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

Hello you to welcome back so this is i think our fifth video on the topic of pricing options and obviously you know if this is the first one you’ve clicked on it might make sense to look at some of the others first what we go through some of the we trying to explain why we use the methods that we do to build these binomial trees so yesterday we looked at multi-step

Binomial trees and in today’s video we’re gonna learn about how to price an american option using binomial trees so an american option just to be clear i have a prior video on the difference between american and european options are actually i think it’s just on i think it might be just on bought our auctions and towards the end of the video i explained that but

Um an american-style option is one that you can exercise early so at any point in the life of the option you’re able to exercise the option and a european-style option is one that can only be exercised at maturity and so we’re not able to price american options using the black scholes model but we can using the binomial tree approach so today we’re going to look

At what modification we have to make to our binomial tree in order to price american-style options and in order to do so i’m gonna price the exact same option that i priced in the last video but this time we’re going to make it an american option and we’re gonna see how the calculation changes and we’ll also see how the price changes now it should be obvious

To you that an american option will be worth more than a european option and not simply because you know there’s this right to early exercise essentially greater flexibility and you’ll always pay more for greater flexibility just if something does more it costs more so let’s take a quick look at it so we’re gonna try and price an american put option here the

Same one as last time and the only difference is that it’s american now so if you haven’t watched the other video it’s probably worth doing so but we’ll push ahead so this is an optimally american put option with a strike price of 20 so that means that the spot price is also at 20 it expires in two years time so we’re making each step one year long and during

Each period the underlying can move up or down by 20% and the risk-free rate is 5% so if you want to if you want to work along with this video you can write down those details i’ll put them up on the screen right now and you could pause the video write them down and do the calculations along with me so the first step in pricing this option is of course to draw a

Binomial tree that’s the first step in all of our pricing approaches so step one draw the tree and write in the information that we know and so all of the information is pretty much the same as in the last video the underlying is at 20 so we write that in it can go up by 20% which means it can go up to 24 in the first step and it can go up again 28.8 in the up up

Node the top node of the tree there it can go down from 20 we multiply 20 by 0.8 that gives a 16 and then we can multiply 16 by 0.8 and that gives us twelve point eight so that’s the downtown node and then for the middle node it can be twenty either times one point two and then times point at eight or times 0.8 and times one point two either way it brings the

Same number which is nineteen point two so we write all that in that’s the three possible price outcomes in this in this option and the next thing we do is we write in what the derivative is worth and so it’s a put option which is the right but not the obligation to sell at the strike price of twenty and so when the underlying is a 28-point age you would never

Want to sell a twenty and so our option our put option is worth zero and not up up scenario in the up down scenario the right to sell at twenty when the underlying is actually trading at nineteen point two would be worth eighty cents and that’s just twenty minus nineteen point two gives us eighty cents and so we write that fu deed that value of the derivatives

In that middle note is worth eighty cents and then finally the the right but not the obligation to sell at twenty when the underlying is trading at twelve point eight is worth twenty minus twelve point eight which is seven dollars and twenty cents so we write that in then we did the same calculation we did in the last video which is p which we calculated in the

Last video as 0.628 two so 0.628 two x 0 is 0 and 1 minus 0.6 to eight two times 80 cents present valued at five percent gives us twenty eight cents that we write fu equals twenty eight cents which you see in the diagram on your screen then we do the same for the the next binomial tree so point six to eight two times eighty cents plus one minus point six two

Eight two times seven dollars and twenty cents all present valued at five percent for a year is worth three dollars and two cents now this is when we stop and do another calculation if we’re doing a european put like we did in the last video we just continue on here but what we do now is we look at the intrinsic value of the options are put with a strike price

Of twenty when the underlying is worth twenty four zero so in the up note there we’re not worried were happy having twenty eight cents in there because that’s more than zero in the download if you had the right but not the obligation to sell a twenty dollars while the underlying is trading at sixteen dollars that would be worth four dollars that’s twenty minus

16 gives us four dollars now our calculated value that we just did give us three dollars and two cents but because this is an american auction it can’t be worth three dollars and two cents simply because if it was worth three dollars and two cents you would just exercise it and get the intrinsic value of four dollars so this is really the difference this is the

Difference between pricing an american and a european auction using a binomial tree and the difference is that at each intermediate node you look at the calculated value and you look at the intrinsic value and you put in the higher of those two values so in our example in the up node we calculated 28 cents while the well the intrinsic value is zero so we put 28

Cents in there and in the down node we calculated three dollars and two cents but the intrinsic value is four dollars so we cross out our calculator you are three dollars and two cents and we sub in four dollars okay so then we just do our calculation the final time which is 0.62 eight two times twenty eight cents at one – 0.628 two times four dollars we press

And value that and we come to a value of one dollar and fifty eight cents now once again we can look at the intrinsic value of the option and it’s the right but not the obligation to sell at twenty when the underlying is at twenty so that’s not really worth anything so the calculated value of one dollar and fifty eight cents is fine in that scenario so that’s

That’s a calculation and we’ve worked out that as an american put this option is worth the dollar in 58 in our last video we press the same option but as a european option and we found it was worth a dollar and twenty four cents so now we’re actually able to see how much extra you would pay for this option if it was american / european and that’s just a dollar

58 – a dollar twenty four gives us thirty four cents so you can use this same approach to price american call options using a binomial tree and so in our next video we’re going to look at pricing an american call option using a binomial tree but we’ll add in one more complication will have it diffident you know okay we’ll see how you can deal with dividends in

Pricing options using binomial trees so see you tomorrow for that video have a great day

Transcribed from video

Pricing American Options using the Binomial Tree Method. – Options Trading Classes By Patrick Boyle