**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.#### Why options are a special type of derivative

Option pricing theory asserts that an active trading strategy in the shares of an asset XYZ can replicate the payoff to an option on XYZ if we know:

- the expiry date of the option

- the strike price

- the current price of XYZ which determines the option’s
*moneyness*or distance from the strike

- the interest and dividends from now until expiry

- the volatility or size of X’s price fluctuations

Our confidence in these inputs varies.

Extremely confident

The strike and expiry are the only inputs we are 100% certain of.

Highly Confident

- We can infer interest rates from the yield curve. If the stock becomes hard to borrow, we have less confidence in the proper discount rate.

- We can estimate dividends based on past behavior and company guidance.

- In normal market conditions, we are highly confident in these inputs. Still, our error bars can widen dramatically during market-wide stress or if there is idiosyncratic news in XYZ we are trading.

- We can estimate the
**current**price by looking at the stock’s bid/ask spread.

Medium to extremely

**low**confidence- We have extremely low confidence in the price of XYZ over the life of the option.

- We have low confidence in the magnitude of future moves in XYZ. The disagreements about what the volatility can be give rise to a very specific form of derivatives trader:

#### The Volatility Trader

The volatility trader, like other derivatives traders (such as so-called Delta One traders), doesn’t make a living based on views on the underlying. Volatility traders have opinions about, well, the volatility. This means a view of how the asset moves, how the option surface changes, and ultimately what it costs to replicate various options or spread options against each other to “leg” into attractive propositions based on an understanding of arbitrage bounds.

While the price of an option is sensitive to all the inputs mentioned above, the volatility trader typically only holds a differentiated opinion on the volatility. When the trader buys or sells the option they are inheriting a sensitivity to drivers they have no opinion about. By analogy, the pharma investor who buys a drug company is exposed to the economy at large, a system that they don’t have a refined view of. The pharma investor may “hedge” out extraneous exposures as best they can since they want to live and die by their drug company analysis, not the wider ripples of national enterprise.

Similarly, the volatility trader wants to sterilize the impact of the non-volatility inputs.

Those other non-volatility inputs tend to be stable over the life of the option — except for the stock price itself which affects the option’s moneyness. The option trader wants to hedge against “directional” moves in the stock.

**They don’t care if the stock goes up or down, they only care about how the magnitude of the moves compares to the magnitude implied in the option price.**#### Dynamic Hedging

The volatility trader sterilizes the sensitivity of the option to direction by “delta hedging”. They want to be delta-neutral just as a long-short stock picker tries to stay “market-neutral”. The analogy is strong. A market-neutral trader is out of balance as soon as the stocks in their portfolio start moving, the “delta-neutral” volatility trader is out of balance as soon as the stock ticks and the option moneyness changes.

A major assumption in the Black-Scholes model is that you can

*hedge the delta. Unfortunately, nobody told the real-world to comply. It is prohibitively expensive to hedge the delta every minute or second, to say nothing of doing so on infinitely small time slices.***continuously**Option traders compromise their desire to “isolate the vol” by staying delta-neutral with the cost of frequent hedging. Said otherwise, the cost of this noise introduced to their p/l is less than the direct costs of minimizing directional risk.

So what do volatility traders do in practice?

They hedge at discrete intervals. This is called “dynamic hedging”.

So what schedule do they hedge on?

Well, you might as well have asked how many religions exist. It’s a giant topic. Taleb even published a book called

*.***Dynamic Hedging**While there are many ways to craft a hedging schedule the key is the tradeoff at the core of the problem:

**We want to isolate our source of edge…but hedging unwanted exposure is always a cost.**

Our tolerance for risk is weighed against the size of our edge and how often we get to take at-bats.

See

*If You Make Money Every Day, You’re Not Maximizing**(A Treatise on Hedging)*#### Getting a feel for the signal and the noise

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?”

**Setting up the simulation**

**I. Simulate the daily logreturns of the stock**

**I. Simulate the daily logreturns of the stock**

The returns are a function of expected return (in the case of B-S, it’s the risk-free-rate) and volatility

**II. Price a European-style option & compute Greeks**

**II. Price a European-style option & compute Greeks**

**We start with an ATM call with the following inputs:**

Initial stock price: $100
Type:

*Call*Strike: 100*RFR and dividend yield = 0% DTE = 1 year Implied volatility =***25%****Option details on T₀**

100 strike call value: $9.95
delta:

**.55**vega:**$.396**gamma: $.016 theta: -$.014**III. Stepping through the strategy**

**III. Stepping through the strategy**

Basics

- Seed a realized vol that will transpire during the 1 year holding period. Each day will draw a random return centered around that realized volatility

- The implied vol of the option of 25% will not change

- Run batches of 500 simulations each. Each run in the batch will be seeded with the same realized vol. Some of the realized vols will be less than 25% and some will be greater than 25%.

Batches:

RV:

*10%*RV: 2

*0%*RV: 25

*%*RV: 26

*%*RV: 27

*%*RV: 30

*%*RV: 40

*%*Day 1

Buy the 1-year ATM call at its theoretical value priced with

**25%**volatility and hedge by shorting the stock in accordance with the option delta (.55).Daily

- Reprice the option with 1 day less to expiration

- Record share and option P/L based on that day’s opening position and the change in stock price

**Note: The simulation assumes the IV on the strike remains constant at 25%!**

- Adjust the hedge by selling more stock or buying some back to remain neutral to the option’s new delta

**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.

- IV is not constant.
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.

- Brownian motion is not reality 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.

- Costs are real 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.

#### 🚀Coming soon…

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!