FIN 376: Binomial Option Pricing and Delta Hedging

Introduction to the binomial option pricing model, delta hedging, and risk-neutral valuation.

[@wolfgangi]

Very intuitive explanation! Thank you! This is indeed a nobel prize worthy idea. The fact that using a risk free (100% hedged position) we can back out the PV using risk free rate as a proxy of the discount rate because they are equivalent. We completely bypass the need to use probability.

[@karthikmurali6667]

Best explanation of binomial hedging and option pricing Ive ever seen. Thanks a lot!!

[@mokichoki1]

Dr. Matt you are simply the best. Thank you for simplifying the topic so logically and in the most comprehendible manner.

[@aronhegedus]

really good video. Teacher is a perfect mix of friendly and explanatory

[@Frizataufik]

THANK YOU SOOOOO MUCH FOR THIS LECTUREE!!! IT HELPS ME A LOTTTTTTTTTT, MY PROF UNI SUCKSS!! HE JUST WANT TO SHOW OFF HOW SMART HE IS!!

[@peerasilpwiwattananon2198]

Thank you so much! Your lecture is easy to understand and help me make sense of complex option pricing formula & Black-Scholes model.

[@thakkarr3]

Excellent .. Made it easier for CFA 2 options pricing topic ..

[@andreavento3427]

Finally, thank you for the intuition, best explanation on youtube

[@king199324]

Great video. Thanks for posting

[@jackFatal]

Hi there i'm from Africa (Côte d'Ivoire), i took a course on pricing options but i didn't understand it well, thanks to your recap it's better now

[@jianxuan38]

Would like to ask: in the vid, you said that we use bonds and stock to replicate the call options. But when you said borrow at time 0, are we referring to buying of bonds? - thast why it is negative so at maturity(t=1), shouldnt it be a plus of 90? since we are getting back our maturity value Thank you so much in advance

[@MattBrigida]

To borrow we *sell* bonds, which is a positive cash flow at time 0 and a negative cash flow at time 1 (when we have to repay the bond).

[@lucanthony2209]

Thanks for the video. Why do we devide 90/105?

[@bat-amgalanbat-erdene2621]

The girl at 11:49 LOL

[@pabloacosta9]

Outstanding professor!!

[@Lukas-cm2b]

what will happen with your long shares if the stock sharply dips

[@heidiw3615]

thanks for the upload!

[@tosinbabasola2963]

Thanks you for the video; I will so glad if you can also make a video on option pricing using hermite polynomial.

[@dodgingdurangos924]

How do you adjust for different number of time periods (e.g., a few hours, a few days, a few weeks, a few months)?

[@zagreus767]

Are we explicitly assuming that the market has no other securities available other than the bond, stock and call option when pricing this option?

[@MattBrigida]

+Fardin Humayun No, we are only assuming the existence of the stock/option/bond.

[@LeilaRmaths]

At 7:29 , could you please someone explain why the portfolio at time 0 equivalent 3 Calls? Thank you

[@LeilaRmaths]

I found it :) Oreka

[@zagreus767]

Sorry I have loads of questions and I will try and post them as I go through the video: 1) Why did you just calculate the PV of $90 when deciding the amount to be borrowed? What is the rationale behind this? 2) Why is the payoff at time 1 = $120 - $90 = $30 ?

[@MattBrigida]

+Fardin Humayun 1) We want the replicating portfolio to pay $0 in the down state. To get this we borrow the PV of $90, so we owe $90 in time 1 and have stock worth $90 => replicating portfolio is worth $0. 2) The stock (which we own) is worth $120, and we owe $90 => our replicating portfolio is worth $30.

[@zagreus767]

+Matt Brigida Thank you so much. That makes a lot of sense. I have some other confusions which I will post later :)

[@williamrussell2008]

This is really helpful. Thanks

[@alirezahaghpanahi1719]

Many thanks, it was very useful.

[@niketankotadiya9542]

Excellent. one doubt. Portfolio 1 => Borrowing and Buying stock= Costs= 14.29 Portfolio 2=> Selling call and Buying stock= ( 33.33-4.76)=28.57 Why there is different cost for same pay-off portfolio?

[@MattBrigida]

+niketan kotadiya The portfolios do not pay the same. The first portfolio replicates 3 call options. It pays $30 in the up-state, and $0 in the down-state. The second portfolio replicates a (default-free) zero coupon bond. It pays $30 in both the up and down states. Both portfolios, however, will imply the *same* price for the option.

[@niketankotadiya9542]

Sir can you explain for 3 periods binomial. I got little bit confusing.

