Monte-Carlo Simulation 101

One of the most powerful tools in any practitioner's toolbox is MC simulation. Every trader/quant that I knows has some kind of MC sim code ready to use for pricing or analysis.

Why is MC such an important piece in derivatives pricing?
The true beauty about MC sim is that we can use it to price literally any type of product we want : vanilla options, 1st/2nd/Nth order exotics, baskets, etc... . you name it, MC can price it, why? because we simulate future price paths with very few parameters...
define the distribution shape (let's face it, we always default to normal distribution), assume drift/volatility, set horizon, and tell me how many paths you want to simulate and I can generate it in lightspeed...

Sound like a no brainer right? this is where it gets bit tricky..
We all know that the underlying process in derivatives pricing is a stochastic process (Brownian Motion to be exact). In short, stochastic process is a process where increments (log-price) are random and independent of each other (i.i.d). This is the core of Brownian Motion
Why is that so important? because it means that our process has NO memory, which means that the future price path is independent of the past (aka Markov process). This allows us to price/simulate the future price from any starting point along the path. Cool right?
Just think what would have happen if this was not the case... think how hard it would to price options on VIX where you need to use the entire history of VIX data from the big bang until today... not very practical imo...

But there is a flip-side to using such process...
Over last few years more and more studies have shown that price dynamic across different asset do carry some kind of long/short memory, which contradicts the Brownian Motion dynamic (which is in the heart of MC simulation). Many have shown highly positive/negative autocorrelation
of log-price (increments), which we usually call them as trend/mean reversion. So how do we model this kind of dynamic? We turn to Fractional Brownian Motion (fBm)

Although sounding quite intimidating, fBm is simply a general case of the "normal" brownian motion
In BM we assume zero-autocorrelation between increments, so we set the autocorrelation parameter (let's call it Hurst exponent) to 0.5 (just think about it as the 0 point).
But what if we want to play around with that and see how price behaves under serious mean-reversion or trend? no problem MC sim for the rescue...

You see, we can use it to simulate pretty much any price dynamic we want (given very few parameters...)
By now you are probably asking "So why are we still using BM and not fBm, if we can get much more flexible with fBm?"

The main problem with fBm is that it lack one very important property that our good old BM has - markovianity . Because our log-prices are autocorrelated
we cannot treat them as independent... so why am I going on and on about fBm, because great minds in quant finance have come up with markovian approximation (which means that with few tweaks we can use fBm in MC sim)

If you are interested check out:…
I would be bold and say that the groundbreaking work in the field of rough volatility is going to change the way we model volatility (both for pricing and trading).

I think that vol traders should definitely take this research field seriously (especially with crypto modeling)

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More from @VolQuant

1 Jun
I frequently get asked by novice traders and quants about essential reading/papers that I would recommend.

While most option/derivatives books have already been covered by others, I thought of comprising a list of "Essential Quants" for volatility trading
Few thing before we begin:

1. the list is broken down into Quant All-Star team (equivalent of best pound-for-found in boxing), and major league quants (essential reading, but more topic specific)

2. the list is in random order. Everyone on this list influenced my work
and way of thinking about vol trading and pricing, so they are well deserved of acknowledgement

3. I left out the OGs of Quant Finance - Bachelier, Black & Scholes

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Very much like in boxing, experimenting different strategies/models/market helps you develop arsenal
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I'm an orthodox fighter, so my default stance a leading left hand, but I do train at least twice/week in a southpaw stance to be able working both stances. This helps me become more versatile as a fighter
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