[throwlaplace]

Holy shit this is literally gold. I just went through this paper ("porting" to python)

https://epubs.siam.org/doi/abs/10.1137/s0036144500378302

I was planning on moving on to his book but now I'll just read this. Kudos to you Nicola for putting in the work and then releasing. If I were you I would typeset this and approach publishers because I bet you would sell many copies.

edit: i'll take this opportunity to poll the audience: does anyone know a good explanation of girsanov? i know a fair amount of measure theory but i'm still looking for a good, detailed, practical explanation of its use in the context of changing to risk-free measure for black-scholes.

[backlash875]

Are you looking for a proof of Girsanov's theorem or an explanation of how it is used to price? A good reference is Oksendal [0], but it's still quite tough going. As for how it's used, a good reference is Shreve's second volume [1], which also contains a proof of Girsanov's theorem. Joshi's book [2] is a little bit lighter on the mathematical rigour.

A quick explanation of how it's used:

Taking the stochastic differential equation for geometric Brownian motion, apply Girsanov's theorem to change measure via a drift change such that we now have a discounted stock price that is a martingale. The discounted stock price is the stock price divided by a short term bond or cash account asset. In this new measure the discounted short term bond/cash account asset is also trivially a martingale since it's being divided by itself. So we have that our two key assets (discounted) are martingales. We then define the time zero price of the option (divided by the time zero price of the bond/cash asset) to be the discounted expected value of its value at maturity in this newly constructed measure. By construction this discounted option price is a martingale and we now have three assets that are all martingales which implies there is no arbitrage possible. With this option price, called the "risk neutral" price, no arbitrage is possible under our newly constructed measure, but, because the original measure is an equivalent measure no arbitrage is possible here in the "real world" either and so this is our actual price.

I appreciate there are a few steps here that seem like a bit of a leap. It took me a while to appreciate them. The key things to appreciate are:

How everything being a martingale implies a lack of arbitrage. Girsanov allows you to make your (discounted) underlying a martingale.

How you can then just make the option price a martingale by construction. And then how lack of arbitrage under one measure means lack of arbitrage in any equivalent measure.

The discounting can also be a little confusing, but it's really just incorporating the time value of money into the calculations.

[0] Oksendal B . Stochastic Differential Equations. [1] Shreve S E. Stochastic Calculus for Finance II. [2] Joshi M S. The Concepts and Practice of Mathematical Finance

[hippich]

From my (a very layman) understanding, the price of the security is stochastic can be assumed only during a very short period, which does not include major moves. If this is correct, does it mean all these models are useful only for market makers earning money from the spread, or they can be useful for retail traders?

[backlash875]

There are other models that take into account jumps in the prices/market. As for retail traders using these models, I think it is not recommended and not practical.

[hippich]

Any names/links/etc to learn more about such models?

[backlash875]

The Heston model has stochastic volatility. You can add jumps to this: https://en.wikipedia.org/wiki/Stochastic_volatility_jump

[euroPoor]

Now THAT’s gold. Thanks for this nice explanation.

[soVeryTired]

I wrote up a simplified explanation of Girsanov's theorem once. You can get it as the limit of the ratio of two N dimensional Gaussian densities as N goes to infinity. I'll see if I can dig out what I wrote up.

[throwlaplace]

i can't edit so i have to reply instead:

for anyone interested these two explanations i found when googling around again today are very good:

https://www.youtube.com/watch?v=omrnRshzHhw

https://www.youtube.com/watch?v=vKjbau2Hlrs

[Cantaro86]

Thanks a lot!

[mjfl]

cool paper, thanks