WOLFRAM|DEMONSTRATIONS PROJECT

Convergence of Binomial Option Pricing under Nonconstant Volatility

This Demonstration is a continuation and test of the validity of a modeling approach presented by the Wolfram Demonstration European Binomial Option Pricing with Nonconstant Volatility. That Demonstration presents a model for pricing an option using binomial option pricing techniques when volatility of the underlying asset is not deemed to be constant through time.
An option gives the holder the right but not the obligation to buy an asset (with value
S
T
) at some point in the future (
time=T
) for a price agreed upon today (strike price
K
). Cox, Ross, and Rubinstein [2] were the first to present a simplified approach of valuing this right. They employ a binomial process for the value of the underlying asset. There are discrete changes in the value of the underlying asset that occur over a set length of time (
dt=
T
N
, where
T
is the time until maturity and
N
is the total number of steps in the binomial tree). The sizes of the expected changes (up and down moves in the binomial tree) during each length of time are driven by the volatility of the underlying asset during that length. When this volatility is constant through each
dt
, the magnitude of the movements is also constant; therefore, the binomial tree recombines. Once this is completed for all
T
, there will be
1+N
possible outcomes. By taking the maximum between
S
T
-K
and
0
for each, and iterating backward through the tree (as in [2]), an estimated value for that option is calculated. As
dt0
, the expected value from the binomial approach converges to the true value.
When volatility is not constant through time, if
dt
is constant, magnitudes of the binomial movements will also not be constant. This makes the binomial tree not recombine, meaning after
N
movements there will be
N
2
possible outcomes. Guthrie [1] presents an approach where rather than having constant
dt
and changing up and down movement magnitudes, there can be constant up and down movements over varying lengths of
dt
. This allows the tree to recombine.
To test the model's validity, we can benchmark the value calculated against the Black–Scholes value (the most generally accepted valuation approach for European options [3]). The Demonstration presents the value calculated by each model (
y
axis) relative to the total number of calculations required after the price process and outcomes have been estimated to calculate it (
x
axis).
The Demonstration highlights the nonlinearity of the nonrecombining approach regarding the size of computation required to approximate a value. It also strongly supports the use and validity of the model developed by [1].