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Interesting that this got voted up. Anyway, here a few notes: 1. Mostly, the goal is not to "price options". There's a liquid market for basic calls/puts, and

by FabHK 8y ago

Interesting that this got voted up. Anyway, here a few notes: 1. Mostly, the goal is not to "price options". There's a liquid market for basic calls/puts, and those prices are used to calibrate a model and then interpolate/extrapolate as well as price more exotic things. So the goal is to "fit the market". 2. Black Scholes is a well defined bijection between a Call price C(K,T) and a BS vol sigma: C(K,T) = BS(K,T,F,df,sigma). However, prices are such that calls at different strike K have different BS vol, thus the vol can't be a description of the underlying stock price. BS is just "the wrong formula to plug in the wrong number (BS vol) to get the right price". In particular, you can't evolve a stock (in Monte Carlo, going forward, or in a PDE, going backward) using BS vol and reprice all options correctly. 2b. But already, BS gives you a means to interpolate and hedge. 3. The next huge step forward was local vol (Dupire), LV. Instead of assuming fixed vol, it assume that the vol is a deterministic function of stock price S and time t. Now you can evolve a stock (in MC or PDE) and reprice vanilla options correctly, by and large. However, two problems remained: 4a. Forward smile. Take prices as they look today, fit a local vol model, and evolve it forward 2 years. You've hit all the 2yr option prices correctly, and you'll hit all the 3yr option prices. However, the 1yr options IN 2 YEARS will look all wrong (in particular, the smile will have decayed unrealistically). 4b. Very short term smile. A gaussian will basically never go more than 3 std devs from its mean, right. So, short term out of the money options should be really worthless. But they aren't, because stock prices in the real world do jump (or move 10 std devs). So, we require enormously high "lognormal" BS or local vols to reproduce observed option prices correctly. 4a. is solved with stochastic vol models, SV. Mix SV and LV and you reprice options perfectly, and go a few years forward, and your forward smile still looks reasonable. 4b. is solved incorporating jumps, JD (jump diffusion). Mix SV, JD, LV and you get a nice model that fits the market, and evolves reasonably. 5. Most exotic products you price have additional features that preclude closed form pricing. If there's path dependency, you often just use Monte Carlo. If there's calculability, you try and use PDEs. If there's both, you have to use advanced methods: either carry state variables with you in the PDE, or use Longstaff-Schwartz like Monte Carlo methods. 6. However, in the last decade or so, after the financial crisis, all the fancy stuff receded in the background, and there was more focus on the basics: rates. Different counter parties have different credit risk, different currencies have different credit, giving rise to cross-currency basis, different LIBOR maturities are at different levels, giving rise to intra-currency basis, etc. All that stuff needs to be captured properly.

keldaris 8y ago
Naive question from a computational physicist with no quant experience - if you forgo closed form pricing anyway, there doesn't seem to be any obstacle to including all of the things you mention in your last paragraph (and more) and just building more sophisticated numerical models to fit the market. There's certainly no shortage of data or computational power at the scale we're talking about. Given that, why has the "fancy stuff" receded in the background?
- FabHK 8y ago
  Two things, really. The models needed to be updated to incorporate all the rates stuff. That takes time. So the focus wasn't so much on innovation on the product front, but on getting all existing models and products to play along with the new reality. (That's a huge undertaking, btw... you need the market data, someone responsible for marking it, put it in the databases, have it flow through the infrastructure, take it into account in the models, etc.) Second, previously there were many fancy products that allowed you to trade, say, vol, mean reversion, correlation, etc. However, trading this rates stuff is fairly straightforward (in particular, you don't need optionality/convexity; linear products (such as forwards or swaps) are enough). Maybe I shouldn't say "recede" - but in the earlier decade the focus was more on fancy exotic products (with huge margins), while in the more recent decade the focus was on simpler products, but modelling them really precisely. (There was also some focus on systems and ops and front-to-back processing.)
seattleeng 8y ago
This is a great post. Im curious if these techniques are widely used in hedge funds/prop shops today. I was an equity options trader many moons ago, and the Chicago prop shop I traded at generally had every trader using a simple vol arb approach. Apart from real time vol calculations per strike in the trading tools using black scholes and historical vol for the equity, nothing more complex was used to execute trades (eg no stochastic vol or jump diffusion). The concept of smiles was understood, so people would generally trade at the money options exclusively. Positions were delta hedged over night, generally at market close. Deep out of the money options were used essentially for gambles. Vega was the main source of edge, and as you can imagine, most traders exited positions once their options drifted in or out of the money since our vol model didn't really account for it.

by FabHK 8y ago

keldaris 8y ago

Naive question from a computational physicist with no quant experience - if you forgo closed form pricing anyway, there doesn't seem to be any obstacle to including all of the things you mention in your last paragraph (and more) and just building more sophisticated numerical models to fit the market. There's certainly no shortage of data or computational power at the scale we're talking about. Given that, why has the "fancy stuff" receded in the background?

FabHK 8y ago
Two things, really. The models needed to be updated to incorporate all the rates stuff. That takes time. So the focus wasn't so much on innovation on the product front, but on getting all existing models and products to play along with the new reality. (That's a huge undertaking, btw... you need the market data, someone responsible for marking it, put it in the databases, have it flow through the infrastructure, take it into account in the models, etc.) Second, previously there were many fancy products that allowed you to trade, say, vol, mean reversion, correlation, etc. However, trading this rates stuff is fairly straightforward (in particular, you don't need optionality/convexity; linear products (such as forwards or swaps) are enough). Maybe I shouldn't say "recede" - but in the earlier decade the focus was more on fancy exotic products (with huge margins), while in the more recent decade the focus was on simpler products, but modelling them really precisely. (There was also some focus on systems and ops and front-to-back processing.)

seattleeng 8y ago

This is a great post. Im curious if these techniques are widely used in hedge funds/prop shops today. I was an equity options trader many moons ago, and the Chicago prop shop I traded at generally had every trader using a simple vol arb approach. Apart from real time vol calculations per strike in the trading tools using black scholes and historical vol for the equity, nothing more complex was used to execute trades (eg no stochastic vol or jump diffusion). The concept of smiles was understood, so people would generally trade at the money options exclusively. Positions were delta hedged over night, generally at market close. Deep out of the money options were used essentially for gambles. Vega was the main source of edge, and as you can imagine, most traders exited positions once their options drifted in or out of the money since our vol model didn't really account for it.