21. Stochastic Differential Equations

MIT 18.S096 Topics in Mathematics with Applications in Finance, Fall 2013
Instructor: Choongbum Lee

This lecture covers the topic of stochastic differential equations, linking probability theory with ordinary and partial differential equations.

00:21 - Stochastic Differential Equations
21:15 - Numerical methods
42:27 - Heat Equation

License: Creative Commons BY-NC-SA

[SeikoVanPaath]

Timestamps:
00:21 Stochastic Differential Equations
21:15 Numerical methods
42:27 Heat Equation

[Boringpenguin]

6:54 Just a side note, for GBM, there is actually another easy way to solve it.
If we consider f(t, X_t) = ln(X_t), then we can apply Ito's lemma and you will see that all the "X_t" will be isolated on the LHS, which means we can now integrate both sides and get the exact same answer as shown in the video.

[dayam]

His voice volume seems to follow Brownian motion.

[elsa]

To solve the Ornstein–Uhlenbeck (OU) SDE: instead of assuming X_t = f(B_t, t), which works for the GMB but not for the OU, you may try:
X_t = f(B_t, t, \int_0^t g(s) dB_s).
Denoting the variables as f(z, t, y) you will then find, by matching terms as you did for the GMB, that f_z must be constant (hence, f_zz must be zero, and then f_z = 0 turns out to be feasible), then, you will find, using the dt term,
f = exp(-\mu t) (y + x_0) where y = \int_0^t g(s) dB_s. After this, you can find g(s) by matching the dB_t term. Done 😊

[wangchong]

The flow of the lecture is wonderful. He covers several of the most important topics in the finMath world. Does he have some other lectures available online?

[Boringpenguin]

38:04 40:26 Just to extend their explanation a bit more, tree method is heavily used to price path-dependent options, as they often don't have close-form solution (depending both on the product and the model), and by its nature the whole price path might be required, unlike the vanilla European options.

A good example would be American style put, since it is well-known that there is no analytical solution for its price, we have no choice but to resort to some sort of tree method (or Monte Carlo simulation) to do the pricing.

[rohankumar]

There are a few comments saying they are not happy with the kind of difficulty involved in this lecture. I would recommend them to go over some of the previous videos of the course to understand what's exactly happening.

[熊育霆]

Thank you, MIT
Thank you, Dr. Choongbum

[Boringpenguin]

33:09 I think there is a crucial link between option pricing and finite difference method that Dr. Lee didn't explicitly state.
Remember your good old Black-Scholes equation? That's a PDE that governs the option price! You can definitely use finite difference method on the Black-Scholes PDE to price options. Although it's a bit silly to do so because we can often derive the closed-form pricing formula for derivatives (even some exotic ones) under the Black-Scholes model. But for some really weird path-dependent ones, this is for sure one way to go.

However, the real power of this technique lies in one observation: Even for models that are more complex than Black-Scholes, we can often write down a PDE that governs the option prices under those models. More remarkably, those PDEs are often of the same form as the Black-Scholes PDE. For example, there is a similar PDE under the Heston model, which is one of the many stochastic volatility models out there, and the channel quantpie has a video about the derivation of that PDE.

What all of this means is that you can still use finite difference method under these advanced models to price exotic derivatives, which is pretty cool imo.

[AndyPayne]

fyi, you can do the finite difference method in microsoft excel just by setting your boundary conditions then 'averaging' neighboring cells. Don't even need to install matlab or octave for this :)

[sjn]

1:35 Whenever a professor says these are not easy problems, I breathe a sigh of relief.

[maxwellsdaemon]

51:50, if this gaussian is normalized to one (integral from -infinity to +infinity is 1), then instead of "4" in the exponent, it should be "2", or, within the square root prefactor, it should be "4" instead of "2"

[ShadabAli-dwsw]

Please, Make a lecture on Existence and uniqueness solution of SDE.

[pedroalonsocazorlasaravia]

1:20 The second equation is the true equation, not the goal. The first one is literally an abuse of notation since dB(t) is nonsense.

[matthewchuang]

have some feeling on what the process would look like. And try this try that, something will succeed.

[ImWriiight]

12:42 I’m confused how f(t, B(t)) became an f(t, x) and became completely deterministic, with no B(t) terms

[RollenJokers]

cant tell if my screen was glitch on your note or if it was me. this is very satisfying stuff

[stevenli]

The guess part can use taylor series to explain it, but the thing of the no terms would be super confusing if he explains like this. Overall I like the course very much, but it could be better.

[drdca]

Is the integral from 0 to t of diracdelta(s - s_0) dW_s equal to deterministic 0 if t < s_0 and equal to the standard normal distribution if t > s_0 ?

[cucciolo]

the introduction of text to video 101