# What exactly is the profit and loss of a portfolio?

When we 'delta-hedge', we make the value of a portfolio 0. Now, we know that an approximation of the change in a portfolios value is given by

THETA * dt + 0.5*GAMMA*S^2

where THETA and GAMMA are the greeks of the portfolio.

Now, here's my problem: everybody knows how to calculate the greeks of a derivative. Just differentiate with respect to some variable.

But, what is the derivative of a portfolio? Because that up there is a portfolio, not a derivative.

For simplicity, we might imagine a portfolio that has holdings in .... a call .... a stock .... and a bank account (to borrow and lend money).

What are the Gamma, Delta, Theta, Vega of such a portfolio?

• You might get better response to this question on quant.stackexchange.com; it's quite a bit more technical than most questions asked here. – Grade 'Eh' Bacon Mar 9 '17 at 20:07
• @Grade'Eh'Bacon - exactly. I've invested for 38 years, including profitable options trades over the long term, and I've managed to avoid learning about the greeks. (Aside from alpha and beta, of course) – JTP - Apologise to Monica Mar 9 '17 at 20:17

When we 'delta-hedge', we make the value of a portfolio 0.

No - you make the risk relative to some underlying 0. The portfolio does have a value, but if whatever underlying you're hedging against changes slightly the value of your portfolio should not change.

But, what is the derivative of a portfolio?

It's the instantaneous rate of change of the portfolio relative to some underlying phenomenon. With a portfolio of many stocks, there's not one single factor that drives the value of your portfolio. You have sensitivity to each underlying stock (price and volatility), interest rates, the market as a whole, etc.

For simplicity, we might imagine a portfolio that has holdings in .... a call .... a stock .... and a bank account (to borrow and lend money).

You will have a delta relative to the stock and a delta relative to the underlying instrument on the option, etc. Those can only be aggregated for each factor (e.g. if the call is an option on the same stock)

Theta is the only one you can calculate for the portfolio as a whole - it will be the aggregate theta of all of your positions (since change in time is constant across all investments). All of the others are not aggregatable since they are measuring sensitivities to different phenomena.