Can the method of Separation of Variables be used solve the Black Scholes PDE?

1. If not, why not, where does it break down?

2. If yes, how do you compute a solution?

It's an interesting quizzie. There's a lot of maths in there.

1. If not, why not, where does it break down?

2. If yes, how do you compute a solution?

It's an interesting quizzie. There's a lot of maths in there.

2. Yes. Start with the easy driftless BS case, where [$]x = \log S[$] and [$]\mu = r - \sigma^2/2 = 0[$]. Then, your question amounts to asking: can we use separation of variables to solve the heat equation for [$]u(t,x)[$] on [$]x \in (-\infty,\infty)[$] with [$]u(0,x) = f(x)[$] given?

Ans: sure, tedious, but start by truncating the problem to [$]x \in (-L,L)[$] with [$]u=0[$] at boundaries. Use standard separation of variables. Following wikipedia, for example, the result has the form:

[$]u_L(t,x) = \int_{-L}^L G_L(t,x,y) f(y) \, dy[$],

where [$]G_L[$] is given by an infinite sum. Then, prove that, as [$]L \rightarrow \infty[$],

[$]G_L \rightarrow \frac{e^{-(x-y)^2/(2 \sigma^2 t)}}{\sqrt{2 \pi \sigma^2 t}}[$].

Surely doable and involves many of the same formulas already found in this thread.

Finally, generalize to non-zero drift, and you're done.

Statistics: Posted by Alan — Today, 5:05 am

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If you go to the continuous spectrum, you might have a payoff $V(S,T)$ which you can write as an inverse Mellin transform

[$]V(S,T)=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}S^{-k}v(k)dk[$]

then for [$]t<T[$] you'd have an exponential in time in there

[$]V(S,t)=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\exp(a(k)(T-t))S^{-k}v(k)dk[$]

Statistics: Posted by ppauper — Yesterday, 4:39 pm

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1. If not, why not, where does it break down?

2. If yes, how do you compute a solution?

It's an interesting quizzie. There's a lot of maths in there.

Statistics: Posted by Cuchulainn — Yesterday, 12:51 pm

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I perfectly agree with your comment on understanding measure theory/ functional analysis ASAP because time isn't our brain's friend.

On the other hand, Alan and bearish are more inclined towards option 2... for reasons I can agree with..

I am still caught in between these two paths...

Statistics: Posted by galvinator — November 13th, 2018, 4:05 pm

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I did about 4 courses on MT from 2nd year one and I hated them. Functional Analysis is much more important.

Statistics: Posted by Cuchulainn — November 13th, 2018, 2:32 pm

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Problem

Let’s assume that there is a market of zero-coupon risk-free bonds.

Price of a bond at time t with maturity at T and face=1 is P(t,T).

Bond’s interest rate is R(t,T) = (1/P(t,T)-1)/(T-t).

Forward interest rate from T_1 to T_2 at t is F(t,T_1,T_2) = (P(t,T_1)/P(t,T_2) – 1)/(T_2 – T_1).

There is a tradable derivative “f” that pays R(T_1, T_2) (interest rate R fixed at T_1) at T_3>T_2.

Question

What is value of “f” at t=0?

(The question is trivial for the case T_3=T_2: f(0) = P(0,T_2)*F(0,T_1,T_2) – “discounted forward rate”)

Actually, I asked this question on another forum and got the following answer from bearish:

So, now I am curious what models do people usually use and how exactly they calculate the adjustment?As long as T_3 is different from T_2 you need a model to calculate what is usually (if somewhat sloppily) referred to as the convexity adjustment. This adjustment would be applied to the zero volatility value of P(0,T_3)*F(0,T_1,T_2).

You might have received a quicker response if posting in the Student forum.

Statistics: Posted by sol2 — November 13th, 2018, 2:27 pm

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@Cuchulainn:

One module I excluded from that long list is, in fact, a module on numerical PDEs.

