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Before we get to today’s meat, here are 2 threads spurred by oil’s advance yesterday.

Delta is God

If you’re reading a Thursday Moontower, “you’ve heard the expression vega wounds but gamma kills.” It’s not quite so cut-and-dry. My pushback to that trope is the recent article vega’s finishing move. However, I’m sympathetic to “gamma kills” mantra. The running joke I’ve used to say on the desk has a similar energy:

“delta is the only greek”

I wouldn’t take this literally, the joke is bowing to the idea that if you have your hard deltas, ie your shares, pointing in the right direction, you tend to win. The Freudian reading of that statement is I’d rather be good at directional trading than a vol monk.

Today, we give delta its due. Delta is god.

No matter what you think it is, you never quite understand it. The best we can do is understand how it will harm or help us based on the thing we can’t know in advance, but know will affect our p/l — path.

While I’ve been meaning to write about this for awhile, this paraphrased question from a moontower user, bumped this post up the editorial queue:

“I’m backtesting delta-hedged straddles and I’m worried the vol I use to compute my hedge delta is ‘wrong.’ Does the choice of hedge vol bias my P&L, and if so, how?”

Pull up a chair, young Padawan.

I’m going to offer 3 perspectives.

1. The quant answer.
2. The quant who speaks “trader” answer
3. The Moontower treatment

Finally, we’ll see how this idea applies to traders and investors who try to structure an options-like payoff to a trade without using options at all.

So much trader mindshare is fixated on delta-hedging for the same reason we are never happy with the quantity we trade in hindsight. The goal here is to create enough clarity that you can not only make better ex-ante decisions but make your peace with them regardless of the outcome.

Onwards.

The Quant Perspective

We’ll start with the mathematical approach. This is not my wheelhouse, so I’ll save my words for later sections, but if you can’t wait to curl up with notation, then this post is for you (h/t to the Moontower Discord where it surfaced).

I couldn’t help but print the acknowledgements section below. I don’t know stochastic calculus, but I suspect the people involved in this paper might.

Which Free Lunch Would You Like Today, Sir?: Delta Hedging, Volatility Arbitrage and Optimal Portfolios by Paul Wilmott & Riaz Ahmad

ABSTRACT

In this paper we examine the statistical properties of the profit to be made from hedging vanilla options that are mispriced by the market and/or hedged using a delta based on different volatilities. We derive formulas for the expected profit and the variance of profit for single options and for portfolios of options on the same underlying. We suggest several ways to choose optimal portfolios.

ACKNOWLEDGMENTS

We would like to thank Hyungsok Ahn and Ed Thorp for their input on the practical application of our results and on portfolio optimization and Peter Carr for his encyclopedic knowledge of the literature.

A Quant Who Talks Like A Trader

The next perspective is a bridge. In the incomparable book, Financial Hacking, quant Philip Maymin breaks things down in terms that your common option flow trader will understand.

On hedging to model (forecast) delta vs implied delta

The short-form intuition is this: you bought a call and hedged it. So you are betting on higher volatility. When volatility ends up higher, even if only for random reasons, you benefit, and when it ends up lower, you lose.

That intuition continues to hold even if you hedge at the wrong vol. If, for example, the true vol is 30 but you hedge to 20, you are just introducing noise. The slope between your P&L and the realized vol is still positive, but not as sharply defined.

Philip brings in the practical concerns of, well, having an employer to answer to who doesn’t like loud “noise”.

If you want to minimize your mark-to-market P&L, you may choose to hedge to the market even if you think the market volatility is wrong.

How do you trade-off these two risks, the mark-to-market risk versus the at-maturity risk? Ultimately, you probably will decide based on the maturity of the option you are hedging.

• If the option will expire in a month or two, you will almost surely be able to weather any intermittent mark-to-market volatility, so you will lean towards hedging to model.
• If the option will expire in many years, you will likely lean towards hedging to market, at least until the expiry gets closer.

And what do people do in practice? They hedge their bets on how to hedge. One common rule of thumb is to hedge halfway between the model and the market delta. Then you’re never exactly hedged, but you’re never too far away either.

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.

This is a fantastic observation to give a sense of proportion:

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.

And remember…the risk from not hedging continuously can be diversified away.

His final point here echoes what I wrote in a misconception about harvesting volatility.

Which brings us to…

The Moontower Treatment

The original paraphrased question once again:

“I’m backtesting delta-hedged straddles and I’m worried the vol I use to compute my hedge delta is ‘wrong.’ Does the choice of hedge vol bias my P&L, and if so, how?”

