Binomial Lattice option pricing pedagogy MBA students

In teaching option pricing theory to MBAs, John Cox at Stanford developed the binomial lattice approach.

DS: Around that time, you also developed the binomial model. How did this relate to the Black-Scholes model, and what made the Cox-Ross-Rubinstein model so popular?

JC: It was easy to use and easy to understand. Its purpose was to simplify the arguments that went into the Black-Scholes derivation, which were initially mysterious enough to everybody, and for masters students totally inaccessible because they were couched in terms of complicated mathematical techniques. It was important to have a way to explain to MBA students what was happening, rather than just presenting the final result in a black box. Sometimes that has to be done, but that’s not really the way education should happen.

The pedagogy of binomial lattices for stochastic processes and dynamic programming had been developed in several books in the 1960s by Stuart E. Dreyfus at the Rand Corporation in Santa Monica.  These books were familiar to the finance professors developing option pricing theory in the 1960s and 1970s.  They were also familiar to Bill Sharpe who was familiar with those people while at UCLA as a student and who suggested the idea to John Cox.

Sharpe was an employee at RAND in 1956.


A binomial lattice for a stock price starts from the current stock price and goes up one step or down one step over the first time interval.  This is repeated from each node.  The lattice can recombine in the simplest case.   Using such lattices, one can teach hedging, stochastic processes, and dynamic programming strategies including hedging and option writing by a dealer.

The alternative to the binomial lattice are Wiener paths in time that have no tangents to them except at isolated points.  These paths are hard to visualize and have properties that are difficult to understand well or at all.

By restricting the path space to paths through a finite lattice, with only two outgoing nodes from each node in the lattice, the problem is simplified to a point that some MBA students will believe they can understand it and are willing to try.  Many do.

The same strategy applies to stepping back from functions on real numbers in algebra one.  Functions on natural numbers are ones students can easily believe they understand.  The same applies to functions on rational numbers.  So they are more willing to learn the new concept using the simpler chassis restricting the complexity of the problem.

Sal Khan got an MBA at Harvard and worked at a hedge fund. So the idea of cross over from finance to math education is already established.  Many of his videos are on finance with some on option pricing.  His put call parity videos are weak compared to standard techniques to teach put call parity.

Put call parity was emphasized as part of the 1970s development of option pricing theory at business schools.  Put call parity was easier to teach conceptually to MBA students than the Black Scholes option pricing model.  However, with calculators, Black Scholes could be taught procedurally.   This included teaching calculating hedge ratios and the Greeks.   This is an example of procedural teaching being easier than conceptual.

The recursion theorem of Dedekind of 1888 is linked to dynamic programming in probability and option pricing theory.  Dynamic programming and recursive algorithms in probability were discovered by Pascal in his correspondence with Fermat on developing a solution to the value of a side in a multiround game of chance popular in 17th century France.  If the game stopped early, the settlement had to be determined.  This is the same conceptually as solving for the price of an option using a binomial lattice working backwards from option maturity.

About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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