Mathematical Finance primer

[DcifrThs]

Since i see there is a small spattering of people online at 3am eastern time, and of those, more would like to see a few MF posts than not, i'll start with a primer about asset pricing.

keeping it relatively simple (this stuff can get pretty dense), the black schoels merton equation is as follows:

exp{} is e^{}.

Delta is greek letter delta. usually stands for a process determined either randomly or held constant. mostly its the # of shares of stock held at any given time t.

r= money market account interest rate.

X(t) is an Ito process that is in and of itself driven by a brownian motion W(t). Properties of W(t) are first that it has a markov condition. that is the expectation of the future is independent of the past, so the best guess we have is the current point. i.e.

let F(t) be a filtration (sequential information) and 0<=s<=t, and W(t) be a standard brownian motion, then E[f(W(t) | F(s)]= f(W(s))

S(t) is a stock price at time t. S(t) is a random variable. S(t)>=0.

S(0) is the initial stock price. S(0) >=0.

Sigma is greek letter sigma which is usually the standard deviation or the diffusion for brownian motion.

Alpha is greek letter alpha. usually stands for the mean return or drift of brownian motion.

if we make Alpha and Sigma >0 constants then the standard brownian motion that defines a stock price is:

S(t)=S(0)exp{Sigma*W(t) + (Alpha - 1/2Sigma^2)t}

this is the asset price model used in BSM (black scholes merton)

this model accumulates quadratic variation at the sate of Sigma^2 per unit time. (normally, in calculus, since models are continuous and differentiable at all points, quadratic variation is 0 and can be ignored. however, since brownian motion has very abrupt jumps, it is not differentiable at all points. it is also way more volatile and the quadratic variation cannot be ignored)

NOW, the BSM model given the above is as follows:

Consider an agent who at each tim t has a portfolio valued at X(t) (note this is the Ito process from before driven by the above mentioned simple geometric brownian motion). This portfolio invests in a money market account and in a stock

dS(t)= Alpha*S(t)dt + Sigma*S(t)dW(t)

suppose at each time t the investor holds Delta(t) shares of stock (delta can be random but must be adapted to (determined by the same information) as the brownian motion that drives the stock). the remainder of the portfolio value, X(t)- Delta(t)S(t) is invested at interest rate r.

The differential dX(t) for the investor's portfolio value at each time t is due to two factors, the capital gain Delta(t)dS(t) on the stock position and the interest earnings r(X(t)-Delta(t)S(t))dt on the cash position.

so,

dX(t)=Delta(t)dS(t) + r(X(t) - Delta(t)S(t))dt
=Delta(t)*(Alpha*S(t)dt + Sigma*S(t)dW(t)) + r(X(t) - Delta(t)S(t))dt
=rX(t)dt + Delta(t)(Alpha - r)S(t)dt + Delta(t)Sigma*S(t)dW(t)

The three terms appearing in the last line of that equation can be interpreted as follows:

(i) an average underlying rate of return r on the portfolio
(ii) a risk premium Alpha - r for investing in the stock
(iii) a volatility term proportional to the size of the stock investment.

now in reality we need to consider the discounted (continuous) stock price (there is some ito-doeblin manipulations that need to be done that i'll skip to just show the final model):

d(exp{-rt}S(t))= (alpha-r)*exp{-rt}*S(t)dt + sigma*exp{-rt}*S(t)dW(t)

those of you interested in following quesitons and comments should bookmark this post as the primer for future discussion.

Barron

(NOTE: the above if from my notes from my reading of Shreve's book)

Barron