My dead-leg-on-the-toilet response:

Here’s the quick answer…the vol that generates your delta introduces bias that you discover after the fact but you can understand how the bias is correlated to your p/l in different scenarios.

For example, if you are long vol and the stock trends, you will wish you hedged on whatever delta was the “lowest” of the reasonable options you could have chosen from…so if the option is ITM you will have wanted to hedge deltas on a high vol, but if it was OTM you will have wish you hedged on a low vol!

I’ve never done this, but you could create a little cheatsheet matrix with:

• option ITM or OTM
• market trends or chops
• preferred vol i wish i would have hedged on = “high” or “low”

By comparing that matrix to your strategy you can see which biases cause you to double down on your implicit exposure vs hedge it (for example, if you are long ITM options and vol expands in a trending market you will hedge on that desirable light delta…but you are already winning on vega so maybe this codependancy is too much “doubling” down which hurts extra if you were short that option)

Of course, I had to make the cheatsheet now that I got a moment to focus on the question. To start, I fed my response to Claude and it whipped something up. I did have to re-work some of its understanding.

[These are Gell-Mann amnesia moments, where it stumbles on things you know well, and wonder about what it tells you in domains you are less equipped to discern.]

Let’s begin with the cheatsheet, memorialized at https://delta-hedging.moontowermeta.com/:

The sheet is self-explanatory, but there are biases we can anticipate. It’s what I referred to as “the doubling-down” in my response to the reader.

Suppose you follow the rule:

“Hedge On Implied Delta”

IF:

[You buy an OTM option because you think IV < forecasted realized]

AND:

[Your vol signal is correct]

THEN:

[Your hedge ratios will be “light”…I buy OTM calls and sell too few shares]

THEREFORE:

If we trend, you will make “extra” p/l beyond the fact that you bought underpriced volatility. This is “doubling-down”.

If we chop, you will make less gamma scalping p/l than you would have with a heavier delta. The forgone p/l will be buffered by the fact that you were right on the vol being cheap.

In this case, hedging on the delta of the implied vol, is doubling down on your vol forecast in the event that we trend, and offsetting some p/l in the event that we chop.

💡Your choice of delta to hedge on begs you to wonder if a high realized vol forecast is more likely to coincide with trend or chop.

Most of the time, options embed a risk premium above the realized vol.

[The bridge between this idea and making money on selling options sways wildly and has a few missing planks. Many have died trying to find the treasure on the other side so take it easy Indiana Jones.]

That said, it’s understandable if you never want to buy an option. But sometimes you want an option like exposure, just like you might want an insurance policy. You want protection against a high-impact event even if you don’t think it will happen.

I discuss this in the Moontower community, where I prefer to hold BTC exposure as options rather than as a hard delta allocation (I actually use a blended approach, but the reasons aren’t germane to this post).

I pick my spots when I buy the options. My most recent call purchases feel validating because I thought the vol was cheap, so despite losing on direction, they were much better buys than the counterfactual of owning hard deltas.

[Welcome to vol trader cope. This is literally what life is like as a vol trader. I lost money but made the right decision. Yay. You only hope that your career lasts long enough to realize the sum of all the right decisions. The alternative of just guessing in a high-variance game and trying to get lucky is good too. If we focus on survivors. And we do. This is America after all.]

But what if you wanted to replicate the call exposure without actually buying the calls?

Replicating a Call When You Think It’s Overpriced

The closest neighbor to the term “portfolio insurance” in a database of vector embedding is “1987” (Did I put those fancy words in the right sequence? Who cares, you get the joke).

Don’t let that taint your mood going into this next section. You know that I know about that history. Calm down, we’ll extract the fruit from replication and point out the poison you can’t eat.

Step-by-step here.

You want BTC call exposure. You look at the options and think they’re overpriced. So you decide to skip the call and instead replicate it dynamically.

How?

You will be delta hedging in reverse. You’re assuming the posture of someone who sold a call and now needs to replicate it. An option market-maker who sells you a call must go out and manufacture it. If they can manufacture it for less than the price they sold it, they make a profit.

In this case, you are taking the role of call buyer, but instead of buying the call, you are going to try to manufacture it yourself, just like the market maker would have if you bought a call from them.

Mechanically, you’ll hold some BTC, intermittently rebalancing your position as spot moves, synthetically tracing the call’s payoff without paying the upfront premium.

How much is some?

You look up the delta of the call you would have bought, and you hold that much BTC.

How does intermittently rebalance work?