The contents include: various numerical integration schemes for solving ordinary differential equations, and (2) finite difference methods for solving various linear partial differential equations. Major topics: (ODE) One-step and linear multistep methods, Runge-Kutta methods, A-stability, convergence; (PDE) Difference calculus, finite difference methods for initial value problems, boundary value problems, and initial-boundary value problems, consistency, stability analysis via von Neumann method and matrix method, convergence, Lax Equivalence Theorem.

Looks pretty good. Convection-diffusion PDE is what universities leave out One module I excluded from that long list is, in fact, a module on numerical PDEs.

The contents include: various numerical integration schemes for solving ordinary differential equations, and (2) finite difference methods for solving various linear partial differential equations. Major topics: (ODE) One-step and linear multistep methods, Runge-Kutta methods, A-stability, convergence; (PDE) Difference calculus, finite difference methods for initial value problems, boundary value problems, and initial-boundary value problems, consistency, stability analysis via von Neumann method and matrix method, convergence, Lax Equivalence Theorem.

Here is an introductory article on FDM for such an equation.

- Duffy.pdf

Statistics: Posted by Cuchulainn — November 13th, 2018, 2:13 pm

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Statistics: Posted by Cuchulainn — November 13th, 2018, 2:12 pm

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Other nice things:

Mathematica/MATLAB/R/LaTeX/WordPress

Statistics: Posted by Alan — November 9th, 2018, 6:11 am

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One module I excluded from that long list is, in fact, a module on numerical PDEs.

The contents include: various numerical integration schemes for solving ordinary differential equations, and (2) finite difference methods for solving various linear partial differential equations. Major topics: (ODE) One-step and linear multistep methods, Runge-Kutta methods, A-stability, convergence; (PDE) Difference calculus, finite difference methods for initial value problems, boundary value problems, and initial-boundary value problems, consistency, stability analysis via von Neumann method and matrix method, convergence, Lax Equivalence Theorem.

Statistics: Posted by galvinator — November 8th, 2018, 4:47 pm

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13. why numerical ODEs?

PDE?

C++/C#/Python

Statistics: Posted by Cuchulainn — November 8th, 2018, 10:28 am

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May I seek your opinion on which electives should I take over the course of my 4-year undergraduate degree?

TLDR: Should I pursue the measure-theoretic probability path or the Statistics/Computer Science path?

Some context about myself:

I am currently a freshman attending a university in Singapore and my major is Quantitative Finance (QF). (Some of you may irk at the sight of a bachelor's degree in QF, I get it...)

A detailed list of my major requirements (the modules I MUST/HAVE/AM/WILL take) includes:

- Calculus (the contents are similar to the United States' Calculus 1 content, plus a tiny bit on Real Analysis)
- Linear Algebra 1 (the usual undergraduate linear algebra stuff; eigenvalues, diagonalization, rank, linear transformation between Euclidean spaces)
- Accounting
- Python programming
- A module on finance which covers: financial statement analysis, long-term financial planning, time value of money, risk and return analysis, capital budgeting methods and applications, common stock valuation, bond valuation, short-term management and financing.
- Multivariable Calculus (the equivalent of US' Calculus 3)
- Real Analysis. Major topics: Basic properties of real numbers, supremum and infimum, completeness axiom. Sequences, limits, monotone convergence theorem, Bolzano-Weierstrass theorem, Cauchy's criterion for convergence. Infinite series, Cauchy's criteria, absolute and conditional convergence, tests for convergence. Limits of functions, fundamental limit theorems, one-sided limits, limits at infinity, monotone functions. Continuity of functions, intermediate-value theorem, extreme-value theorem, inverse functions
- Numerical Analysis
- Probability (calculus-based probability)
- Investment Instruments: Theory and Computation which focuses on the basic paradigms of modern financial investment theory, to provide a foundation for analysing risks in financial markets and to study the pricing of financial securities. Topics will include the pricing of forward and futures contracts, swaps, interest rate and currency derivatives, hedging of risk exposures using these instruments, option trading strategies and value-at-risk computation for core financial instruments. A programming project will provide students with hands-on experience with real market instruments and data.
- Mathematical Finance 1 covering: the basics of financial mathematics and targets all students who have an interest in building a foundation in financial mathematics. Topics include basic mathematical theory of interest, term structure of interest rates, fixed income securities, risk aversion, basic utility theory, single-period portfolio optimization, basic option theory, emphasizing on mathematical rigour.
- Regression Analysis
- Ordinary Differential Equation, that includes the mathematical analysis of ODE
- Linear and Network Optimization or Nonlinear programming
- A couple of Business oriented modules
- Financial Modelling: which equips me with the knowledge of modelling financial process for the purpose of pricing financial derivatives, hedging derivatives, and managing financial risks. The emphasis of this module will be on numerical methods and implementation of models which includes: implied trinomial trees, finite difference lattices, Monte Carlo methods, model risk, discrete implementations of short rate models, credit risk and value-at-risk.
- Mathematical Finance 2: which provides me with in-depth knowledge of pricing and hedging of financial derivatives in equity, currency and fixed income markets. Major topics include fundamental of asset pricing, basic stochastic calculus, Ito’s formula, Black-Scholes models for European, American, path-dependent options such as Barrier, Asian and Lookback options, as well as multi-asset options and American exchange options.