[@tylerdupont1811]

awesome video!!! Extremely useful and good teaching

[@fleurashiyanova5552]

Good day! When you said that we have to pay 90 back, don't we also have to pay the interest at that time too? So it will be 94.5... thank you!

[@MattBrigida]

We borrow the present value of 90 (which is 90/1.05 = 85.71) so that we have to pay back 90 in time 1. This ensures the payoff on our replicating portfolio is 0 in the down state at time 1.

[@MattBrigida]

A call option is a thing that pays $10 in the up state and $0 in the down state at time 1. We can replicate three of these things for a cost of $14.29 today. So the cost of replicating one option today is $14.29/3 = $4.76. In short, we are not replicating a net gain, but an option's value at time 1.

[@bardhimtolka1765]

How did you know that you can replicate three call options?

[@myurathankajendran2526]

Your video is super useful. I am wondering about american option with dividend. How we can derive option price for them? and about risk neutral pricing?

[@MattBrigida]

+Myurathan Kajendran Great question. We can use binomial trees, and the backward induction algorithm. I'll post a video soon outlining the method.

[@yuss6494]

+Matt Brigida waiting for the video! This one was very interesting. Great work Matt!

[@patrickzhao6335]

Hi Sir the formula to get the call option price at 13:59 is incorrect. You did it right at the first place but you then changed it to a wrong one. The formula should be 100/3-Call=28.57 hence Call=100/3-28.57=4.76

[@MattBrigida]

Thanks for your comment, but both 100/3 - 28.57 = 4.76 and 28.57 - 100/3 = -4.76 are "correct". They are opposite sides of the same trade. It is perfectly fine to price the option as either the person who (1) sells the call and buys delta stock, or (2) buys the call and shorts delta shares of stock.

[@brankobajcetic4290]

He means that the equation you changed it to was incorrect, since initially you wrote: 100/3 - C = 28.57, for which C = 4.76 is correct, but then you changed it to: -100/3 + C = 28.57, for which C is equal to 61.90, and that is an incorrect price of C, it's just a minor sign change though, you explain concepts well!

[@aichamarzougui1545]

@@brankobajcetic4290 I don't know wether or not you still need an answer for this but I'm sure someone does. When he changed the equation, he should've added a minus sign to 28.57 so that it becomes C-100/3= -28.57 which gives C=4.76 The logic behind it is that you add a minus to anything you buy because it's a cost that you incur or cash out. The 28.57 dollars are the cost of the portfolio or the bond that you have bought at time 0. It's the same logic we used to add a negative sign to 100/3 (because we bought the stock). I hope I got it right and it's clear.

[@NoNonsense_01]

​@@aichamarzougui1545 Thanks for this explanation. This was driving me crazy for weeks. Although it is only a multiplication with negative one on both sides, assigning negative value to C, which is a cash inflow, bugged me to no end.

[@caverac]

This is great! Thanks so much for uploading this. I have a question, in the second model when you replicate B, why would anyone want to sell a call? My point is that if I sell a call, in the up-state I loose 10, and in the down-state I gain nothing. So, there's no benefit in selling the call. Am I missing something?

[@kishanmarvania7155]

Well, you are getting premium from selling the put option and hence that would reduce the amount of investment being made in the portfolio of call option and stock.

[@christina6788]

Why do you call stock prices in period 1 stock returns? isn't stock return equal to the appreciation in the price divided by the original price of the stock?

[@AnhHLe-gw5es]

what happen if i assume in the way that i buy call and sell delta stock? my calculation is that the risk free cash flow is a negative value, why i get the negative value which is -30 ?

[@swaroopkj8441]

Yes, since you have sold ∆ shares, it will be a short sale as you do not hold such shares today. So, at the end of the option period you will have to buy shares and settle your position which results into negative cash flow i.e., -30(+10-40 or -40). Also, your final answer of option premium would be -4.76 since you bought the option and it'd be a cash outflow.

[@joshpiper2508]

you said the badly of the portfolio is 100 - 85.71. where did you get 100 from and what do you mean by "portfolio value". thank a lot. wouldn't the portfolio just be worth what you bought it for (85.71)?

[@joshpiper2508]

meant "value" not "badly" sorry

[@MattBrigida]

You borrow 85.71 and buy the stock for 100. This short bond and long stock portfolio costs you 100 - 85.71 = $14.29.

[@davide.lionetti]

Awesome video. Wish my teacher was like that. By the way, why do we assume that the delta stays constant for whatever change in price?

[@MacGuffin1234]

Linear relationship is easy. Delta would be changing according to gamma, would need to re-hedge when delta changes.

