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In my textbook that I am self-studying from it is given that (assuming the Black-Scholes framework):

  1. $\Delta_{stock} = \partial S / \partial S = 1$
  2. All other Greeks for the underlying stock = 0

I can see why $\Gamma_{stock} = 0$, from taking the partial derivative of $\Delta_{stock}$.

But why is there not some significance to $\theta_{stock}$ $\rho_{stock}$, $\psi_{stock}$, etc. and why are those necessarily 0?

It would seem to me that it could be useful taking the partial derivative of the stock price with respect to the risk free rate, the continuously compounded return on the stock, and the variance of the stock.

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  • $\begingroup$ What is your model ?? $\endgroup$
    – user16651
    Jun 28, 2016 at 15:57
  • $\begingroup$ Sorry, forgot to say - Black-Scholes $\endgroup$
    – user2521987
    Jun 28, 2016 at 16:02
  • $\begingroup$ Right, but my textbook is implying that would define $\Delta_{stock} = \frac{\partial S}{\partial S} = 1$, which would make $\Gamma_{stock} = \frac{\partial S^2}{\partial^2 S} = 0$...but why couldn't we do the same thing for the partial derivative of the stock with respect to $r, \delta, \sigma$, etc? $\endgroup$
    – user2521987
    Jun 28, 2016 at 16:09
  • $\begingroup$ What is your textbook and in which section? $\endgroup$
    – Gordon
    Jun 28, 2016 at 17:46
  • $\begingroup$ Ask yourself this: you have an option position with a Theta of 3.14. How many shares of the stock do you need to buy (or sell) to bring the Theta to the value 2.718? You can't, because the option is going to expire regardless and holding some stock is not going to change that. That's why we say that a stock has no Theta (or zero Theta, if you prefer). The Theta comes from the fact tat "the clock is ticking" on the option. $\endgroup$
    – Alex C
    Jun 29, 2016 at 1:07

1 Answer 1

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In the black scholes model, today's stock price, risk free rate and stock volatility are considered independent variables. They are inputs to the model. Hence the cross partial derivatives are zero.

You could invent a model where you tried to explain the current stock price in terms of risk free rate and volatility. Then indeed the partial derivatives would not have to be zero.

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