As BTC rises, delta increases, you buy more. As BTC falls, delta decreases, you sell some. You’re manufacturing the call’s convex payoff with a series of linear trades.

How often?

How often does a market-maker hedge? This is the question we’ve tackled many times. It’s a trade-off between the “noise” Maymin alludes to as you sample volatility. If you are a market-maker hedging a short option and the market trends, you’ll wish you hedged often (sampling a lower vol than experience from point-to-point).

If it chops, you’ll wish you hedged weekly, sampling a much lower vol than the daily ranges suggest. Both you and the market-maker face the same problem. You are both trying to manufacture an option whereby each time you trade you “sample” a realized volatility. The more you sample, the closer you get to the real vol. The less you sample, the more likely your replication strategy will differ from the real vol and you could get lucky or unlucky to the platonic (and non-existent) continuous vol.

The cost of this replication comes from the adjustments. To replicate a call, you buy more as the market rallies because the option for the strike you’re trying to mimic increases. You sell as the market falls. You are always buying high and selling low. The sum of those round-trips is your premium. You’re just paying it in installments instead of upfront. If you think these installments net of all transaction and slippage costs would exceed the call premium, you should just buy the call.

To feel good about this strategy, you’re rooting for the options to have been overpriced. If realized vol comes in lower than implied, your rebalancing costs less than the call premium would have. You built the same payoff for cheaper.

To determine how much stock you need to buy, you’re computing your delta at some vol, and that choice determines whether your delta is heavy or light. If you hedge at a high vol (say, the implied you think is too rich), you’re holding more BTC than you “should” — heavy delta. If you hedge at a lower vol (your realized estimate), you’re holding less — light delta.

The cheatsheet as an aid to your hedging strategy

The sheet has the posture of someone long an option, who by replicating is manufacturing an equivalent short option. They paid a premium upfront, but hope the sum of their gamma scalp stream exceeds the premium they paid. In other words, their replication posture is the opposite of yours. You are trying to replicate a long option because you think it will cost less than actually buying a call.

So you invert the logic of the sheet!

If BTC chops you want a light delta. Fewer round-trips means less friction eating into the savings you’re generating by not paying the full premium. If you are right about the IV being overpriced but you hedged using the implied delta, then you will suffer a bit because your delta will have been heavy. But this will partially offset the profitable decision to not buy the call outright. If you hedge on your “model” delta (ie the vol based on your realized forecast), then you are doubling down on your prediction that the vol is cheap in the event we chop.

Again, the idea of vol and its coincidence with trending or chopping is lurking beneath but now you are aware of it.

Restriking Your Synthetic Call

Say BTC has run from 70k to 90k. You’ve been replicating a 100k-strike call, but you want to “roll” it up, taking profit and starting fresh with a 130k-strike call.

You can just look up the 130k call at your chosen vol and adjust the delta to match. That will result in monetizing some of your BTC as the 130k call will have a lower delta than the 100k call.

Notice that if you don’t roll your 100k call is closer to ATM with the spot BTC now up to 90k. It has more gamma than your old deep ITM 90k call. More gamma means your rebalancing is more frequent and more costly. You’re “long” a more expensive option. There’s no free lunch. If you substitute your replicated call for a real call, that call’s theta will reflect the higher rebalancing costs you tried to avoid.

So….

What Makes You Wish You’d Just Bought The Call?

This question strikes at the heart of the Black-Scholes assumption of continuity.

Gaps.

The call buyer pays implied vol upfront and owns the path, for better or worse, for the duration of its life. If a stock gaps up 20% over the weekend, the call captures the full move. The gamma which you prepaid for, ensures your delta adjusts automatically.

The synthetic call you tried to manufacture missed buying deltas in the gap. You are not as long as you should be and to make it up you need to buy all your shortfall deltas up 12% as opposed to prices along the way.

Hard optionality is valuable and impossible to replicate. This is why Option Market Maker 101 class teaches you that the only way to hedge an OTM option is with another OTM option. Nobody knows what the SPX down 25% put is actually worth.* You can reason about a relatively tight put spread only because the error is bounded in proportion to the risk you know you are taking beforehand.

(Although we can reason that it commands a premium and likely trades for more than its actuarial value which is not really known. It’s all a bit circular. And you are still left to contend with the fact that the people, as a category, who buy those teenies know a lot more about vol trading than you. There is no non-vol trader buying that option. Also, this paragraph was written in invisible ink to reveal the VIX basis traders on the mailing list.)

Portfolio insurance failed because it was crowded thus blowing up the cost of put replication by feeding on itself. Meanwhile, the owners of the actual puts went on to start the trading firms you know of today.

