LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Rough volatility models need the whole past to take the next step, so simulation is slow and the usual pricing shortcuts break. Markovian lifting replaces that memory with a finite stack of mean-reverting factors. Each factor's next step depends only on where it is now, plus a new shock. The path dependence is approximated, not stored.
Why it matters
Brownian motion has the Markov property. The next step depends on the current point, not on the trail behind it, so simulation stays cheap and many option calculations stay available.
Some features fit the data better when they remember. Volatility is the example. Since Gatheral's 2014 paper, roughness has been a live topic, and Heston has a rough extension. The memory sits in a Volterra kernel, so the next variance move depends on the path. The affine structure that makes transforms fast goes with it. Putting the entire history into the state keeps the memory and does not buy the speed back.
The idea in plain words
You can enlarge a state without keeping the whole trail. Package the last two points as one vector and, conditional on that vector, older points drop out. The lecture also notes you can enlarge across kinds of path: a Brownian piece and a mean-reverting piece can share a state if each piece stays Markov on its own.
Rough Heston puts the memory in a kernel. The lecture writes it as time to the power alpha minus one, over the gamma function at alpha. Alpha is the Hurst parameter plus one half, and Hurst sits strictly between zero and one. A power does not factor across a time step. Shift the clock and the kernel is not a function of the step alone times a function of the time already elapsed. The increment still sees the whole integral.
Exponentials do factor: a sum inside the exponent becomes a product of two terms. That split lets an Ornstein-Uhlenbeck factor step forward from its current value plus a new shock. A Laplace-style rewrite turns the power law into an integral of those exponentials, an infinite family of mean-reverting factors. Euler's reflection formula then replaces the product of the two gamma values with the sine of pi alpha, divided by pi, which is cheaper to evaluate.
The integral is cut into n bins. The lecture spaces the speeds in a geometric progression, a constant to the power i, because the kernel blows up as time approaches zero and some factors must revert extremely fast. Inside each bin the exponential is frozen at the bin's speed and pulled out. What remains integrates to a weight. The kernel is then approximately the sum of weight times e to the minus speed times time.
Swap the sum and the integral in the variance equation, which linearity allows. Each term is an auxiliary factor. Apply Ito's formula and the lecture's differential is a shock minus speed times the factor's current level, times dt. Nothing in that increment reaches back to time zero. N such factors replace one infinite-dimensional Volterra process. The lecture says the scheme is affine, fast Fourier methods for vanilla prices come back, and simulation cost moves from N squared to linear. That is the point of calibrating a rough model to liquid options.
How to use it
Keep the fractional kernel if roughness is the point. Rewrite the power as an integral of exponentials, discretize on the geometric grid, and step the factors like ordinary mean-reverting shocks. More factors track the theoretical covariance more closely, and the match is still an approximation.
The lecture cites a former colleague's quadratic rough Heston model for zero-day options, where the skew is steep. An extra boosting term removes the Fourier shortcut. The factors still make the simulation cheaper.
What usually goes wrong
Calling the entire path a state reproduces the memory and does not buy a Markov step. A uniform grid then hits the blow-up at time zero. The geometric speeds exist to cover the fast factors. The power-law kernel itself does not split across a time step, which is why the integral rewrite comes first.
Questions people ask
What does rough mean here?
The lecture ties it to Gatheral's 2014 claim and to a power-law kernel whose exponent depends on a Hurst parameter strictly between zero and one. The process needs its past, not only its current value.
Why do exponentials restore the Markov property?
Because they split across a time step. After the split, the next value of each factor is the current value, decayed, plus a new shock. The power-law kernel does not split that way.
What do you get back computationally?
A stack of N affine factors, Fourier pricing for vanillas, and simulation the lecture describes as linear rather than N squared. The kernel is approximated, and a model with an extra boosting term can lose the Fourier piece while keeping the faster simulation.