Having two degrees in mathematics and having taught university math (calculus, linear algebra, differential equations, topology, combinatorics, abstract algebra, and so on) for twenty years, I’m pretty sure I already knew that. (Yup. I checked: I did already know that.)
Now . . . here’s the killer: stock prices aren’t constant.
When a stock’s price (S) increases by $1, then . . . wait for it . . . the stock’s price increases by . . . you guessed it! . . . $1. So the derivative of the stock’s price with respect to the stock’s price – dS/dS – is 1.
I’m not trying to be a smart-aleck here . . . well, only a little; that relationship is actually important enough to recognize explicitly. When you’re hedging a portfolio it’s good to know how the price of everything in your portfolio will change when one variable (or, perhaps, more than one variable: partial derivatives, anyone?) changes. clever and I were just pointing out one of the everythings.
(I’m not challenging anyone’s intelligence either, Aether: you’re obviously quite sharp. Just having a little fun.)
Point taken, however, I inferred that you were talking about the derivative of the stock price outright.
Overall, you still have to think of it terms of real-life trading and discrete vs. continuous variables. For me, majority of the stock valuation models are discrete in nature. Options valuation takes us into continuous variables, which is where the above argument comes into play. Maybe I’m missing something, but you can point me to continuous stock valuation models that define the delta, gamma, vega, and other greeks of stock prices - AND that are actually used in practice? Good luck finding 'em in the CFAI books :).
And I completely understand what you’re trying to suggest with regards to portfolio hedging… I didn’t have any arguments on that topic to begin with. Even with portfolios though, only thing I’ll say is that majortiy of the theory/practice out there makes simplified assumptions about the time continuum and how things are normally distributed.
I think you’re too hung up on models Aether. You don’t need a model or a quant to tell you that the delta of a stock is 1.
At the end of day we can sit here and argue all day long about this. From my 11 years of working in derivatives in institutional S&T, market-making prop shops, and a funds management context in England and Australia, the delta of the underlying has always been 1 as far as can remember. Maybe in America it’s different, who knows.
I do agree with you though that if you tell a cash trader about stock having delta, he may look at you like you’re an alien. Cash traders and Option traders are two different breeds of people with different lingo.