How to Price Options using a Binomial Tree (The Portfolio Approach)

How to Price Options using a Binomial Tree. The portfolio approach.

In today’s video we’re going to learn about option pricing and in particular option pricing using the binomial tree and in the textbook that we use said there’s a link to that in the description below if you’re interesting in the textbook we look at two approaches to the binomial tree and so for this video we’re gonna look at the first one which is the portfolio

Approach to pricing options using a binomial tree now it’s worth noting that the black-scholes model was actually developed before the binomial tree approach but usually i try and teach the binomial tree model first and then we follow up with the black-scholes model but the reason for this is just that the binomial trees may be a little bit easier to understand

The intuition of and both the black-scholes and the binomial tree models actually will rely on an awful lot of the same assumptions you know there there are very much that they’re quite similar to each other in when it’s just a discrete time model and the other is a continuous time model but the beauty of the binomial tree model is that it can actually handle a

Variety of conditions that other models cannot and so often people think well the black-scholes model it uses more sophisticated mathematics and thus it’s a better model and that’s not really the case actually both both models are really good models and they’re often just suited to different things so the binomial tree model is slower computationally than the

Black-scholes model but it’s able to handle things like american options dividends and it’s just more flexible and it can be more accurate at times and requires fewer assumptions so actually both models are really useful and really good so we’re going to learn about the portfolio approach today and the first step in in any binomial tree is simply to is to draw

A diagram of the possible parts of the underline so we start by just drawing a binomial tree and in this example we’re going to start out with a very basic and unrealistic model we’re gonna we’re gonna make an awful lot of assumptions that probably if you’re watching this for the first time you’re gonna sort of say well this is entirely unrealistic you know this

This doesn’t seem like a good approach at all but don’t worry we’ll start with this approach and then what we’ll do is once we understand we’ll use this sort of really basic model to understand the intuition and material they’re lying the models and then we’ll start to add more and more realism so by the end of our you know little class here on binomial trees

Hopefully they’ll make a lot of sense to you in your field that they they actually are not really quite quite as unrealistic as maybe they’re the first appear to be so the first step as i said is to draw a diagram of the possible parts of the underlying and then what you do is you calculate the present value of the cash flows of the auctions so that seems like a

Reasonable approach to pricing anything so we have to start with two assumptions in order to build our first binomial tree one is quite a reasonable assumption it’s that no arbitrage is are freely available in the marketplace and that’s not such a crazy assumption to make most of the time you would expect that if there were obvious arbitrage is available there’s

An awful lot of smart people in the finance world that are out there looking for them and they probably start trading upon it and try and profit and actually even in the option space there’s particularly kind of a smart group of mathematically inclined people who be looking for this sort of thing so that’s not an unreasonable assumption our next assumption is

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Kind of unreasonable but it’s one that we’ll be able to fix over time and that is the assumption that we know with certainty that there are only two possible price outcomes for our underlying so we’re basically saying that we know the price right now and at some point in the future we’ll say in a month’s time it’ll be at one of two prices and it can’t possibly

Be at any other price okay so that doesn’t i imagine you’re watching this and thinking well that’s not at all reasonable is this approach reasonable it is a reasonable approach but it gets reasonable when we when we add more to it but this will this will hopefully explain the theory to you so we’re going to just start with an example we’re gonna try and price an

Option we’re gonna say that the underlying is trading at $50 and at the end of one month it’s gonna either be at 70 dollars or 30 dollars okay so it can’t go to any other prices it can’t be at 70 dollars and one penny it can’t be at $29 it will only be a one of two prices not seventy or thirty okay and then what we’re gonna try and do is value a european call

Option honored with a strike of fifty so that’s an after money call option one month to expiration where the interest rate is at five percent so the first thing we’re gonna do is draw a tree okay and so you can see up here on the screen our first tree and we fill in the information we know and so what we know is that the underlying is at fifty right now so we

Write that in and then we know at expiration that the underlying will be at seventy or thirty so we write in in the upside economic scenario st equals 70 the underlying is at 70 and in the down side economic scenario that st equals 30 okay so that’s the two possible prices it can be add the next thing we have to do now is to work out what the option will be worth

It’s a call option what will it be worth in these two scenarios so if you have a call option which is the right but not the obligation to buy the underlying at the strike price which is 50 and the onion is 70 our expiration of call option has to be worth the difference between 70 and 50 which is $20 so we’re going to write that in as well a c equals 20 in the

Upside scenario now in the downside economic scenario of the right but not the obligation to buy the underlying of 50 but the underlying is actually trading at 30 so you would not exercise that option you wouldn’t buy at 50 when when you can just go out in the market and buy a 30 so in that scenario you would just tear up your options contract throw it away it’s

