{ij}}$

where

${\beta _{ij}}={\frac {Cov(r_{i},d_{j})}{Var(d_{j})}}$

and

$y_{j}$ is the weighting multiple for factor $j$

$Cov(r_{i},d_{j})$ is the covariance between the return on the ith asset and the jth factor,

$Var(d_{j})$ is the variance of the jth factor

Black-Scholes option pricing model

The fair price,P, of a call option on a security is given by:

$P=CN(d_{1})-Xe^{-rt}N(d_{2})\,$

where:

C is the current price of the security;

$N(d_{1})$ is the cumulative probability distribution for the standard normal variate from -∞ to $d_{1}$;

X is the exercise price (see options definition);

r is the risk-free interest rate;

t is the time to expiry of the option;

$d_{1}$ and $d_{2}$ are given by the equations:

$d_{1}={\frac {\ln(C/X)+(r+\sigma ^{2}/2)t}{\sigma {\sqrt {t}}}}$;

$d_{2}=d_{1}-\sigma {\sqrt {t}}$;

and

$\sigma$ is the standard deviation (or volatility) of the price of the asset.

The first expression, $CN(d_{1})$,   of the equation is the expected benefit from acquiring a stock outright, obtained by multiplying the asset price by the change in the call premium with respect to a change in the underlying asset price. The second expression,  $Xe^{-rt}N(d_{2})$, is the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts

The underlying assumptions include:

• Dividend payments are not included;
• Options cannot be exercise before the stipulated date;
• Markets are efficient;
• No commissions are paid;
• Volatility is constant;
• The interest rate is constant; and,
• Returns are log-normally distributed.

Gambler's ruin

If q is the risk of losing one throw in a win-or-lose winner-takes-all game in which an amount c is repeatedly staked, and k is the amount with which the gambler starts, then the risk, r, of losing it all is given by:

r  =  (q/p)^(k/c)

where p  =  (1 - q),  and q  ≠  1/2

(for a fuller exposition, see Miller & Starr Executive Decisions and Operations Research Chapter 12, Prentice Hall 1960)