The “Bridge of Asses”

📺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.

Related learning:

🔗 Understanding Risk-Neutral Probability | Moontower

🖥️Moontower Presentation on Black Scholes “As a Trading Strategy” | Slides

• The slides for that presentation are based on this post: The Intuition Behind The Black-Scholes Equation

[UPDATED]

As I expected, the post how to get arbed with perfect info would trip people up. I didn’t expect confusion because I thought Professor Doug Costa, whose explanation is featured in that post, was itself confusing. But because the concept of replication is hard and feels like it violates the good. It’s triggering. It means you can know the truth and still get arbed. Again, this is why I call it the pons asinorum of finance.

A reader brought it up in our Discord so I’m going to share the discussion here as he felt like our back and forth helped.

Before getting to the conversation, let’s refresh the problem Doug set up:

A company issues contracts based on a provably fair coin. The contract pays $150 on heads, $75 on tails. It trades at $100. Interest rate is 0%.

You calculate the true expected value of the 110 call using the true probability of 50%.

It’s worth $20 because it has a 50% chance of being $40 in-the-money.

You pay $20 for it (but even if you paid a bit less for an unambiguously positive EV trade, this analysis will hold. I just want to stay with Professor’s example)

The dealer sells it to you, hedges with 8/15 of the underlying contract, and locks in $6.67 profit in both states. Pure arb.

You had perfect information about the true probability and you still got arbed. The dealer made money in all scenarios, trading the call at fair value with you.

Doug is showing how the real-world probability doesn’t matter to the derivatives trader if they can also trade the underlying. In this case, the underlying is mispriced, but the dealer doesn’t know that. All the dealer cares about is whether the relationship between the derivative price and the underlying price is mispriced. In this contrived example, the mispricing was more profitable than knowing the true probabilities.

And to add something Doug doesn’t mention…if the investor knew the stock was underpriced and bought that instead, they’d have a positive EV trade (the fair price of the stock is $112.50) but they are still worse off than the dealer who knows the relative value of the 2 securities is wrong and gets to make a profit in all scenarios.

This is a good place to insert the chat.

Reader: I see, so the main point is we can converge a spread by trading two things instead of betting on one.

Kris: In a world with no derivatives, you’re left with having to be good at guessing real-world probabilities, but derivatives are their own source of possible edge that doesn’t inherit from knowledge of the future but from relative mispricings between the derivative and the underlying.

It’s obvious that being able to handicap probabilities would be a source of edge, but it’s quite subtle that once you introduce derivatives and the idea of replication, there becomes a source of profit that doesn’t rely on such an ability.

Reader: Right, so it’s instructive in giving one more spread to look at. If you used a put in your example, then the dealer would lose because they’d be too short. Then a dealer that actually has no information and sells both sides ends up $0. This example is picking the (long) side where it wins.

Kris: Yeah, the underlying in this example is too cheap RELATIVE to the call option.

If the call option was $13.33, then from the vantage point of real-world probability, both the underlying and call are too cheap, but they are priced correctly with respect to each other.

Which makes the point — if a derivative and underlying are correctly priced to each other, then the real-world probability is not important to the dealer. The dealer only cares about the relative values.

You can just compute the Sharpe of buying one vs the other I suppose to see which is better (that’s one lens). The call is more underpriced in % terms, 3.33 when it’s worth 20. But it’s more volatile as it will lose all of its value when it loses.

I’d just stick them both in a Kelly calculator in Claude or something and whichever one it says bet more on is the better one lol.

There are some important implications here. And brain damage — investor brain and derivatives brain collision.

The goal is that derivativesbrainskill.md becomes something one calls as needed, like Neo downloading kung fu. But you don’t wanna get carried away with it and shoot it at everything in life. It’s this weird artificial thing that works in a replication context, but it’s also not artificial in that its violation presents hard cash arbitrage!

That’s the end of the chat, but let me add one more thing to make you feel better if it’s still foggy.

I’ve seen this subtlety trip up seasoned options traders where they take B-S pricing to mean that the forward for a stock is stock grown at the risk-free rate (RFR), but this is ONLY true in a world where you can trade the underlying AND the options. Outside the context of replication, you cannot make that assumption.

Struggling with this idea is entirely forgivable. I mean, the realization that you could use RFR as the discount rate was a revolutionary breakthrough. Bachelier figured out option pricing in the early 1900s, but he and his contemporaries were stumped by what rate to discount the payoffs.

