# Pricing Options Using Multi Step Binomial Trees

##### These classes are all based on the book Trading and Pricing Financial Derivatives, available on Amazon at this link.

Hello youtube and welcome back today’s video we’re going to learn about multi-step binomial trees now if you haven’t watched my other videos you probably should watch them first because essentially they try to explain you know why we use binomial trees for pricing options and why they might make sense so in the last two videos we looked at the portfolio approach

And then the risk-neutral approaches to building binomial trees which were just two different approaches to really getting to the same point and there were some very interesting theoretical ideas in there but you know at the end of both videos i pointed out that the problem was that we were in still quite an unrealistic scenario where we were saying that we

Could build it we could price an option if we knew the two next steps in the price of the underlying and of course the problem is that it’s probably impossible to know the two next steps in anything in particular when we’re saying there’s only two possible outcomes in a price over the next month period so it was interesting but we weren’t really very realistic so

In today’s video we have to find a way of making all of that theory kind of worked out and seem more realistic so what we’re gonna do now is we’re gonna build our by a risk neutral binomial tree and just add some additional steps to it and adding more steps really just involves doing the same calculation a few more times it’s this there’s no really additional

Mathematics involved here we’re just building more steps to our tree and as you’ll see the formulas are all the same but we end up with with more possible outcomes which is well begin to make a little bit more sense and be more real-world so the two-step binomial tree what we’re gonna do is we’re going to build firstly a we build a wooden step binomial tree in

Our private we’re gonna now build a two-step binomial tree and once you know how to build a two-step binomial tree the theory is the same for building a three step four step five step and frankly as many steps as you want to binomial tree so all you have to do is you do the same calculation for a two step by no military but this time we’re gonna do three times

Instead of once okay and what you do is you calculate the value of the furthest sole four steps in the binomial tree that gives you do you and d for the next binomial trees and you just keep calculating them back till you get to the start of the the tree and that is the price for derivative so we’re going to use the same formula we used before which is the value

Of the derivative is the present value of the risk-neutral probability time of an up move times paired in enough scenario plus the risk-neutral probability of the down move times the payout of the derivatives in the down scenario present valued and so that’s all we have to do we just have to do it for each step of the binomial tree now one thing it’s worth noting

Here is it would but you see the formula up on the screen the difference between this one and the last one is it says you know e to the minus or t2 minus t1 what that really means is just that we have to we have to divide up the time the life of the derivative so we’ll say if it’s a one-month derivative and we’re doing a two step binomial tree or the time period

For each step has to be half of a month and the reason for that is that otherwise we’d be over discounting the derivative we be making a mistake in our present valuing so essentially let’s let’s take a look at an example and we will price our will price of derivative so we’re going to use our binomial tree it’s able to value both puts and calls and so for this

Example i think the last time we did call so this time we’ll do a part we’re gonna price and add the money european part with a strike of 20 and so it’s at the money that means that the underlying is also trading at 20 and we’re going to use a two-step binomial tree where at each node the underlying is able to move either up or down 20% the risk-free rate we’re

Going to use is 5% and so off we go so the first step is simply to build the tree so we much like in the last video we’re just gonna put in the information we have so step one draw the tree you can see the the tree up there on the screen and we write in all of the information we know so we know that the underline is at twenty dollars right now so we write that

In and we then know that it can go up by 20% or down by 20% at each step so 20 times 1.2 gives us 24 so we write that in at the next step on the tree and then 24 times 1.2 is twenty eight point eight so we write that in at the very top at the up-up node on the tree and then let’s calculate the down and a downtown node so 20 times 0.8 gives us \$16 so we write

That at the down node and then 16 times 0.8 gives us it gives us twelve point eight so we write that in again now for the middle node that we can call that either the up down or the down up node you take 20 and you multiply it by one point 2 and then by 0.8 or you can multiply it by 0.8 and then by or two either way you’ll come to the same number which is nineteen

Point two and so that’s the price of the underlying in the middle scenario so we’ve now got last time we just have two possible price outcomes now we’ve got three possible price outcomes so ad expiration the underlying can either be in twenty eight point eight nineteen point two or twelve point eight so the next thing we have to do is put in what the derivative

Will be worth at expiration in each of those scenarios so it’s a put option so it’s worth more if the price of the underlying falls and it’s got a strike price of twenty which means it only takes on value if the underlying is below the price of twenty so in our up up scenario the underlying is at twenty eight point eight and we’ve the right but not the obligation

To sell at twenty you’re not gonna sell at twenty when you’re able to sell at twenty eight point eight so you would just allow that contract to expire worthless in that scenario so in the fu u the value of the derivative or the payout of derivatives at the up up note is zero then in the middle scenario the underlying is at nineteen point two and you’ve the right

But not the obligation to sell at twenty so obviously you’d happily sell at twenty when the underlying is at nineteen point two and the right to sell at twenty would obviously both be worth the difference between twenty and nineteen point two which is eighty cents so we write that in f you do you can call that fd you if you want to um is worth eighty cents and

Then finally in the downtown scenario the underlying is a twelve point eight the strike price of the put option was \$20 so twenty minus twelve point eight gives us seven dollars and twenty cents so we write fdd the value of the derivative at the down-down node is worth seven dollars and twenty cents so that’s all the information we need written in to our binomial

