Friends,
Let’s stay with the education theme as we’re still with a week of the new school year. We’ll lean towards finance though…
This is a re-print from the Gappy (head of quant research at Balyasny) LinkedIn:
Yesterday I was talking to a very smart high-schooler interested in finance, and the topic of college major came up (he was undecided between math and physics). My preference for finance jobs is to hire applied mathematicians/physicists over pure math major. Consider the simple example of PDEs. A mathematician will study properties of wave equation, KdV, Navier-Stokes, by taking the formulas as given. If they had to derive even the wave equation from physical principles, I bet most mathematicians would be stumped (unless they have to teach the derivation to undergrads). Conversely, for a physicist getting to a realistic equation from first principles is 95% of the thing. This often involves making a toy model, expanding some terms, etc. It involves linking different layers of reality. We take these equations for granted, but it took a gigantic effort to Euler to write down his equation, or to Einstein to write down General Relativity equations.
I would argue that finance is much more than Physics than Math. The real challenge is the modeling part. In certain famous cases (like Black-Scholes) modeling was the *only* challenge, since the solution was trivial. And it is revealing that extensions to B-S, which were attempted mostly by mathematicians, were less than successful. Also: it is very interesting and revealing that there are no courses in mathematical modeling (*). Applied math is usually “numerical methods to solve problems.” Statistics (as taught) is increasingly moving away from model thinking. I believe that the reason for this is that modeling can only be taught by examples, but examples are sometimes domain-specific. However, many can be taught with minimal assumptions. It would be useful for real-world people if there were a course going over such cases. Here is my current list:
1. Granovetter’s Threshold Model
2. Volterra-Lotka predator-prey
3. Heating and Cooling
4. Ising Model
5. Polya’s Urn/Kirman-Folmer
6. Schelling’s Chessboard model
7. Sandpile model
8. Logistic growth and exponential growth
9. Concentration of measure emergence in the real world
10. Emergence processes of heavy tails
11. Supply and demand
12. Ricardian trade
13. SEIR model
14. Reynolds Boids and swarm models
15. Hooke’s law and (forced) oscillators
16. Hardy-Weinberg’s principle
17. Morphogenetic models
18. Replicator models in evolution
In related fun, Gappy had some thoughts about whether Terence Tao, Fields Medalist and if there’s a conversation about the smartest person alive his name’s coming up, would thrive in finance — thread. I can tell it’s a thougtful take even if I don’t feel like I know enough to have an opinion myself. I do know this reply got me to spit my drink.
Gappy’s LinkedIn post spurred me to ping Justin about whether Math Academy would ever be able to let you specify a topic and it lay out the knowledge tree that you would need to traverse to get there with a properly backfilled understanding. I won’t give away his answer, but I’ll just say I love what they’re doing over at Math Academy.
Anyway in the spirit of a Justin/LinkedIn incantation here’s some love for actually knowing your craft:
I feel like the 2010s were an era of fake experts or parlaying some extraordinary experimental finding that doesn’t replicate into an airport book and speaking career. Generally ick.
Well, be careful what you complain about because it feels like there’s no pretense about expertise to even hold up. You can be an expert on anything you want bro. Imposter syndrome was the thing holding you back. Not the whole “not knowing stuff” part.
Oops, sorry about that, my opinions were leaking out.
Moving on…
Money Angle
I can’t remember which of the 3 Todd Simkin interviews on my blog I summarized where he mentions it but Todd was asked if SIG’s secretiveness has been an advantage. He said in trading, it’s been good, but when it comes to recruiting technologists or researchers, it’s been a hindrance. The FAANG companies are household names and since trading firms compete for some of the same talent, you’d want more people to know what SIG is.
I figure this recognition is behind their increased public outreach. Like this awesome video that recently dropped from the lecture series where Professor Costa teaches their trainees about the GFC.
It starts assuming you don’t even know what a bond is and proceeds to cover an unbelievable amount of distance in one hour. The narrative and history going back to the 80s is fantastic and I even learned (or reviewed) a lot of basic market knowledge.
#teaching_goals
While this video is loaded, here’s 5 bits that stood out for me. There’s also a very SIG-esque lesson in there about anchoring bias.