Expired worthless okay so we write that information in as well c equals 0 in the downside scenario or college and it’s worth nothing okay so that is some information so far we’ve just drawn a tree and filled in the information we know and we’ve done a tiny calculation which is what the call option is worth in either scenario so now for the theory now for the bit

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That that this video is all about what we want to do here is we want to see if we can set up a portfolio with some amount of the underlying and the derivative in this case a call option though that allows us to know the value of ports of our portfolio expiration by that i mean is there an amount of the underline that you could own along with the along with the

Call option that in both the upside and the downside economic scenario you’d end up with the exact same amount of money now considering there are only two possible outcomes right there’s we’re in a world in which this underlying can only be in one of two prices at expiration essentially if we can set up a portfolio like this we’re essentially saying that we have

A portfolio that pays us the exact same amount no matter what happens in the economy right and if we had a portfolio like that that’s a risk-free portfolio right because if if if you have like guaranteed and – money no matter what happens in the economy that is risk free there’s no risk there’s no question as to how much money you’re gonna get and therefore if

It is risk-free we can then we obviously have to present value it and hopefully it’s obvious to you that the rate that we would use to present value it is the risk-free rate simply because it’s a risk free cash flow so if there’s some amount of the underlying and the call option that we can have in our portfolio that gives us the same amount of money in either

Scenario we’re then able to discount it at the risk-free rate and we’re able to work out the fair value of our call option so let’s set up a portfolio that has some amount of the underlying and we’ll call that delta x s so delta just means some amount of an s is the underlying – si which is – – call option now the reason that we have – the call option what if i

Were long the underlying in short the call option is quite simply that the call option being the right but not the obligation to buy a call option in the stock will both go up in value if the underlying goes up in value and about fall in value is the underlying false in value right so if we need a portfolio that has the same amount in both the up and the down

Side scenarios we need when one is going up for the other to be going down and vice versa we need them to move inversely to each other so therefore it needs to be some amount of stock – to call option or you could do it the other way around that have minus some amount of the stock plus a call option but one has to hedge the other they have to move opposite to

Each other otherwise there’d be no value in which the portfolio is worth the same at expiration so let’s write that in on our binomial tree so here on you can now see on screen our next next little piece of information in there so in the upside economic scenario our portfolio cash flow is going to be 70 times delta because it’s some amount of the they’re lying

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And the underlined is where it’s 70 in the outside and in the downside it’s gonna be dealt at times 30 which is some amount of the underlying and the underlying is worth 30 minus in the upside scenarios got minus 20 which is the value of the call option in the upside scenario and in the downside scenario got minus 0 which is the value of the call option in the

In the downside economic scenario so in order to work out if there’s an amount of the underline that we can hold that will make it that that will give us the same payout in either scenario we just have to make those two portfolios equal to each other and solve for delta so we write in delta times 70 minus 20 equals delta times 30 and then we solve for delta

Now in this example of when we solved the delta we find that the answer is 0.5 so let’s look at that point 5 times 70 it’s 35 minus 20 gives us 15 in the upside scenario in the downside scenario 0.5 times 30 gives us 15 minus 0 is still 15 right so in either scenario we have a payout of 15 so we’ve now worked out that there’s some amount of it that we can up the

Underlined we can hold that will actually make our portfolio riskless so if there is a value for delta where the two portfolios are identical at maturity in all possible scenarios we can then just press and value that and then of course because it’s a portfolio it contains both the stock and it contains some amount of the stock in our example houses a share of

The stock and minus a call option we then just have to strip out the amount of the stock and we have to flip the side on the call option to have the value of the call option so we take our 15 and we present value with to get the the price of it today that comes to $14.95 we just present value that with our our interest rate that we specified early on and then

We work out the the stock our present is worth $50 we knew that 0.5 times $50 is 25 so then 25 minus 14 94 gives us 10 dollars and six cents which is the value of our car auction so that’s actually all we have to do that is the portfolio approach to pricing an option using a binomial tree and so what we’ve learned here is that there is an interesting result that

If we knew the next two possible steps in an underlying assets price and we assume no arbitrage we comprised the derivative so in our next video we’re going to look at another approach to doing the same calculation and we’ll also try and find ways of making our assumptions much much more realistic so we’re gonna we’re gonna have to solve the problem of the this

Big glaring issue that we we can’t possibly know that the underlying will be at one of two possible prices at some point in the future but we’ll be able to work through that and get to a reasonable conclusion and so tune in tomorrow and and watch that video see you later bye

Transcribed from video
How to Price Options using a Binomial Tree (The Portfolio Approach) By Patrick Boyle

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