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Call options are more valuable if the underlying security is more volatile (because there's higher chance of ending in the money) Whatever the volatility is

by camjw 7y ago

Call options are more valuable if the underlying security is more volatile (because there's higher chance of ending in the money) Whatever the volatility is you still (in Black-Scholes) have a 50/50 of the option ending ITM (stock returns are normally distributed, higher volatility just means higher std. deviation). One reason why ATM options are more valuable with higher volatility is because there's a greater chance of the option ending far away from the strike price (either in the positive OR negative direction), so there's a lot of time value on those. This doesn't discount the rest of your post though, clear that VCs go for high risk options.

H8crilA 7y ago
Oh yes, you're completely right. I oversimplified :) And, to nitpick back: the logarithm of stock return is normally distributed under B-S.
- camjw 7y ago
  Sorry to be that guy again but B-S assumes stock prices are log-normally distributed but stock returns are normally distributed :) [0] [0] https://www.investopedia.com/terms/b/blackscholes.asp https://www.investopedia.com/terms/b/blackscholes.asp
  - H8crilA 7y ago
    It's all fine :). I thought that "stock return" is the [exit price]/[entry price], for an asset that does not pay dividends, no? exit/entry still requires a log() to be normally distributed, for example exit/entry is non-negative, wile gaussian is of course sometimes negative, no matter what the mean is.
    - camjw 7y ago
      I think your formula is slightly off. Stock return = (exit_price - entry_price + dividends) / entry_price. It’s clear that then the mean return is the dividends paid and can be negative if the exit price is sufficiently low. I think by a bit of squinting (using the central limit theorem) you can say that this should be normally distributed as long as entry_price and exit_price have the same distribution
      - H8crilA 7y ago
        I still think it should be log()ed explicitly. Coming back to options world, entry_price is a constant when opening the contract, let's ignore dividends, the formula is (exit_price - entry_price) / entry_price = exit_price/entry_price - 1 = exit_price/constant - 1. This is normally distributed if, and only if exit_price is normally distributed. You'd want to to add back the 1, log() it, add back the log(constant) to cancel it out and just work on the log(exit_price) normally distributed random variable. Stock return really is not normally distributed. Log(stock return) is normally distributed (under B-S, it's an assumption after all). Stock return is log-normally distributed. Multiply by 1/entry_price and subtract 1 to get your version of stock returns.

by camjw 7y ago

H8crilA 7y ago

Oh yes, you're completely right. I oversimplified :) And, to nitpick back: the logarithm of stock return is normally distributed under B-S.

camjw 7y ago
Sorry to be that guy again but B-S assumes stock prices are log-normally distributed but stock returns are normally distributed :) [0] [0] https://www.investopedia.com/terms/b/blackscholes.asp https://www.investopedia.com/terms/b/blackscholes.asp
- H8crilA 7y ago
  It's all fine :). I thought that "stock return" is the [exit price]/[entry price], for an asset that does not pay dividends, no? exit/entry still requires a log() to be normally distributed, for example exit/entry is non-negative, wile gaussian is of course sometimes negative, no matter what the mean is.
  - camjw 7y ago
    I think your formula is slightly off. Stock return = (exit_price - entry_price + dividends) / entry_price. It’s clear that then the mean return is the dividends paid and can be negative if the exit price is sufficiently low. I think by a bit of squinting (using the central limit theorem) you can say that this should be normally distributed as long as entry_price and exit_price have the same distribution
    - H8crilA 7y ago
      I still think it should be log()ed explicitly. Coming back to options world, entry_price is a constant when opening the contract, let's ignore dividends, the formula is (exit_price - entry_price) / entry_price = exit_price/entry_price - 1 = exit_price/constant - 1. This is normally distributed if, and only if exit_price is normally distributed. You'd want to to add back the 1, log() it, add back the log(constant) to cancel it out and just work on the log(exit_price) normally distributed random variable. Stock return really is not normally distributed. Log(stock return) is normally distributed (under B-S, it's an assumption after all). Stock return is log-normally distributed. Multiply by 1/entry_price and subtract 1 to get your version of stock returns.