Now I want to discuss the concept of self financing, and

in particular self financing trading strategies.

Before we do that, we just need to define what we mean by the value process,

Vt of theta, that is associated with the trading strategy theta t equals xt, yt.

We define Vt to be equal to xtSt plus ytBt for t greater than or

equal to 1, and of course, this is just the value of the portfolio time t.

Xt, xt's the number of units of the stock that we hold, so

xtSt is our stock position, and ytBt is our position in the cash account,

our value of our cash account position, so this is the total value of the portfolio.

We need a slightly different def, definition of t equal to 0 and

that's because we don't have an x0, y0.

If you recall,

xt is the number of units of shares that we purchased at time t minus 1.

So, if we had an x0, we would be referring to time minus 1, but

we don't have a time minus 1.

So we have a slightly different definition for t equal to 0, but

we still end up with the value of the strategy times 0.

So when t equal 0, we get Vt equals x1 times s0 plus y1 times b0.

But of course x1 has chosen at times 0, and y1 has chosen at times 0.

So if you like,

this is the value of the portfolio immediately after trading at times 0.

Okay, so that's the value process.

We now want to define what we mean by a self-financing trading strategy.

A self-financing trading strategy is a trading

strategy where changes in Vt are due entirely to trading gains or losses,

rather than the addition or withdrawal of cash funds.

In particular, a self-financing trading strategy satisfies

Vt equals xt plus 1 st plus yt plus 1 Bt.

And if you notice, Vt is therefore the value of the portfolio immediately

after trading at time t, because xt plus 1 is the number of units of shared

help between times t and t plus 1 and yt plus 1 is the number of the units of

the cash account held between times t and t plus 1.

So this is the value of the portfolio immediately after trading at time t.

Up here, this is the value of the portfolio immediately before trading

at time t.

We now have the following proposition.

If a trading strategy, theta t is self-financing

then the corresponding value process Vt satisfies this relationship here.

So what's going on on this relationship here?

Well the left hand side is just the change in value of the portfolio,

and on the right hand side we have xt plus 1 which is the number of units

of the share held between times t and t plus 1 times the gain on the stock.