Why Black–Scholes Stock Greeks Except Delta Are Zero

Summary

The document explains why the underlying stock has delta equal to one and other Greeks equal to zero in the standard Black–Scholes setup. The key point is that the stock price is treated as an input variable independent of the model inputs such as the risk-free rate and volatility. Differentiating the stock price with respect to those separate inputs therefore gives zero, while differentiating it with respect to itself gives one; its delta does not change with the stock price, so its gamma is zero.

The answer distinguishes this modeling convention from a claim about real-world economic relationships. A different model could explain the current stock price using rates or volatility, in which case those sensitivities need not be zero. The discussion is conceptual and does not derive the full set of Greeks or address options, where sensitivities to time and market parameters are generally relevant. Its conclusion applies to the underlying asset as represented in the stated framework.

Key ideas

• In the stated framework, the stock price is an input independent of the risk-free rate and volatility.
• The stock’s derivative with respect to itself is one, giving it unit delta.
• The stock’s gamma is zero because its delta does not vary with the stock price.
• Other partial derivatives are zero under the model’s independent-input assumption.
• A model that explains stock price using rates or volatility could produce different sensitivities.

Tags

# Why are the greeks for the underlying stock 0 with the exception of delta?


# Why are the greeks for the underlying stock 0 with the exception of delta?












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

- $\Delta_{stock} = \partial S / \partial S = 1$

- 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.

## Answer by dm63 (score 4, accepted)

https://quant.stackexchange.com/a/27845

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.