[@marycrawley4721]

very helpful, thanks

[@sibyliaputrilingga7687]

thank you

[@liamk7440]

I understand everything except where the 90 and 120 come from. Are those just arbitrary numbers (in the real world based on maybe standard deviation) or are they some way consistent?

[@MattBrigida]

Yes, they are functions of the standard deviation of the asset's returns. For example, e^(vol * sqrt(change in time)) and e^(-vol * sqrt(change in time)) are used in Cox, Ross, and Rubinstein (1979). These values are 'consistent' in that they do not allow an arbitrage. If you just chose arbitrary values, you could allow an arbitrage.

[@niketankotadiya9542]

sir can you explain multi period in this way?

[@Lukas-cm2b]

10:42 you have to pay $1k not $10

[@andreaeverson9083]

Are you still playing soccer?! Damn baby you were so good?!

[@mohammedalbanna9413]

sell call and pay 10$, it doesn't make sense to me tbh, i think he meant it to be the other way around,

[@nicolasdupin7254]

11:50

[@marcosg2201]

Excelente, lamentable la traducción se pierde mucha explicación interesante. Thank you

[@RealSoundNow]

The lecture is incorrect. You can not solve a hedge for two specific points and then assume you're properly hedged for any points (ie. not caring about the return of the stock). What if the stock went up 50%? You'd have $50 in stock at 1/3 share where share price $150, but be short an option worth $40 ($150-$110). You'd have $10 in ending value from your $28.xx investment and have a loss. The reason for this is the non-linear return of the call option, the delta hedge is only valid for the points solved.

[@MattBrigida]

The lecture is correct, and is the standard way of introducing option pricing. See Hull, Bodie, Kane, and Markus, or any introductory investments/derivatives text. In the setup of the problem we clearly assume the stock can *only* do two things, increase to 120 or decline to 90. So it is not possible for the stock to increase to 150. In this simplified setup you can understand the core concepts in option pricing. Once this is understood we can move to setups where the stock can have take whatever value in the future.

[@RealSoundNow]

The explanation of Black Scholes at the end of the lecture extrapolates the solved problem to imply the rate of return is irrelevant - that you are hedged for any rate of return. That is false. The delta hedge *only works for the specific points solved (only those specific rates of return). You can not pick a risk free rate, arbitrarily, and claim to be hedged.

[@MattBrigida]

The whole idea of Black-Scholes (that won the Nobel) is that I write a call, and buy delta shares of stock, so over the next instant my loss on one is a gain on the other and the portfolio is risk-free. This is over an instant, where I then have to re-delta hedge. It does not matter at all what the expected return on the stock is. The idea is you hedge out any source of uncertainty -- in the Black-Scholes world stock is the only source of uncertainty. Note this is nonsense in a Black-Scholes world, "only works for the specific points solved". Black-Scholes is continuous time, so each point has probability 0. You can't solve for points.

[@RealSoundNow]

I meant that your specific delta hedge only works for the points solved -- which is accurate - those are the only returns that are hedged risk free.I am not arguing about Black Sholes here (or whether it won a Nobel). I am pointing out that your statements from 15.05-15.38 are not true. The return of the stock indeed matters and you can not simply say "I can just assume that the stock increases at whatever rate .." The rate is specifically tied to the system of equations you solved for the hedge, ie where the end values are 120 or 90. Those are the only points that are hedged risk-free.

[@RealSoundNow]

So you can not have it both ways, where you say in your first response the stock can only do two things, increase to 120 or decline to 90, two specific rates of return solved by the two cases of linear equations to get your hedge portfolio, and then also say in your later discussion that the return does not matter, the portfolio is hedged for any assumed rate of return.

[@عادلالشعلانالغامدي-ك9ل]

قال الله تعالى (..... إِنْ هُمْ إِلَّا كَالْأَنْعَامِ ۖ بَلْ هُمْ أَضَلُّ سَبِيلًا .)الايه أعترض. فيتو ✋ حبيبي عادل حل عادل منصف . البيبان تختلف 'وين الفطرة الآدمية ؟

[@sulavshrestha8316]

kzbin.info/www/bejne/hovVnoKeZ9ONq9Um51s I think it's stock minus call. Your correction I reckon is wrong. BTW thank you for your explanation.

[@noueruz-zaman7894]

lol..for some reason it looked like for a bit that they had lesson in an attic

[@mohammedalbanna9413]

whos chasing him!

[@bmwman5]

Confusing.... needs better explanation.

[@sulavshrestha8316]

kzbin.info/www/bejne/hovVnoKeZ9ONq9Um39s your students are bad at maths I guess😂