One of our trading friends asked me a very good question a while back. The question was related to POS (delta) and risk. “If one chooses the same delta, say .20, (80% POS), for two different stocks, is that not the same level of risk? Say you choose a strike on TSLA at .20 delta a week out, then do the same with IWM. The TSLA strike, since TSLA has much more volatility, is many strikes away from the money, while the IWM strike is only a few. So isn’t the risk of assignment the same? And since the TSLA choice pays more in premium as a percentage of stock price, why aren’t we selling options on high-volatility stocks instead of IWM?”
That’s a great question and I didn’t know.
So I started asking people who might know, (thanks again John!) and reading a little. No, reading a LOT. Finally I think I understand what I read and what John told me enough now to actually formulate an answer. Much fun! Here goes:
The explanation, which I will keep as simple as I can:
Remember the greeks? Not the ones in history class who kicked the Persian’s asses in 480 BC but the ones in option trading. Delta, gamma, vega, rho? The simple answer is based on the idea that delta is a first order variable while gamma is a second order variable. Gamma influences the rate of change of delta, or in other words, the rate of change of the rate of change. We can pick delta, but not gamma, a part of the Black-Sholes option equation. And it turns out different stocks have different gammas that then influence their delta. Therefore, when we select a .20 delta strike for one stock, it is not the same thing as .20 delta with another stock. That option value can and will change at a different rate even when both start out the same. Because, gamma.
Delta, as we know, measures the change in option price with regard to the change in share price. However, delta only maps the first order change. There is still a lot of the change in the option price that isn’t captured by delta. This is where the second order change comes in i.e., gamma. Now if you think about it, you can take it a step further and calculate the third order, fourth order, etc. But the first and second order (delta and gamma) greeks generally capture a majority of the price change such that the other orders are all negligible.
For example: Look again at both TSLA and IWM puts, one week out, both at .20 delta or 80% POS. The gamma for IWM is currently .047, while the gamma for TSLA is .0058. That’s an entire magnitude of difference. And these are inverse, in the same way that delta and POS are inverse of each other, so TSLA’s gamma is a magnitude larger than that of IWM. That’s why TSLA pays more: there is more volatility when you factor in both delta and gamma.
So. The reason a .20 delta strike is not the same for two stocks is because gamma is different for different stocks, which then influences the rate of change of delta as time passes. So yes, there is a difference in risk between two stocks, even at the same delta. John confirms this.
Check out the attached chart. Imagine two charts with different curves of the blue line. That in turn would change the shape of the red line, the one we are familiar with.
That’s not necessarily a bad thing, we still have to look at all the other parameters we look at and make a decision, but it does explain why selling options on a high-volatility stock carries more risk than on a low volatility stock, even when you pick the same delta.
“The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return.”
