<aside> ℹ️ About
Active traders and investors buy and sell bonds, stocks, and commodities because they have a view that the asset is cheap or expensive. Sometimes they will use options to express their bets. They are experts in discerning value in these “underlying” securities.
Derivative traders who make up a tiny overall portion of the investing landscape are not experts in the underlying, but instead, they are focused on pricing the securities that “derive” their value from the price behavior of the underlying. Futures, forwards, ETFs, and options are common examples of derivative securities.
The derivatives trader doesn’t hold a high conviction view on the direction of the underlying. Instead, the trader pencils out how much it would cost to “replicate” the payoffs of the derivative in various states of the world using a mix of cash (and interest) + the underlying securities. This entire framework is referred to as arbitrage pricing.
Arbitrage pricing, unlike other pricing frameworks such as CAPM (capital asset pricing model) starts with The Law of One Price. This law is not like a law that comes from Congress. It’s the idea that identical future payoffs should have the same cost today regardless of the portfolio that generated them. It is the basis of option pricing theory.
This document uses simulations to provide intuition for how realized volatility drives the p/l of delta-hedged position when hedging is done on a regular but discrete interval.
It answers questions of the variety:
“If I buy an ATM option for 20% and the stock realizes 25% what does my p/l look like?”
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<aside> ⌨️ Setting up the simulation
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<aside> 📊 Results
Zooming in on individual years
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<aside> 💡 Concluding thoughts
This exercise used many assumptions. Some of them are quite benign in a real-world setting. For example, we assumed a constant RFR. But if you are an options trader who gets hosed by rho risk swamping your book, you have bigger issues.
There are 3 assumptions embedded in this work that should dominate your thinking as you translate the lessons to reality.
Even if you don’t care about mark-to-market because you are dead-set on scalping the option to expiry and letting RV play out…the IV determines the delta, gamma, and theta. There is a terrific discussion in Financial Hacking about the biases of hedging on IVs that are both above or below the volatility that is eventually realized.
Real-world stock distributions differ from the assumptions of Black-Scholes. A lognormally distributed asset is a favorite to go down in price but has positive skew. This is the opposite of empirical observations in many markets! Option skews created butterfly prices which imply distributions that should match the real-world better (and if they didn’t do a fairly good job of that there would be easy alpha everywhere).
All of the stats generated in these simulations are downstream of a Brownian motion diffusion process.
The cost to hedge. The funding spread between your long and short rates on stock. The cost to trade the options themselves. In these simulations a 1-year option needs to be mispriced by 5% (ie you make $.50 edge on a $10 option) to have a 1 Sharpe ratio trade assuming NO TRADING OR FUNDING SPREADS.
Despite these deficiencies in the analysis, there’s a lot of basic intuition for delta-neutral vol trading buried within.
As a matter of personal taste, I think the most useful revelation is in the appreciation of path and its focal question: Did the moves that diverged from implied happen when my gamma was big or small?
The noise in vanilla options trading in many ways make it more complex than variance swaps which were invented to smooth the gamma profile of a vol position over both price range and time. The irreducible problem with variance swaps is their liquidity providers, in replicating the swap payoffs, cannot themselves escape the path dependence of their more liquid building blocks.
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We will ship the monte carlo simulator used to generate these studies on moontower.ai so you can explore the paths of hedged option positions for various expiries, strikes, and vol levels yourself!