Later academics wondered if you should use something like the required return from CAPM or something, but it was the whole idea that if you trade a derivative vs the underlying against one another, then you can have equivalent payoffs and therefore it’s riskless to go long one and short the other. If it’s riskless, then RFR is the appropriate discount rate.

Warren Buffett sees the necessity of agnostic dealers using the RFR to price options in arbitrage-free ways as an opportunity. He asserts that put options are overpriced because they use too low of a discount rate, but the dealers don’t care so long as they can trade the underlying, they can arb any other rate assumption. Again, so long as “they can trade the underlying.”

This single idea allows derivatives traders who know nothing about the fundamentals of securities to make money in a sea of people who do. It’s quite profound and not a small part behind why I think vol trading is easier than directional trading.

A final follow-up

A reader asked:

How did I get to the 8/15 hedge ratio?

It came from Prof Costa’s setup:

A stock is 50/50 to go to $150 (up) or $75 (down) from $100. What is the no-arbitrage price of a 110-strike call on a one-period binomial?

p̂ = (100 − 75) / (150 − 75) = 1/3

Call value = 1/3 × $40 + 2/3 × $0 = $13.33

Delta = (40 − 0) / (150 − 75) = 8/15 of a contract

Let’s take this apart.

Risk-neutral probability of stock going up

p̂ is the risk-neutral probability of the stock going up.

How do we get that intuitively?

Start with the payoffs:

• Up_payoff = 1.5x
• Dn_payoff = .75x

The risk-neutral probability is the one that makes the stock price fairly priced given the possible payoffs. In other words, if you buy the stock, the expected return is 0. It must satisfy this equation:

p̂ (Up_payoff) – (1-p̂ ) (Dn_payoff) = 0

Solve for p̂:

p̂ (Up_payoff) – Dn_payoff + p̂(Dn_payoff) = 0

p̂ (Up_payoff + Dn_payoff) = Dn_payoff

p̂ = Dn_payoff / (Up_payoff + Dn_payoff)

Concisely stated:

p̂ = d/(u + d)

p̂ = .75/(1.5+.75)

p̂ = 1/3

When in doubt, you can always set up the expected value equation and solve the algebra. You don’t have to memorize a formula.

But you can also put on your gambler goggles.

If something pays 2-1 odds like a money line of +200 or a prediction market trading at 33, then the implied risk-neutral probability is 1/3.

If you buy this stock, you risk $25 to make $50.

Just remember the intuitive odds to probability converter:

x-to-y odds = y / (x + y) probability

2 to 1 odds = 1/(2+1) = 1/3

This is just symbols for “If I get paid 2-1 when I win, then I must win 1 out of 3 times for this to be fair”

You can practice some more in these posts:

• the arbitrage reflex is more profitable than the opinion reflex
• Binary Straddle Example Based On The 2016 Election

The hedge ratio

Back to the original example…if you’re short one call, you buy 8/15 of a share to be delta-neutral. The numerator is the spread in the call’s payoff across the two states ($40 vs $0). The denominator is the spread in the stock’s payoff ($150 vs $75).

Delta is just option change over stock change. How much the derivative moves for a move in the underlying.

The formula in terms of return:

delta = (C_up – C-down) / S (Up_payoff – Dn_payoff)

delta of the 110 call= (40 – 0) / 100(1.5-.75) = 40/75 = 8/15 = .533

Relative value

To reinforce the main point of Prof Costa’s talk, risk-neutral probabilities are enough to make money if you can find an inconsistency between option prices and the underlying. The real-world probability doesn’t matter in a relative framework.

By understanding the distribution of the stock, we were able to compute a delta for any contract. That distribution implies some probability embedded in the underlying. This is not the same as the real-world probability, which is decreed to be 50/50.

Professor Costa showed that if you buy the stock which was underpriced (although you didn’t know that because you computed it must be fair with a 1/3 probability of going up) as a hedge on the call delta, then if someone paid anymore than the risk-neutral fair value of the call, even if they paid less than what the real-world implied probability price is, the dealer makes free money!

Work through the logic for a bunch of strikes and this is what you get if you sell the calls at the risk-neutral fair value (green) or real-world fair value (purple).

The dealer always wins against the real-world probability!

It is also true that the call buyer who pays some price between risk-neutral and real-world has positive expectancy but they don’t have an arbitrage.

So who loses?

The people selling the stock at $100 when the real-world probability is that it’s 50/50 to be 150 or 75. The stock should be $112.50.

Learn more:

🔗 Kellogg lecture notes that walks through exactly this kind of binomial pricing and hedging