Questions about martingales and measures

[egondragon]

Hello

I am reading Hull chapter concerning Martingales.
I have some difficulty to understand martingale concept :
Let say i have a variable theta, that follow a process :
dtheta/theta = m * dt + s * dz

Theta is my underlying
From this i build 2 derivatives f1 and f2

We can build a riskless portfolio by doing pi = sigma2 * f2 * f1 - sigma1 * f1 * f2
For me sigma is the standard deviation, how can i buy sigma * something ?

In hull calculation, delta pi = (mu1 * sigma2 * f1 * f2 - mu2 * sigma1 * f1 * f2) * delta_t

How is this delta_t obtained ? how is this formula obtained ?

A martingale is defined as a stochastic process with zero drift
What is the signification of the drift here ?
I mean what will be the exact materialization of a drift in the real world for let's say a future ?

Another point that does not sound clear at all for me :
df/f = mu * dt + sum(sigma_i * dz_i) for i ={0, 1, ..., n}
mu = trend
sigma = drift

A variable following a martingale means that its expected value in n days is the same as today

I don't see the relation between this affirmation and the Hull formula's.

Thanks for any help
Regards

 

[DStahl]

First note that Hull does not do anywhere near a mathematically rigorous description of martingales.

You can buy sigma of something because sigma is simply a number. If sigma = .5, then I can buy .5 of a stock, for instance. In Hull's book, you buy sigma of assets in order to eliminate the delta z term so that there is no risk in the portfolio. From there you can derive that the sharpe ratio (u-r)/sigma must be equal across different maturities.

The essence for Hull's description of market risk is to provide a reason for changing the measure of the brownian motion such that the drift is the risk free rate, and not the actual drift of the asset. Under this measure, any discounted derivative security is a martingale (ie, e^{-rt}f(S, t) is a martingale).