- Diversification has literal monetary value – Great demonstration of how portfolio theory translates directly into pricing and risk management
- Reflexivity in credit markets – Default rates weren’t actuarial constants but depended on loan originators’ incentives. Once originators became divorced from risk while retaining pricing/underwriting control, the system became unstable. A systems thinker would have spotted this disconnect.
- Misaligned incentives drove market distortion – Traders focused narrowly on derivatives markets where the CDS market dwarfed the underlying bond market. Unlike bond issuance (limited by actual capital formation needs), derivative trading appetite was essentially unlimited.
- Good ideas taken too far become dangerous – Diversification through low correlation assets is sound in principle, but this conceptual acceptance prevented people from asking the critical follow-up: “To what degree is this still safe?” (The opposite is hormesis – sometimes a little of a bad thing is actually beneficial. As the old saying goes “the posion is in the dose”.)
- “This would turn out to be a fateful decision” – The final section on implied correlation reveals how trading desks completely inverted their hedge ratios between tranches, fundamentally misunderstanding how correlation affects different credits.
Money Angle For Masochists
Paid subs will recall my story of Doug teaching Black-Scholes to my cohort at SIG back in 2001. Four hours in one day to explain the assumptions and four hours the next day to derive the equation. I tried to keep up but dropped off embarrassingly quickly.
I did that webinar to explain how I eventually came to understand the formula. The recording is paywalled but these are the slides for the talk.
Here’s the distilled version:
Start with what we know.
At expiry, a call option is worth the stock price minus the strike price (or zero if the call is “out-of-the-money”)
So today, the call price equals
“the current expected value of the stock given the call is exercised”
minus
“the discounted strike price”
[The strike price gets discounted for both the time value of money AND the probability of exercise.]
Let’s work through this with common sense.
You’re looking at a 1-year $50 strike call. The stock trades at $50 today, risk-free rate is 5%.
Say the call has a 50% chance of being in-the-money.
Let’s also assert that in the state of the world where the call gets exercised, the stock is on average $58*. That happens 50% of the time, so the expected value is 0.50 × $58 = $29.
*Think of this like rolling a die: given that you roll greater than 3, what’s the expected value? It’s 5 (the average of 4, 5, 6).
What about the discounted strike price?
The $50 strike discounted to present value is $50 × e^(-0.05) = $47.56. With a 50% exercise probability: 0.50 × $47.56 = $23.78.
The call value from our definition
“the current expected value of the stock given the call is exercised”
minus
“the discounted strike price”
maps to
$29 – $23.78 = $5.22.
The key insight: we can replicate a call option with a portfolio of stock and cash
You can replicate a call’s payoff by owning some amount of stock. This amount is more commonly referred to as the “delta” (or hedge ratio).
This delta changes as the stock becomes more or less likely to finish in-the-money. As the stock rises, you buy more shares to replicate the call’s potential payoff. As it falls, you sell shares since exercise becomes less likely. You’re buying high and selling low—creating negative cash flows. That sum of negative P&L should is what the option is worth.
You can either buy the option (pre-paying these cash flows) or manufacture it yourself through this delta hedging strategy.
In an arbitrage-free world, the option price must equal the present value of these replicating cash flows. If the option were priced with higher volatility than actual, you could short it, hedge with shares, and pocket the difference.
The self-financing part is elegant.
To replicate the call, you need to buy the “delta” quantity of shares. With what cash? You borrow it—specifically, you borrow $23.78 and use that cash to buy the shares today. This is why the strategy is self-financing: we’re simply borrowing against a future cash flow.
Why does this work?
At expiration, if the call gets exercised, you sell your stock at $50 to the call owner. With 50% exercise probability, your mathematical expectation is to receive $25 in one year. So you can borrow the present value of $25 today (ie $23.78), use that borrowed money to buy the shares, knowing you can repay the loan at expiry with the proceeds from selling those shares.
Notice why call values increase with interest rate:
a call is ultimately the difference in value between the number of shares you need to buy (delta shares) and the number of shares you can afford to buy via the loan. The higher the interest rate, the less you can borrow, the fewer shares you can buy, so the call value—which bridges that gap—increases.
In a sentence…
a call value represents the difference between how much stock you need to buy and how much you can afford to buy to achieve that hockey stick payoff.
Fwiw…
One of the webinar attendees says this diagram made it click. I’ve never seen it anywhere else and came up with it when I wrote A Visual Appreciation for Black-Scholes Delta
Stay groovy
☮️
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