Assume I have only basic math knowledge, what specific areas of math would I need to learn in order to understand the following webpage:


Many thanks.

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    $\begingroup$ How "basic" are we talking? Certainly you would have to know PDEs and stochastic calculus to grok Black-Scholes. $\endgroup$ Oct 20, 2010 at 17:26
  • 4
    $\begingroup$ Short answer: Quite a lot! (Partial differential equations, random processes, etc. And to learn that, you need a very solid foundation in calculus, probability, etc.) $\endgroup$ Oct 20, 2010 at 17:39
  • $\begingroup$ @J.M.isn'tamathematician in Rosenthal's 'First Look At Rigorous Probability' it is derived without pdes, just sde+Ito's formula+normal CDF $\endgroup$
    – Alex
    Aug 14, 2020 at 10:46

6 Answers 6


The standard low technology argument for Black-Scholes (the famous "binomial tree") requires only basic material, though there is also a standard medium technology approach using stochastic calculus (informally) and an advanced approach using the rigorous mathematical apparatus of stochastic processes, Brownian motion, and diffusion equations. I think summation of a finite geometric series, very basic probability, and knowledge of how to take a limit of the binomial distribution to get the Gaussian normal distribution, is enough for the binary tree approach. This is in almost every elementary text on options pricing, such as the ones used in business schools or quant basic training, where by elementary I mean not starting immediately from stochastic calculus.

(This relatively simple derivation applies to the Black-Scholes formula for pricing the simplest types of options and, in some developments of the theory, also to the Black-Scholes model giving a more general formalism for pricing arbitrary European options as expected destinations of a random walk. The binomial tree is not ordinarily presented as a method for producing the Black-Scholes partial differential equation satisfied by prices in their model, although in theory it could do that. The differential equation is usually arrived at by a simple heuristic argument using stochastic calculus (given in all finance books that introduce stochastic calculus). A fully rigorous derivation is quite complicated and does require a lot of mathematical background, but this is not necessary for understanding the formula and its use in finance. An elementary mathematical derivation of the differential equation not explicitly using the binomial tree was posted at Terry Tao's webpage, though it does not dwell on financial intuitions or interpretations.)

The problem can be discretized, as pricing of an option to buy a security whose value goes up and down in an (exponentiated) random walk, where there are a finite number of time steps and at each step the price is multiplied by $r$ or $1/r$ with a constant probability of up- or down- motion in the price at each step. Taking a limit you get the Black-Scholes formula. The key point is to consider a single time step and show that the price is determined by the relationship between the payoffs for the security and for the option. The rest is a matter of algebraically propagating this known answer back up through the binary tree (the $2^n$ possible paths for the price of the security between time 0 and time $n$) until you reach the root. This gives the answer for the finite problem, then taking the continuous limit of the problem "derives" Black-Scholes for the continuous time case.

To show that the continuous time theory makes sense mathematically involves more than just suggesting that such a theory might exist as a limit of a well-defined discrete theory. For that you need to construct the limit theory directly, which requires the machinery of stochastic analysis. To understand the limitations of the finite or continuous theory requires knowledge of finance in addition to the mathematics.

[References added: low-tech approach (Cox-Ross-Rubinstein binomial trees): http://en.wikipedia.org/wiki/Binomial_options_pricing_model ; medium-tech is in most introductory books on derivatives pricing that cover stochastic calculus -- Hull and its competitors; high-tech is in books on stochastic PDE such as Oksendal or advanced finance books like Shreve. ]

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    $\begingroup$ Here's the link to Terry Tao's post, to save others the trouble of Googling: terrytao.wordpress.com/2008/07/01/the-black-scholes-equation $\endgroup$
    – user856
    Oct 20, 2010 at 19:20
  • $\begingroup$ Dear T, I have already read about Black-Scholes in passing, but have never seen the treatment of it being related to a binary tree. May I know what book(s) I can peer at for this? $\endgroup$ Oct 21, 2010 at 0:34
  • $\begingroup$ @J.M. : posting now expanded and with references. $\endgroup$
    – T..
    Oct 21, 2010 at 1:07
  • $\begingroup$ Thanks a lot, T; a shame I cannot upvote an answer twice... $\endgroup$ Oct 21, 2010 at 1:18
  • $\begingroup$ Second the reference to Shreve's book: it has two volumes, the first doing the theory on binomial trees and the second in continuous time. He also has some lecture notes math.cmu.edu/users/shreve/LectureNotes.pdf $\endgroup$ Oct 21, 2010 at 3:33

I found this book quite useful. But it does take the background listed. The basic result that the equation is the same as one dimensional heat flow and the graphs of roughly how the value flows are not too tough. There are also free calculators on the web that let you play with it.


Take a look at the syllabi for the first three actuarial preliminary examinations. If you study for these tests, in order, you will learn everything you need in order to understand and apply the Black-Scholes formula.

Each exam typically requires three hundred hours of study to do well. If you pace yourself, and you take one exam every six months, you will be studying 11 to 12 hours per week for one and a half years to master this material. Of course, you could cover this material much faster, if you're more interested in breadth than depth.

The first three exams are 2.5 to 3 hours each. They are

  1. 1/P - The Probability Exam - A thorough command of calculus and probability topics is assumed.
  2. 2/FM - Financial Mathematics Exam - Interest theory (discrete and continuous) and an introduction to derivative securities.
  3. 3F/MFE - Financial Economics Exam - Interest rate models, rational valuation of derivative securities, simulation, and financial risk management techniques.

The syllabi list the textbooks and papers you should study. Additionally, there are study guides that cover each test.

Exam 3F/MFE covers Black-Scholes. Specifically, you must be able to

  • Calculate the value of European and American options using the Black-Scholes option-pricing model.
  • Interpret the option Greeks.
  • Explain the properties of a lognormal distribution and explain the Black-Scholes formula as a limited expected value for a lognormal distribution.

Well, it depends if you want to understand esp. this derivation or an derivation (so what is going on there).

The easiest and most intuitive derivation I have seen, ever - is this one:

Intuitive Proof of Black-Scholes Formula Based on Arbitrage and Properties of Lognormal Distribution by Alexei Krouglov
Link: http://arxiv.org/abs/physics/0612022

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    $\begingroup$ The derivation in that paper is circular. It simply assumes that the same type of "interest discount" arbitrage argument that relates the value of a stock owned today and at time T in the future, can also leap over T time steps to price (today) an option to buy stock at that time. To derive Black-Scholes you need to relate the present and the future along paths of small time steps. Having some sort of exponential discount and taking expected values of a random walk are consequences of the Black-Scholes theory. The arxiv paper just assumes that these things are true for the option price. $\endgroup$
    – T..
    Oct 21, 2010 at 12:43

I found this concise paper one of the most accessible ones giving an overview of the traditional approach:


For an entirely stochastic and non-financial approach, see Rosenthal's 'A First Look at Rigorous Probability', Chapter 15.8. It goes this way:

  1. Stochastic Differential Equations (SDEs), Ito's integral and formula.
  2. Black-Scholes SDE: $$ dP_t = \sigma P_t d B_t + \mu P_t dt $$
  3. Derivation of the closed-form expression for $P_t$ using Ito's formula as a function of $B_t$.
  4. Finally, derivation of the expected value of the European call option at time $T$ given value at $t=0$, risk-free interest rate $r$: $$ \mathbf{E}[e^{-rT}\max(P_T-q, 0)|P_0] $$

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