Option 1: Follow up with 3 more mathematical analysis modules, which covers differentiability, Riemann integrals, metric spaces, an analysis treatise on multivariable calculus, measure theory?

OR

Option 2: Follow up with more statistics and computer science modules which exposes me to mathematical statistics, applied time series analysis, statistical learning, machine learning, designing/ analysis of algorithms, data analysis etc.

I also plan to pursue postgraduate studies, either a Masters in Financial Engineering or PhD (that's quite some time away but I want to be in a position best suited to further my studies).

Correct me if I am wrong, but from my premature understanding of QF, I understand that stochastic calculus/ stochastic analysis are used for areas such as option pricing and model validation whereas one can apply his/her knowledge of statistics and Computer Science on algorithmic trading/ HFT. Currently, I am interested in algorithmic trading - I'm working on a mean reversion theory project in university. However, since I am unsure of which area of QF I am interested in, I do not want to close my doors unnecessarily....

I think I have already said quite a bit so I'll stop here for now.

Nevertheless, thank you for reading and I would like to hear what you have to say!

Galvinator

Statistics: Posted by galvinator — November 7th, 2018, 7:43 am

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Well, I am presently taking a science communication module which requires me to put across the findings of a mathematics research paper in the form of a popular science article. I am not compelled to write about a mathematical finance popular science article but I wanted to write about it since I am interested in mathematical finance (I am also pursuing an undergraduate degree in mathematics and finance)

But as you said, I would rather be spending the time to focus on the foundations of mathematical finance.

@Alan, agreed, I have decided to drop the finance and I am currently writing on voting theory. I do hope it goes well.

Thanks all

Statistics: Posted by galvinator — September 22nd, 2018, 3:21 pm

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Hi,

My name is Galvin and I am a university Freshman.

As part of my university's curriculum, I have been tasked to prepare a popular science article on a mathematics research paper. The target audience of the popular science article is the general public. I would like to write a science article based on a topic from Quantitative Finance/ Mathematical Finance.

I think your best shot it to drop the Finance part. My name is Galvin and I am a university Freshman.

As part of my university's curriculum, I have been tasked to prepare a popular science article on a mathematics research paper. The target audience of the popular science article is the general public. I would like to write a science article based on a topic from Quantitative Finance/ Mathematical Finance.

There are annual popularizations of what's been going on in Mathematics generally:

"The best writing on Mathematics in 19nn" (edited) by M. Pitici

"What's happening in the Mathematical Sciences", vol n, by D. Mackenzie.

To write or edit this kind of thing, you have to be a PhD mathematician and a good writer or a sense for it. However, your task can be to start with Pitici or Mackenzie, and then dumb it down even further. That should be doable. But you'll find very little finance in those sources -- although I do see an article 'Mathematics and the Financial Crisis' in Mackenzie vol 8, copyright 2011.

Statistics: Posted by Alan — September 19th, 2018, 5:36 pm

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