Derivation Of The Convergence Of The Binomial Option Formula To The Black-Scholes Equation

Derivation of the convergence of the binomial option formula to the Black-Scholes equation
Jaime Rodrigo Rojas Palomino

The binomial option pricing model assumes that a stock will follow a binomial process from one period to the next: it can only go upwards by a factor of “u” with probability “p” or go downwards by a factor of “d” with probability “1-p”. Since this model converges (as thenumber of periods increase to infinite) to the Black-Scholes model, it provides a discrete time approximation to the continuous time process which underlies the Black–Scholes model.
An important limitation to the binomial option pricing model is that they become much less practical due to several difficulties for options with several sources of uncertainty and more complex features. In this case,Monte Carlo option models are commonly used instead.
To develop the Black-Scholes-type pricing model, we take the following assumptions:
* There are no transactions costs, taxes, or problems with indivisibilities of assets.
* There is no arbitrage opportunity.
* There are a sufficient number of investors with comparable wealth levels so that each investor believes that he can buy andsell as much of an asset as he wants at the market price.
* There exists an exchange market for borrowing and lending at the same rate of interest.
* Short-sales of all assets, with full use of the proceeds, are allowed.
* Trading in assets takes place continuously in time.
* The M-M theorem that the value of the firm is invariant to its capital structure obtains.
* Theterm-structure is “flat” and known with certainly.

Derivation
We start the derivation with the Black-Scholes equation.
S | current stock price |
X | strike price |
rc: | continuously compounded riskfree interest rate |
T | time to maturity |
σ2 | variance of the continuously compounded return of the stock |
N(di) | Accumulative normal probability for i= 1 and 2 |
C = SN(d1) – Xe-rcTN(d2)Where:
d1=log(S/X) + (rc + σ2/2)Tσ√T
d2= log(S/X) + (rc- σ2/2)Tσ√T
The equation for the binomial option model for “n” periods is:
C= j=0nnj pj (1-p)n-j max{0 , uj dn-j S – X} / ron (1)
p | The risk neutral probability of an up move. The “subjective probability” |
u | one plus the return per period on the stockif it goes up |
d | one plus the return per period on the stock if it goes down |
1/ron | Discounts the denominator to present value. |
Where:
nj = n! (n-j)! Represents the number of paths the stock can take to reach a certain point in a binomial tree.


Now, we can simplify equation (1) by removing those periods where max {…} = 0. Let “a” represent the minimum number of upward movesfor the call to “finish in the money”. So for all (j < a), max {…} = 0 and for all (j > a), max {…} = ua dn-a S – X. now we can rewrite equation (1):
C= j=annj pj (1-p)n-j uj dn-j S/ ron – j=annj pj (1-p)n-j X] / ron
C= SB1 - X ro-n B2 (2)Notice that B2 has a binomial probability function.
Now we have to make equation (2) converge to the Black-Scholes formula given at the beginning. We will divide the procedure in six steps
Step 1: Redefining B1
We need to rewrite B1 in order to use the DeMoivre-LaPlace limit theorem.
B1 = pj (1-p)n-jujdn-j / ron = [(u/ro)p]j[(d/ro)(1-p)]n-j p*j(1-p*)n-j
p* = (u/ro)p ; 1- p* =(d/ro)(1-p)
Now B1 has a binomial distribution (just like B2), but with the probability of each trial being p*.
Step 2: Transforming ro-n to e-rcT.
ro-n is the present value factor for “n” periods where the “per period” rate is ro. The “per period” rate can be related to an annual rate applied for T years by the relationship ro = r1/na, where na is the number of periods per year. Then:
ro =...