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The Normal distribution is part of the parametric derivation of the Black-Scholes pricing model [1]. It really does seem to be part of the model's key assumpti

The Normal distribution is part of the parametric derivation of the Black-Scholes pricing model [1]. It really does seem to be part of the model's key assumptions. Can you expand on how the Volatility Smile demonstrates that traders model fat tails in their options pricing strategies. It wasn't clear to me from the wiki page you linked. [1] http://en.wikipedia.org/wiki/Black-Scholes#Notation http://en.wikipedia.org/wiki/Black-Scholes#Notation

tedunangst 14y ago
If you calculate the implied volatility (volatility is not readily observable) from the option prices at different strikes, you will get different values. Ergo, the traders are not simply plugging a single volatility value into the formula for pricing. BS does assume a normal distribution; the point is that people do not strictly take the output from BS and say "tada, here's the price."
omonra 14y ago
Sure, I'll try. Here is an example: Assume that you want to buy an option on Apple stock - 3 months in the future. Assume that price today is $500. If you want to buy an option with strike=500, you will pay 50 vol (BS as a formula will just convert the price in vol terms into a dollar cash price). But - if you want to buy an option struck at 400 - you might pay 75 vol. And if you want a 300 strike - it will be 100 vol. As you see, volatility will RISE the further you move away from spot price (ie where it is today). So while BS calculation assumes constant volatility to convert vol to dollars, traders always ask for higher vol, as you move out to less likely scenarios. That is how they deal with the world being non-normal. It really is options 101 - so any writer who talks about BS assuming Gaussian probabilities is either lying or doesn't know anything about the subject.
- jfager 14y ago
  The most common derivation of Black-Scholes absolutely does assume a standard normal distribution. There are other models that don't (i.e., Mandelbrot tried to get people to use Levy distributions, and a lot of newer models are built around martingale theory), and traders have adopted rules-of-thumb that pull prices out of line with what BS says they should be, but it's certainly not the case that anyone who speaks of BS assuming Gaussians is lying or speaking from ignorance.
  - omonra 14y ago
    Uggh, no. Let me try and explain. BS is not used as a model that explains how the world works. It's just a formula to convert volatility to a dollar price of an option. But the actual vol surface that people use is anything but Gaussian. So the job of the trader is to construct a vol surface. However in order for 2 people to trade an option, they have to agree to a cash price - which is when BS is used. But the entire trading universe revolves around vol surfaces. Think of it as making a painting. BS is just a single paint pigment - whereas the painter will take 20 pigments and mix them up to produce millions of colours. So to accuse those who use BS formula of assuming Gaussian probability distribution is akin to saying that painters use one color to paint.

tedunangst 14y ago

If you calculate the implied volatility (volatility is not readily observable) from the option prices at different strikes, you will get different values. Ergo, the traders are not simply plugging a single volatility value into the formula for pricing. BS does assume a normal distribution; the point is that people do not strictly take the output from BS and say "tada, here's the price."

omonra 14y ago

Sure, I'll try. Here is an example: Assume that you want to buy an option on Apple stock - 3 months in the future. Assume that price today is $500. If you want to buy an option with strike=500, you will pay 50 vol (BS as a formula will just convert the price in vol terms into a dollar cash price). But - if you want to buy an option struck at 400 - you might pay 75 vol. And if you want a 300 strike - it will be 100 vol. As you see, volatility will RISE the further you move away from spot price (ie where it is today). So while BS calculation assumes constant volatility to convert vol to dollars, traders always ask for higher vol, as you move out to less likely scenarios. That is how they deal with the world being non-normal. It really is options 101 - so any writer who talks about BS assuming Gaussian probabilities is either lying or doesn't know anything about the subject.

jfager 14y ago
The most common derivation of Black-Scholes absolutely does assume a standard normal distribution. There are other models that don't (i.e., Mandelbrot tried to get people to use Levy distributions, and a lot of newer models are built around martingale theory), and traders have adopted rules-of-thumb that pull prices out of line with what BS says they should be, but it's certainly not the case that anyone who speaks of BS assuming Gaussians is lying or speaking from ignorance.
- omonra 14y ago
  Uggh, no. Let me try and explain. BS is not used as a model that explains how the world works. It's just a formula to convert volatility to a dollar price of an option. But the actual vol surface that people use is anything but Gaussian. So the job of the trader is to construct a vol surface. However in order for 2 people to trade an option, they have to agree to a cash price - which is when BS is used. But the entire trading universe revolves around vol surfaces. Think of it as making a painting. BS is just a single paint pigment - whereas the painter will take 20 pigments and mix them up to produce millions of colours. So to accuse those who use BS formula of assuming Gaussian probability distribution is akin to saying that painters use one color to paint.