Black Scholes Explained - A Mathematical Breakdown

options they're used all around the world across a variety of assets many of you probably have even tried your hand at trading options however the mechanism behind how options are priced is often mysterious to investors most people in finance have heard of black shols and might have even built the model once or twice but have you ever stopped to wonder what the individual components mean and how they work together to determine the output and what kind of assumptions were built into the model to arrive at the conclusion in today's video we're going to be answering just Those Questions by taking this now famous financial model and breaking it down so that we can understand the finer details of the black schs formula as a warning there will be a bit of math in today's video but don't worry we'll be breaking it down and discussing the arithmetic in detail also there will be an Excel file linked to today's description with a breakdown of the topics discussed in the video which will help you better understand the concepts at work and assist you in building your own options model so let's dive in we're going to assume that you have a basic understanding of options you know that options are linked to an Associated asset called the underline you also know that all options have a strike price and expiration and can be bought long or sold short you should also understand the concepts which differentiate calls and puts so let's begin the black shows formula was first articulated by fiser black and myam scholes in their paper title the pricing of options and corporate liabilities a link to which can be found in the description if you're interested to begin we'll focus on the value of a call option then we'll simplify it to an easier to understand formula breaking down the components at play then reincorporating the simplified components to build a full understanding once we understand the call option formula we will use our knowledge to easily derive a formula for puts and then prove out our put formula using our financial knowledge the formula for a call option is as follows where D1 equals and D2 equals if we plug D1 and D2 into our first equation we get this which is a rather cumbersome equation but if we break it down it becomes much more conceptually simple let's assign each each of the variables a name so we better understand what they represent C is the price of our option this is what we're solving for S is our underlyings current price for example today we will be assuming the underlying is a stock but in reality an underlying can be any asset T is our time component and is assumed to be in years unless otherwise noted where t equal 1 is equal to one year Sigma is our underlying volatility this is an annualized number unless otherwise noted R is our annualized risk free rate and is used for discounting e is Oilers number and is like Pi in that it represents a fixed number Oiler number is equal to approximately 2.71828 and is a constant used for continuous discounting N is a function for a normal continuous distribution we will get to this later to discuss what this function means Ln is the log normal function and K is our option strike price with this information you can easily take all the variables put them into a calculator and get an answer for C however what we're interested in today's video is not so so much what the answer to C is but how all these pieces come together to compute C in other words what is going on inside the formula in math when applicable it is often useful to simplify a form into smaller structures when analyzing it cutting the form into smaller bite-sized chunks will let us do that first let's simplify D2 doing some algebra we find that our initial D2 value is actually equal to this which looks very similar to our D1 value we can work our way through the various algebraic steps to prove this relationship now that we've simplified the D2 value let's assume that our option expires in one year so that t is set to one additionally we will be assuming that we're living in a zero interest rate world where money today is equal to money tomorrow so we will set our R to zero if we do this we find that our equation simplifies significantly this is a much less complicated equation and we can now consider this simpler equation in order to gain an understanding of what's going on first let's break down the nd1 value the first thing we should look at is the log of s over K what is this component and what is it saying also why are we taking the log of these two values simply stated s over K is the ratio of our stock price to our strike price we are taking the log of this ratio to normalize the relationship with respect to zero when the price of the stock is less than the strike price the call option is out of the money and the sign of our ratio is negative when the price of the stock is greater than the strike the call option is in the money and the ratio is positive this Dynamic will be the driving force behind our nd1 relationship but it is not the entire picture now let's assume volatility is very small virtually zero when s equals K or the call option is at the money then we see that as Sigma approaches zero the value of our D1 value is zero now if we adjust S and K under these assumptions then we see that as s becomes larger than K the equation quickly tends towards towards positive Infinity while as K becomes larger than S the equation quickly tends towards negative Infinity so let's use this understanding to help build out what nd1 is doing what is the in function and what does it mean the in function is a standin for the cumulative normal distribution at our value of D1 the cumulative normal distribution is given as this integral we will not be diving into the Nuance of the integral and why Pi is a part of it however 3B 1B has already made made a fantastic video covering this exact topic and we highly recommend you check it out if you're more interested in learning about normal distributions and why this is the equation for it what is important to know is that for a normal distribution a value of D1 corresponding to zero is equal to5 or 50% of the normal distribution if we consider a normal distribution curve what this means is that at a value of zero we are at the exact mean of the distribution whereby 50% or5 of our values under the the carve fall at this point or less so in the case of our limit we see that when s equals K or when the call is at the money and volatility approaches zero our value of nd1 is .5 what we're gauging here is how Sensi of our option is in relation to changes in the underlying stocks price this is often called Delta by option Traders and is one of the first order Greeks the Greeks are important Concepts and we will be covering them further in future videos expanding upon our knowledge of nd1 we see that the function is still subject to the sigma variable which is our stocks volatility to better understand the impact volatility has on our nd1 value let's consider what happens when volatility moves away from zero to a more realistic number like 0.2 or 20% annualized volatility keeping s equal to K we can see that our nd1 value has increased to about 54% meaning that for every $1 change in stock