<aside> ℹ️ About
This lesson will build an understanding of risk-neutral probability and why it’s such an important concept in reasoning about the value of investments.
Through a progression of questions, you will develop an intuition for the concept. From that base of understanding, you will be able to make novel interpretations of what asset prices imply.
This is a key skill in being able to identify assets that “disagree” with each other — a disconnect that may signal opportunity.
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<aside> 🧗🏾 A step-by-step progression
Imagine a game that pays:
The coin will be flipped immediately. 1. What is the expected value of the game? - Answer 2. How much would you pay to play this game? - Answer
Let’s add an obstacle.
Suppose you have to pay to play today, but the coin won’t be flipped for a year. Your money will be held in escrow.
It helps to imagine that the amount of money is meaningful but if you lost it you’d be ok. Without peering into your soul, I’ll just pick something arbitrary. Let’s say you’re making $100k/yr of net income and heads pays out $1,000 and tails nothing.
[Look at you noticing how we switched the word from “play” to “invest”.]
<aside> 🏕️ Risk Neutral Probability
It may have felt like an easy stroll so far but I have great news — you’ve climbed to base camp. You’ve discovered a foundational investing concept called risk-neutral probability.
You might not recognize it just yet, but let’s recap what you’ve done:
Let’s conjugate that statement differently to internalize an action that smart investors constantly do:
You discount possible outcomes to present value using a riskless rate.
In other words — we are answering the previous questions in reverse! By observing the price being paid today relative to the present value of the outcomes, we are solving for the probability of getting heads.
This is important because most propositions we care about are not coin flips where we know the probabilities.
We need to listen to the probabilities that the prices whisper.
<aside> 🧠 Advanced Topic: Compute risk-neutral probabilities from a binomial tree
In the Practice above you computed your first risk-neutral probabilities.
Believe it or not, that practice is the basis of derivatives pricing. ** We can price options by repeating the Practice above by just adding more steps.
The General Formula to Back Out The Risk-Neutral Probability
<aside> 🔑 The key insight
Risk-neutral probabilities are the probabilities that imply no arbitrage
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<aside> 🧠 Advanced Topic: Replication
You might recognize the topic of risk-neutral probability from the options pricing world. It is a load-bearing concept underpinning replication.
Replication is the foundation of no-arbitrage derivatives pricing.
A comprehensive walk-through of option replication and Black-Scholes is out of scope for this post (and also not playing to my strengths as a teacher here). However, you are about to learn:
Continue:
Real World vs Risk-Neutral Worlds
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<aside> 🧽 Risk absorption
If you ventured into the advanced replication topics, you learned a neat bit of financial theory:
If there is an arbitrage that allows you to risklessly replicate another asset or derivative, then you should not expect to earn more than the risk-free rate. If you could then someone would buy the mispriced asset, offset its risk via replication, and earn more than the risk-free rate for a portfolio that is neutralized from risk.
That idea is an instance of a broader principle:
A proposition that has no risk, should not earn more than the risk-free rate.
Replication is not the only way to de-risk. There are 2 major ways that you are already familiar with.
We can demonstrate them by staying with simple examples.
Ways to absorb risk
The key takeaway:
The more risk you can absorb, the closer your bid approaches the risk-neutral (ie arbitrage-free) price.
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<aside> 🌐 Tying It Altogether: Applications to Investing
<aside> ⏮️ Recap
<aside> 🏗️ Why did we slowly build all this theoretical scaffolding?
The 2 broad categories I generally file reasons under:
<aside> 🗝️ If you remember nothing else…
“No risk no premium” does not imply the converse — that risk will earn you a premium. There are strong reasons to believe idiosyncratic risk will not earn you a proportional return.
Why?
Markets are evolutionary systems. There are invisible forces that stimulate survivors to adapt. Those adaptations are unrelenting pressures that push markets closer to theory. You might even think of efficiency as the removal of frictions (and sources of edge) such as opacity or communication costs. As these are removed, the system approaches a theoretical vacuum. It looks more like theory.
The theory of markets is deeper and more fundamental than object-level strategies. In fact, the meta-idea of a “strategy getting priced in” is an example of a deeper-level market phenomenon.
Adaptations that survivors internalize become part of the native intelligence of “risk absorption”. The efficient transfer of risk absorption amongst the willing and able is the very function markets and must be considered by any challengers.
Some of the deepest roots of that distributed intelligence are captured in these ideas:
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