Tree now what we have to do is price the option so we then of course need to calculate p so we’re using the same formula we used in the last video to calculate p p equals e to the rt minus d over u minus d so p is equal to e to the 0.05 we’ve made this a one year expiration option so we don’t need to do any more with that some e to the 0.05 minus 0.8 over 1.2

Minus 0.8 and so once again i explained in the last video that we’re getting u and d from the magnitude of the upper down low so it can either move up by 1.2 from what was it moved from 20 to 24 per step or it moved from 20 to 60 in which to get to 60 and you have to multiply 20 by 0.8 so that’s where und came from and when we do that calculation we calculate

That p equals zero point six two eight two okay so then we just do our calculation we say fu so the value of do two derivatives at the up node rather do po not the middle of node is p times zero which is the power of the derivative of the up node anything multiplied by zero is what zero so we’ll leave that out and we’ll just say 1 minus p so 1 minus 0.6 to a 2

Multiplied by 0.8 which is the value of the derivative in that middle node comes to ones present value comes to 28 cents so fu then is 28 cents fd then is calculated as p multiplied by 0.8 so p is 0.6 to a 2 times 0.8 plus 1 minus 0.6 to 8 2 times 7 dollars and 20 cents which is the value of the dura in the downtown node present valued at 5% for one year gives

Us three dollars and two cents so then we just plug our 28 cents in our three dollars and two cents into perform now and do the calculation again so p 0.628 2 times 28 cents plus one minus point 6 2 8 2 multiplied by three dollars and two cents gives us a dollar in twenty four cents or put in this example is worth \$1 and twenty four cents so i should concede

Now we’ve gotten more realistic with our calculations we haven’t gotten hugely more realistic we’ve gotten a little bit more realistic in that we moved from saying there are two possible outcomes to saying that there are three possible outcomes so that’s a bit better but it’s it’s hardly amazing it’s hardly real-world but the beauty of this is that we can just

Build more and more steps into our tree now you’re not gonna sit down with a pen and paper and calculate a massive binomial tree with a thousand steps but luckily computers are good at that right so you can either build a spreadsheet or you can code something up and we’ll say c++ to build as many steps to a binomial tree as you want to do and just two more steps

You build the more time it takes to do the calculation you’ll also find that after a certain number of steps it doesn’t necessarily refine the price and awful lot more so you don’t have to build ten million steps to a binomial tree but if we build more steps what happens is we have more possible price outcomes and the more possible price outcomes we have the

More it starts to look like the real world because while it might be really unrealistic to say that we know the price of an underlying and months time they’ll either be seventy or thirty like in our first example it’s not so crazy to say that over the next you know second or microsecond that the underline will move up or down by will say one penny like whatever

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One tick is on the underline that you’re looking at and so once you move in that direction would you add in enough steps you end up with an awful lot of possible price outcomes and so as you can see up on the screen here now this is a big binomial tree and then you’ll notice that there’s only really one part that will bring you to the left and right tails off

Of that you know to the down down down down down up up up up up node but there’s many parts that will bring you towards the middle so what does that mean it means that we’re starting to look at where it we’re essentially saying here with this binomial tree with a big enough binomial tree that the price of the underlying has a distribution kind of like a normal

Distribution now we’ll talk later in further future videos about actual distributions of stock prices but the point of this is really just to say that we’ve moved in a much more realistic direction now we’ve moved away from the idea that the price of the underlying can be seventy or thirty and now it’s essentially there’s a distribution of possible as a price

Outcomes for the underlying and so now we’re much closer to the real world than we were early on but we’ve got all of the same set of finance theory built in there so now you can see how the binomial tree approach might not be so crazy that it might actually make sense and be a good way of pricing options so we’ve got one little problem left that maybe has been

Needling you is that even though we’ve now moved to lots of steps that lots of possible price outcomes the question is where did we get u and d from you know when we said it could move up or down 20% or up and down by whatever amount you want to say well that actually is calculated from the volatility of the underlying so early on i said that we really only need

To know two things to price a derivative honor on an underlying and that is the price of the underlying right now and the volatility of the underlying so we get you indeed from the volatility of the underlying and here’s the formula on the screen right now so u equals e to the sigma which is the standard deviation of the underlying times the square root of time

And that’s time per step in our binomial tree and then d is just 1 over u so essentially we’ve now worked out that if we know the price of something and how much it wiggles around its standard deviation how much it moves were able to come up with a reasonable price for an option on it a derivative on it and in fact it doesn’t even have to be something like a call

Or a put option it can just be a payout like so for example if we said you’ll get paid \$1 as it closes above a hundred and nothing if it closes below or even if you’ve got a dollar if it’s above and 50 cents if it’s below a given price all we have to do is build a binomial tree put in the payouts that you would receive from the derivative at those final nodes do

Our calculations which we’ve just gone through you value it all and that that tells us the value of that derivative are not bad and cidade is binomial trees and so our next video we’re going to look at how we can do even more with binomial trees we’re going to look at pricing american options there options where you’re allowed to sighs early so far we’ve just

Looked at european option so see you tomorrow for a video on pricing american options using binomial trees

Transcribed from video
Pricing Options Using Multi Step Binomial Trees By Patrick Boyle

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