price up or down the value of our options price will change by5 4 cents up or down this is very useful for Traders looking to hedge their positions as it determines the number of options needed to hedge the underlying asset as volatility increases so does the sensitivity of our stock if volatility were to become say 300% or Sigma equals 3 we see our nd1 value which we will from now on refer to as Delta becomes 93% meaning a $1 shift in the stock price will change the value of the call option by 93 this is logical because the stock is now riskier and this risk is reflected in the options price sensitivity making the option riskier now we understand our equation for Delta which we did by simplifying some variables but what about if we add in a new variable T what does this do to our Delta value simply put if we keep all things equal a larger T causes our Delta value to become larger and a smaller T causes our Delta value to become smaller why is this the case theoretically if we have more time to wait a larger T then there is more risk in the option giving more things can happen between now and maturity in other words the impact of our volatility is heightened but what happens as T becomes very small let's consider a new scenario an option is set to expire 1 second in the future and you quickly pause time now you can analyze the option at your leisure when examining the options Delta you see that one of either two things has occurred either the option is in the money where the stock price is higher than the strike price or the option is out of the money where the stock price is lower than the strike price in the in the money case we will see the Delta is of 100% meaning that the option has essentially become valued in lock step with the price of the stock in the out ofth money case the option Delta will be 0% meaning that the option is worthless given there isn't enough time remaining for the option to enter into money this makes te a very powerful variable especially at the end of the options life when any changes in SK or even the volatility have little to know no impact on the Delta values as T begins to approach expiration at Time Zero at expiration the option is either in the money or out of the money as there's no time remaining to change the ultimate outcome this inevitability is reflected in the Delta value with the inclusion of the time variable now we understand D1 but what about the rest of the formula from here the understanding becomes easier to explain as the fundamentals have already been flushed out let's now break out our black shols into two pieces the left hand of the equation is much of what we've already discussed given a stock price s what is the relative payout of the stock price against Delta we can think of s in D1 as our payoff component or reward portion of the black shs equation then what is K * nd2 this is our cost portion of the function or what we're giving up by owning the option D2 is similar to D1 but has an inverse relationship to volatility and represents the option sensitivity to our strike price K as volatility increases the sensitivity of the option strike decreases all else equal an interesting aspect of options is the relationship between nd1 and nd2 many new options Traders often learn that Delta nd1 can be used as a proxy to determine the odds that an option will finish in the money based on your understanding so far do you believe this is true if you're close to expiration then Delta is more accurate in determining the estimated probability finishing in the money but what if we were at the money with 90% volatility in 5e until expiration would you believe me if I said the option had an 87% chance of finishing in the money probably not 90% volatility is a lot and 5 years is a while to wait the stock price could go anywhere in that time so what is a good way of estimating the probability of the option expiring in the money the answer is to take the average of the indie1 and indd 2 values by taking the average you are balancing the return and cost factors to establish the likely expected probability as a rule of thumb at zero interest rates regardless of volatility or time at the money options will have a 50% nd1 nd2 average this is because the distribution profile in a standard black scholes model is assumed to be normal remember the 50% probability may or may not reflect economic reality and you need to conduct your own analysis to understand if the model is appropriate for your scenario all models are wrong but some models are useful the normal distribution assumption is often pointed to by critics as one of the major drawbacks of the black sches model as asset returns rarely reflect a normal distribution so now that we understand the two components of our model we can put them together to find our call options price lastly let's examine why we're discounting K the strike price by the interest rate with respect to time why do we discount the strike price value and not the stock price consider that we're looking at two prices one which exists in time now the stock price and one which exists only in the future the strike price in finance we have to pay special attention to prices with respect to where where they live in time the strike price is not reflective of a price now it is reflective of some price in the future which must be discounted back to its present value today but what about put options this answer is simple using our knowledge of put options we know that a put option is the inverse of a call option logically our formula should look something like this that was easy but do we believe this is there a way we can prove this formula is correct well we could run through our previous example with calls to determine the Dynamics with respect puts but let's do a fun and easy proof to conceptually understand the Dynamics of put pricing in the option space there's a concept known as put call parody this idea states that the value of the put should be offset against the value of the call with respect to the discounted strike and stock prices why is this so if we were to take on a long call and a short put both at the money the payout structure of this position should look identical to a stock less the discount of the strike the position is discounted as this is seen as the cost of money to have entered the position as compared to a stock which is paid in cash allowing for no Arbitrage between the actual stock and the synthetic put call position the put call parody is shown as the following if we were to assume that all our options were at the money options then we could set this parody to equal zero while setting all s equal to K which we'll denote further as X to solve we can set the equation equal to zero and we can show that the two sides of our equation are in fact equal there's a bit of algebra involved but if you work your way through it you should be able to prove that calls and puts exist in an arbitrage-free parody that was a fair amount of math and we recommend you take a look at the attached Excel sheet which demonstrates many of the concepts detailed in this video if you want to see more content like this then be sure to like And subscribe now please enjoy one of these quality videos on finance recommended just for you have a great day