The Math of 'The Trillion Dollar Equation'

Here are my notes from I was a PhD student on this stuff (we were allowed to bring in short notes to the exam)

0:00 The Trillion Dollar Equation
0:18 Veritasium's Example of the Call Option
2:13 The function v(s t) for the value of the option
3:29 Veritasium's Delta hedged portfolio
4:19 Veritasium's slide by derivation
5:24 Derivation of the Black-Scholes equation from scratch
8:26 Change in portfolio value d Pi t
9:47 Approximate changes by derivatives of v a la Taylor series
15:08 Model for the stock price S t Geometric Brownian Motion
19:42 d S t squared is just a dt a la Itos Lemma
25:44 Set delta to di v di s to perfectly hedge
27:23 Equate to the risk free interest rate equation
29:10 But isnt delta a function of time? Why does this work?

[joeeeee]

Great vid. Next one: "Solving the trillion dollar equation"

[gertjan]

I'm coincidentally currently taking a course on finance right now. The answer to why this delta is a function of s and t is a bit silly, but the idea is that delta is a constant which is set by your portfolio. To set delta equal to this dv/ds you simply have to short delta times the amount of options you have in stock (shorting or buying depends on the type of option and whether you're shorting or buying the option). Setting this delta equal to dv/ds is thus not a mathematical trick, but a trading one called "delta hedging". The point here is that now the price of your portfolio becomes independent of the price of your stock, as an increase in stock price (which you shorted, so is bad for you) will be absorbed by the increase in options price and vice versa. You then make money by "dynamic hedging", which makes use of the volatility (which was briefly mentioned here) and continuous delta hedging to make money. Interestingly, this is why option trading is also sometimes called "volatility trading" as you are trading on your guess of the volatility versus that of the market. This is also why the price of an option is dependent on this volatility.

[DanielKRui]

LMAO this is exactly what I thought when I watched Derek’s video. Like the title and thumbnail were about the equation but the actually equation got like 10 seconds of screen time and 0 seconds of explanation.

[stevenlarson]

The assumptions behind the Black-Scholes model include:
1. Constant volatility: The model assumes that volatility remains constant over time, although this is not the case in reality.
2. Efficient markets: It assumes that markets are efficient, and that people cannot consistently predict market directions.
3. No dividends: The model assumes that the underlying stock does not pay dividends during the option's life.
4. Constant and known interest rates: Interest rates are assumed to be constant and known in the model, typically represented by the risk-free rate.
5. Lognormally distributed returns: The model assumes that returns on the underlying stock follow a normal distribution.
6. European-style options: It assumes European-style options that can only be exercised on the expiration date.
7. No commissions and transaction costs: The model assumes no fees for buying or selling options and stocks, as well as no barriers to trading.
8. Liquidity: It assumes perfectly liquid markets where any amount of stock can be bought or sold without difficulty.

[DistortedV]

Probably one of the best vids here. This kind of reminds me of machine learning and reinforcement learning, but have to finish watching it.

[donaldlee]

So glad someone on YouTube is doing actual math rather than telling a kindergarten story.

[niklas]

There is actually also an easier way to directly derive the solution to the Black Scholes equation. You can immediately use the formula at 6:00 to do a calculation.

The idea is that the value of our option should just be equal to the expected profit spread (compared to leaving the money in the bank) if we don't account for risk. For that switch to a risk neutral measure (here S_t will be a martingale/have constant expectation).

Write E[ exp{-r(T-t)} K(S_T) | S_t ] = v(S_t, t) and plug in the formulas for K() and S and you're done

[pandabearguy]

I did the continuous time arbitrage theory course last semester so the math is still pretty fresh. PDEs are fascinating, but Christ on a bike, what a pain in the ass to work with most of the time.

[cassiojp]

I learnt so much from this man, more than my calculus teacher could teach in 4 months.

[rishabhsemwal]

Thats what i am looking for. I love maths behind things. Subscribed you

[Polyamathematics]

Great video.

Btw to fix the audio channel issue, when you create an audio track make sure you set it to "mono" / 1 audio channel (rathere than stereo / 2 audio channels). That just makes the audio come through both headphone channels equally.

[mickeymaples]

Currently I'm taking time series econometrics and so Its really nice that this is practical use of it

[nobodysfool]

In the chart at 1:00, the bands would be more aesthetically pleasing is they were drawn concave to reflect the square root diffusion of brownian motion with respect to time.

[Budha]

12 hrs later and I’m still wondering about how we can ignore derivatives of Delta. Great vid!!!

[benpierce]

I think Delta might be assumed constant across time intervals because we put residual cash into a hypothetical bond holding. The value of the hedging portfolio only changes based on changes in the financial instruments rather that from changes in the hypothetical bond. This is the self-financing assumption of Black-Scholes and the basis for the risk-free return. Great vid!

[zaydmohammed]

Thank you for this video! I had decided to do my assignment for financial economics on the black Scholes Merton Model but was struggling to understand all the mathematics. Thankfully the timing was such that Veritasium and you made easy to follow explanations while I was working on it.

[SamuelLiJ]

The reason that d(Delta) is excluded is that you want to account for the total profit/loss achievable to the investor (to equate to the risk-free return), not the change in the portfolio per se. Your profit/loss cannot be affected by simply buying or selling shares at market price. It is affected only by changes in the value of your holdings (the option and the underlying security).

[VoidFame]

A "mono audio output" option would save lives

[fsthomson]

Loved this!! Thanks for creating it and sharing!

[franckherve]

Good explanation 🎉🎉But in pratice we don't use black and Sholes to hedge or price our products ...we use LV (local volatility), LSV (local stochastic volatility), or SV stochastic model instead, simply because of the assumptions of the volatility (constant) in the BS model's but the implied volatility is derived from the BS...