If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:10:24

we're now going to talk about probably the most famous formula in all of finance and that's the black Scholes formula sometimes called the black Scholes Merton formula and it's named after these gentlemen this right over here is Fischer black this is Myron Scholes and they really laid the foundation for what led to the black Scholes model and black Scholes formula and that's why it has their name and this is Bob Merton who really took what black Scholes did and took it to another level to really get to our modern interpretations of the black Scholes model and the black Scholes formula and all three of these gentlemen would have won the Nobel Prize in Economics except for the unfortunate fact that Fischer black passed away before the award was given but Myron Scholes and Bob Merton did get the Nobel Prize for the work and the reason why this is such a big deal why it was his Nobel prize-worthy and actually there's many reasons I could do a whole series of videos on that is that people had been trading stock option or they've been trading options for a very very very very long time they had been trading them they had been buying them they had been selling them it was a major part of financial markets already but there was no really good way of putting our mathematical minds around around how to value an option people had a sense of the things that they cared about and I would assume especially options traders had a sense of the things that they cared about when they were trading options but we really didn't have an analytical framework for it and that's what the black Scholes formula gave us but let's just before we actually dive into this seemingly hairy formula but the more we talk about it hopefully it'll start to see him a lot friendlier than it looks right now let's start to get an intuition for the things that we would care about if we were thinking about the price of a stock option so you would care about the stock price you would care about the exercise price you would actually especially care about how much higher or lower the stock price is relative to the exercise price you would care about the risk-free interest rate the risk-free interest rate keep showing up when we think about taking a present value of something if we want to discount the value of something back to today you would of course think about well how long do I have to actually exercise this option and then finally this might look a little bit bizarre at first but we'll talk about in a second you would care about how volatile that stock is and we measure volatility as the standard deviation of log returns for that security so that's it seems very fancy and we'll talk about that in more depth in future videos but at just a intuitive level just think about two stocks so let's say that this is stock one right over here and it jumps around and I'll make them go flat just so we make no judgment about whether it's a good investment or so you have one stock that kind of does that and then you have another stock so let me actually draw them on the same so let's say that is stock one and then you have stock two that does this that does this it jumps around all over the place so this green one right over here is stock two so you could imagine stock two just in the way we use the word volatile it is more volatile it's a Wilder ride also if you were looking at how dispersed the returns are away from their mean you see it has it has the the returns have more dispersion it'll have a higher standard deviation so stock two will have a higher will have a higher volatility or a higher standard deviation of logarithmic returns and in a future video we'll talk about why we care about log returns and stock one would have a lower volatility would have a lower volatility so you can imagine options are more valuable when you're dealing with or if you're dealing with the stock that has higher volatility that has higher signal like this this feels like it would drive the value of an option up you would rather have an option when you have something like this because look if you're owning the stock man you have to go off this is a wild ride but if you have the option you don't you can ignore the wildness and then it might actually mate and then you can you can you can exercise the option if it seems like the right time to do it so it feels like if you're just trading it that while the most the more volatile something is the more valuable and option would be on that so with now that we've talked about this let's actually look at the black Scholes formula and the variety that I have right over here this is for a European call option European call option we could do something very similar for a European put option so this right over here is a European call option and remember European call option it's it's mathematically simpler than an American call option and that there's only one time which you can exercise it on the exercise date on an American call option you can exercise it at any point but with that said let's try to at least intuitively dissect the black-scholes for me a little bit so the first thing you have here you have this term that in video involves the current stock price and then you're multiplying it times this function that's taking this as an input and this is how we define that input and then you have minus the exercise price discounted back this discounts back the exercise price times that function again and now the input is slightly different into that function and just so that we have a little bit of background about what this function n is n is the cumulative distribution function for a standard normal distribution I know that seems might seem a little bit daunting but you could look at the statistics playlist and it shouldn't be that bad this is essentially saying for a standard normal distribution the probability that your random variable is less than or equal to X and another way of thinking about that if that sounds a little and it's all explained in arched in our statistics playlist if that was confusing but if you want to think about a little bit mathematically you also know that this is going to be it's a probability it's always going to be greater than 0 and it is going to be less than 1 so with that out of the way let's think about what these pieces are telling us so this right over here is dealing with the it's the current stock of the price it's car it's the current stock price and it's being weighted by some type of a some type of a a probability and so this is essentially one way of thinking about in very rough terms is this is what you're going to get you're going to get the stock and it's kind of being weighted by the probability that you're actually going to do this thing and I'm speaking in very rough terms and then this term right over here is what you pay this is what you pay this is your exercise price discounted back somewhat being weighted and I'm speak what's in I'm hand-waving a lot of the mathematics but look what what's like are we actually going to do this thing are we actually going to exercise our option so that makes sense right over there and it makes sense if the stock price is worth a lot more than the exercise price and if we're definitely going to do this let's say that d1 and d2 were very very large numbers we're definitely going to do this at some point in time then it makes sense that the value of the call option would be the value of the would be the value of the stock minus the exercise price discounted back to today so this right over here this is the discounting kind of giving us the present value of the Esk exercise price and we have videos on discounting and present value if you find that a little bit daunting and it also makes sense that the more the higher the stock prices so we see that right over here relative to the exercise price the more that the option would be worth but it also makes sense that the higher the stock price relative to the exercise price the more likely that we will actually exercise the option and you see that in both of these terms right over here you have the ratio of the stock price to the exercise price ratio of the stock price to the exercise price we're taking the natural log of it but the higher this ratio is the larger d1 or d2 is so that means the larger the input into the cumulative distribution function is which means the higher probabilities we're going to get and so it's it's a it's a higher chance we're going to exercise this price and it makes sense that then this is actually going to have some value so that makes sense the relationship between the stock price and the exercise price the other thing I will focus on because this tends to be a deep focus of people who operate with options is the volatility so we already had an intuition that the higher the volatility the higher the option price so let's see where this factors into this equation here so we don't see it at this first level but it definitely factors into d1 and d2 so in d1 the higher your standard deviation of log nor of your log returns so the higher sigma we have a sigma in the numerator and the denominator but in the numerator we're squaring it so the higher sigma will make d1 go up so Sigma Sigma goes up D one will go up let's think about what's happening here well here we have a Sigma it's still squared it's in the numerator but we're subtracting it it's growing boy this is going to grow faster than this but we're subtracting it now so for D to a higher Sigma is going to make D to go down because we are subtracting it so will this actually make can we actually say that this is going to make a higher Sigma is going to make the value of our call option higher well let's look at it if the value of our Sigma goes up then d1 will go up then this input this input goes up if that input goes up our cumulative distribution function of that input is going to go up and so this term this whole term is going to drive this whole term up now what's going to happen here well if d2 goes down then our cumulative distribution function evaluated there is going to go down and so this whole thing is going to be lower and so we're going to have to pay less so if we get more and pay less than I'm speaking in very hand wavy terms but this is just to understand this isn't as intuitively daunting as you might think but it looks definitively that if the standard deviation if the standard deviation of our log returns or for volatility goes up the value of our call option the value of our European call option goes up likewise using the same logic if our volatility were to be lower then the value of our call option would go down I'll leave you there in future videos we'll think about this in a little bit more depth