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Path-Dependent Volatility (PDV)

Every model on this site assumes that vol depends on where the price is now -- the current level, maybe the current vol state. Path-Dependent Volatility (Guyon & Lekeufack, 2023) says that is not enough. Vol also depends on where the price has been. A coin that crashed 10% and recovered to 100doesnottradethesameasonethatsatat100 does not trade the same as one that sat at 100 the whole time. The crash-and-recovery coin has elevated implied vol, steeper skew, and wider wings -- because the market remembers the crash.

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Markets have memory

If BTC just had a 15% drawdown, vol stays elevated even after the price recovers. PDV makes vol a function of two things: recent realized vol and recent price trend. That is the entire model. The vol surface shifts in response to the path, not just the current price.

See It in Action​

Toggle between a crash-recovery path and a flat path. Both end at the same price, but produce different vol smiles. Drag the memory slider to see how the lookback window changes the effect.

Path-Dependent Volatility

Price dropped 10% and bounced back. The path memory keeps vol elevated even after the price recovers.
Recent price path
memory window10091Time
Resulting vol smile
34%45%56%OTM PutATMOTM CallCrash & recoveryFlat market
Path memory (lookback window)30 days
1 day (short memory)90 days (long memory)

Toggle between scenarios to see how the same current price produces different smiles based on the recent path. Drag the memory slider to see how the lookback window changes the effect.

How It Works​

1. Two inputs from the price path​

PDV distills the recent price history into two numbers:

Input
What it captures
Trader intuition
Recent realized vol
How much the price has been moving over the lookback window.
You already check this on any vol dashboard. High recent RV = elevated IV.
Recent trend
Net price change over the lookback window (up or down).
A big down-move steepens the skew. A rally flattens it. You see this daily.

2. Vol is a function of these two inputs​

The model says: implied vol at any strike is a function of the current spot plus these two path summaries. No stochastic vol state variable, no fractional calculus, no hidden Markov chain. Just: "where is the price, how much has it been moving, and which direction?"

3. Rough vol behavior without rough models​

This setup reproduces several "hard" phenomena:

  • Vol clustering -- high vol begets high vol, because recent realized vol stays elevated
  • Leverage effect -- down-moves increase vol more than up-moves, because the trend input skews the function. Produces skew that varies with recent returns.
  • Rough-vol-like scaling -- the apparent roughness of vol paths emerges naturally from path dependence, without needing fractional Brownian motion
  • Joint SPX/VIX calibration -- the model calibrates to both index options and VIX options simultaneously, which most models cannot do
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Why this matters for crypto

Crypto markets have extreme path dependence. After a liquidation cascade, vol stays elevated for days even if the price recovers. After a long grind up, vol compresses. PDV captures this directly. Traditional models treat every 60k BTC the same -- PDV treats "60k after a crash from 70k" differently from "60k after a rally from 50k." That distinction matters for pricing and delta hedging.

PDV vs. Other Models​

Feature
Heston / SABR
Rough Bergomi
PDV
Vol depends on
Current state only
Full vol history (fractional)
Recent RV + trend
Path memory
None (Markov)
Infinite (power law)
Finite (lookback window)
Complexity
Low
High (non-Markov)
Low
Joint SPX/VIX fit
Poor
Moderate
Good
Vol clustering
Partial
Yes
Yes
Simulation speed
Fast
Slow
Fast (Markov)
Maturity
Decades
~10 years
New (2023)

Strengths and Limitations​

Strength
What it means for you
Intuitive inputs
Recent realized vol and trend are things every trader watches. No abstract state variables.
Markov (fast to simulate)
Despite capturing path effects, the model is Markov in (S, realized vol, trend). Monte Carlo is standard speed.
Rough vol without rough math
Reproduces the scaling properties of rough vol models without fractional calculus or non-Markov simulation.
Joint calibration
Calibrates to both vanilla options and vol-of-vol products (VIX options, vol swaps) simultaneously.
Limitation
What it means for you
Lookback window choice
The memory parameter matters and must be chosen or fitted. Different windows produce different surfaces.
No closed-form pricing
Option prices require Monte Carlo simulation. Slower than Heston or SABR closed-form approximations.
New (2023)
Limited production experience. Edge cases and failure modes not fully documented.
Needs price history
Cannot price options on a brand-new token with no trading history. Needs enough data to compute realized vol and trend.
💡
Simplest path-dependent vol model

PDV uses recent realized vol and recent trend to explain smile dynamics that stochastic vol models miss. Reproduces rough vol, vol clustering, and leverage effects without exotic math. Vega under PDV differs from Black-Scholes because the path state changes the smile shape. The tradeoff: new, requires Monte Carlo, and depends on the choice of lookback window.

Equation Explorer​

Convert between implied vol, total variance, log-moneyness, and option prices.

Equation Explorer

w = σ2 × Ttotal variance = IV2 × time
%
The implied volatility
days
Calendar days to expiration
Total Variance (w)
0.022225
Annualized Variance (σ²)
0.2704
Round-trip IV
52.00%
Total variance is what SVI and other models fit. It scales with time, so a 50% vol for 30 days has less total variance than 50% vol for 90 days.

Test your understanding before moving on.

Q: BTC is at $65k. It got here by crashing from $72k and recovering. Under PDV, how does the vol surface differ from a scenario where BTC drifted up slowly from $60k?
Q: Why does PDV reproduce rough-vol-like behavior without using fractional Brownian motion?
Q: You are choosing a lookback window for PDV on ETH options. What are the tradeoffs between 7 days and 60 days?

💡 Tip: Try answering each question yourself before revealing the answer.

See also:

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