In this issue:
📺Option Pricing Explained: No Arbitrage + Financial Mathematics from a Quant | 52 min watch
Doug Costa (SIG quant, former math professor, and the teacher I learned Black-Scholes from 25 years ago) builds no-arbitrage derivatives pricing from scratch using a binomial tree. No calculus, pure replication.
The thing I want to point you to is the profound role of the no-arbitrage axiom. It is the basis of derivatives replication and, by my assertion, represents the “bridge of asses” in investing education.
As a reminder, since nobody clicks links, Wikipedia says the pons asinorum or “bridge of asses” is:
used metaphorically for a problem or challenge which acts as a test of critical thinking, referring to the “ass’ bridge’s” ability to separate capable and incapable reasoners
The notion of replication is the pons asinorum of investing education because it is:
the conceptual rails of looking at a web of branching future payoffs, seeing how they could be replicated, and measuring the cost of that replicating portfolio today. It is the formalization of finance’s deepest truth — you cannot eradicate risk, but only change its shape.
You could make an even stronger claim that it lies at the core of decision-making itself, as it formalizes opportunity cost.
And I say this without being able to appreciate its deeper impact. Doug pauses for a moment in the video to marvel: when you add no-arbitrage condition to the standard axioms of mathematics, he says, the entire field of financial engineering “blossoms” out.
His colleague frames the no-arbitrage axiom joyfully:
Either we get a formula [so we win mathematically]. Or it’s violated and we make free money. Either way, we win.
Towards the end of the video, Doug discusses reflexive pushbacks he’s encountered after teaching this.
“One piece of pushback is typically, well, maybe it’s just that with stock prices, you don’t really know the probabilities. So it’s just a matter of knowing the right probabilities— if you could really discover somehow what the true probabilities were, then it would be better to use them [than the risk neutral probabilities].”
Doug’s rebuttal shows how you would still be arbed.
“I’m going to give you an example to debunk that idea. And I call this example the coin flip contract. So I’m going to postulate that there’s a company, a corporation, that finances itself, not by selling stock, but by selling what they call coin flip contracts. And the corporation has gone to great trouble and expense to manufacture a perfect coin, meaning a coin that is exactly 50% to be heads and 50% to be tails every time it’s flipped. So the probabilities are always 1 half and 1 half guaranteed…
You can watch the video, but I paraphrased it here as well. Here’s how it works.
A company issues coin-flip contracts based on a provably fair coin. The contract pays $150 on heads, $75 on tails. These trade in a secondary market at $100. Interest rate is 0%.
So we know everything. The probabilities aren’t hidden or estimated. They’re printed on the coin: p = ½.
Now: what’s the no-arbitrage price of a 110-strike call on this contract?
p̂ = (100 − 75) / (150 − 75) = ⅓
Call value = ⅓ × $40 + ⅔ × $0 = $13.33
Delta = (40 − 0) / (150 − 75) = 8/15 of a contract
Now suppose you say: I know better. The real probabilities are ½ and ½, and I’m not going to ignore them. Expected payoff is ½ × $40 = $20. So you buy the call from me at $20.
Here’s what I do next. I’m short the call. I immediately buy 8/15 of a contract to hedge.
Heads: My 8/15 position gains 8/15 × $50 = $26.67. Plus your $20 premium, I have $46.67. I owe you $40 (I have to buy the contract at $150 and sell it to you at $110). Net: +$6.67.
Tails: My 8/15 position loses 8/15 × $25 = $13.33. But I have your $20 premium. Net: +$6.67.
Every time. Both states. Guaranteed $6.67. I haven’t predicted anything. I don’t care what the coin does.
What did you get? Heads: gain $40 on the option, paid $20, net +$20. Tails: lose your $20 premium, net −$20. You’ve turned a coin flip into a coin flip — a $20 bet where you win or lose based on what the coin does.
If you try to hedge back? Doesn’t matter how you move delta. Win more on heads, lose more on tails. Move it down: vice versa. The best you can do is lock in a guaranteed $6.67 loss.
You had perfect information about the true probability….and you still got arbed buying the calls (you should have bought the contract!).
The market-maker doesn’t need a view on the coin, just the ability to trade the underlying and the derivative simultaneously. And acquiring the knowledge to cross the “bridge of asses.”
A random personal thought:
I suspect is kind of triggering for some people. It offends one’s sensibilities to think
that understanding derivative pricing ends up trumping knowledge about the true odds of things.
It’s like you spend all this time researching and learning and at the end of the day some market-maker knows just enough to not trade at the wrong price with you anyway. I’m overstating that reality, getting picked-off is real and market-makers are rightfully paranoid. But I guess that’s why I’m drawn to replication as a way of thinking. A trader is just looking for some free money when your bid or offer presents a contradiction. And that hunt makes all prices a little smarter, which, is a public good (but also a frustrating result for traders themselves, which is why the job is always uphill. A byproduct of your success is a smaller TAM).
Just to be thorough, this replication thing applies mostly to derivatives. The arb needs to be able to trade the derivative and the underlying and all advantage comes from the relationship between the two. The arb is useless without relative value.
🔗 Understanding Risk-Neutral Probability | Moontower
🖥️Moontower Presentation on Black Scholes “As a Trading Strategy” | Slides
In a misconception about harvesting volatility, you learn that you do NOT need to scalp the gamma to isolate the vol of an option trade.
If you buy options implying a daily vol of 2% per day and it moves 4% per day, your expectancy is positive regardless of whether you hedge or not. That doesn’t mean you will win any more than it means you will win if you flip a fair coin and receive 2-1 odds. You have made Sklansky bucks, not necessarily real bucks.
