Financial economics/Tutorials: Difference between revisions
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:::<math>Var(d_j)</math> is the variance of the jth factor | :::<math>Var(d_j)</math> is the variance of the jth factor | ||
==Black-Scholes== | ==Black-Scholes option pricing theorem== | ||
The fair price,P, of a call option on a security is given by: | |||
:<math> P = CN(d_1) - Xe^{-rt}N(d_2) \, </math> | :<math> P = CN(d_1) - Xe^{-rt}N(d_2) \, </math> | ||
:<math> | where: | ||
:C is the current price of the security; | |||
:<math>N(d_1)</math> is the cumulative probability distribution for the standard normal variate from -∞ to <math> d_1 </math>; | |||
:X is the exercise price (see ''options'' definition); | |||
:r is the risk-free interest rate; | |||
:t is the time to expiry of the option; | |||
:<math> d_1 </math> and <math> d_2 </math> are given by the equations: | |||
:<math> d_1 = \frac{\ln(C/X) + (r + \sigma^2/2)t}{\sigma\sqrt{t}} </math>; | |||
:<math> d_2 = d_1 - \sigma\sqrt{t}</math>; | |||
and | |||
:<math>\sigma</math> is the ''standard deviation (or volatility) of the price of the asset. | |||
The underlying assumptions are that: | |||
:* 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== | ==Gambler's ruin== |
Revision as of 03:47, 14 March 2008
The Capital Asset Pricing Model
The rate of return, r, from an equity asset is given by
- r = rf β(rm - rf)
'
where
rf is the risk-free rate of return
rm is the equity market rate of return
(and rm - rf 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
where
and
- is the weighting multiple for factor
- is the covariance between the return on the ith asset and the jth factor,
- is the variance of the jth factor
Black-Scholes option pricing theorem
The fair price,P, of a call option on a security is given by:
where:
- C is the current price of the security;
- is the cumulative probability distribution for the standard normal variate from -∞ to ;
- X is the exercise price (see options definition);
- r is the risk-free interest rate;
- t is the time to expiry of the option;
- and are given by the equations:
- ;
- ;
and
- is the standard deviation (or volatility) of the price of the asset.
The underlying assumptions are that:
- 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)