Hedging reduces the p/l variation around the expectancy.
In Financial Hacking, Philip Maymin explains
The inability to hedge perfectly continuously impacts your trading by introducing random risk. This risk decreases if you hedge more frequently, but only as fast as the square root. Therefore, if you want to halve your risk, you have to hedge four times as often.
He makes this tangible and practical when he says:
Noise from hedging a one-year option on a daily basis instead of continuously is about the same as one volatility point. If you make one volatility point in expected profit and the standard deviation of your profit is one volatility point, then your Sharpe ratio is about one.
His final point echoes my argument that a requirement to hedge to isolate vol is a misconception:
The risk from not hedging continuously can be diversified away.
I built a simulator so you can see this scaling law in action.
An oblique insight can be witnessed if you set up the simulation with negative expectancy, ie pay 24% vol for a stock that realizes 20%. The more you hedge the more certain you lock in negative expectancy.
Doug Costa actually showed that happen in the toy example above. The investor who bought the 110 calls based on the real-world probability but then hedged by shorting the mispriced security actually assured themselves of a loss.
If you have no edge, variance is your friend. Not financial advice.
🎮Moontower Discrete Hedging Simulator
With LLMs lowering the effort on info-gathering I figured why not include citations to literature on this topic for those who think purely in math.
I had Claude devise a Research Collector Skill.
Its purpose:
When building an interactive tool or educational artifact that demonstrates a quantitative concept, this skill gathers the academic and practitioner literature behind the concept and produces a research lineage panel — not a flat bibliography, but a structured narrative showing how ideas built on each other.
This is what it produced and what you can find in the dropdown at the end of the tool:
Bertsimas, Kogan & Lo — Journal of Financial Economics, 2000
The foundational paper. Characterizes the asymptotic distribution of replication errors from delta-hedging in discrete time. Introduces “temporal granularity” — a measure of how well discrete hedging tracks a derivative’s payoff. Proves that for European options under GBM, the hedging error standard deviation scales as 1/√n where n is the number of rebalancing intervals. Derives closed-form expressions for calls and puts.
web.mit.edu/lkogan2/www/Papers/WITC.pdf (free PDF)
Evaluating Hedging Errors: An Asymptotic Approach
Hayashi & Mykland — Mathematical Finance, 2005
Generalizes Bertsimas et al. (2000) from one-dimensional diffusions to continuous Itô processes driven by multidimensional Brownian motion — covering stochastic volatility models, non-Markovian settings, and data-driven hedging strategies where the true model is unknown. Shows the hedging error converges to a time-changed Brownian motion.
galton.uchicago.edu/~mykland/paperlinks/hedgeerrors.pdf (free PDF)
↳ addresses a limitation of the foundation paper…
Discrete Time Hedging Errors for Options with Irregular Payoffs
Gobet & Temam — Finance and Stochastics, 2001
Shows the convergence rate depends on payoff smoothness. For standard European calls/puts (smooth payoff), the L² error converges at rate 1/√n. But for digital options (discontinuous payoff), the rate drops to n^(1/4). This matters practically — hedging binary options is fundamentally harder than hedging vanillas, and more frequent hedging buys you less improvement.
↳ extends to delta-gamma hedging…
The Tracking Error Rate of the Delta-Gamma Hedging Strategy
Gobet & Makhlouf — Mathematical Finance, 2012
Shows that adding gamma hedging (hedging with a second option) can improve the convergence rate from 1/√n to 1/n for smooth payoffs. The tracking error is driven by the third derivative of the price function rather than the second (gamma). Gives conditions on trading dates to achieve optimal convergence.
hal.science/hal-00401182/document (free PDF)
Which Free Lunch Would You Like Today, Sir?
Ahmad & Wilmott — Wilmott Magazine, 2005
The practitioner bridge. Derives closed-form expected profit and variance of profit for delta-hedging mispriced options. Key insight: hedging with implied vol gives path-dependent but always-positive daily P&L when on the right side (RV > IV for longs). Hedging with realized vol gives path-independent total P&L but wild daily swings. Also covers optimal portfolio construction across multiple mispriced options.
spekulant.com.pl/…/DeltaHedgingVolatility.pdf (free PDF)
Colin Bennett — Santander, 2014
Comprehensive practitioner reference. Page 95 states the normalization coefficient for the hedging error formula is √π, giving the full result: σ(P&L) ∝ ½ S² σ² T Γ × √(1/N). Covers the full landscape of volatility trading — skew, term structure, and practical hedging mechanics. The standard desk reference for vol traders.
trading-volatility.com/Trading-Volatility.pdf (free PDF)
Lihong — The Logbook (Substack), 2024
Clear, modern walkthrough of the hedging error framework. Connects the discrete hedging variance to the diffusion scaling intuition (price variance ∝ √time, so hedging error ∝ √(1/frequency)). Also covers the impact of return autocorrelation on optimal hedge frequency — negative correlation (mean-reversion) reduces the benefit of frequent hedging, while positive correlation increases it. References Ahmad & Wilmott and Bennett.
freeportlogbook.substack.com/p/hedging-errors
QuantLib: DiscreteHedging Example
QuantLib Project (C++)
Production-grade C++ implementation of the discrete hedging Monte Carlo in the QuantLib open source library. Simulates replication error across random scenarios, directly implementing the Bertsimas-Kogan-Lo framework. Useful reference for verifying simulation logic against an independent codebase.
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