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Comparing Binomial and Monte Carlo with Black Scholes

karafrylee

karafrylee

Joined
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I have a couple of questions to ask regarding the above and I hope you can help me out.

The option values obtained from both Binomial option pricing model and Monte Carlo simulation can be compared to the value obtained from Black-Scholes formula. In fact, the option prices for Monte Carlo converges to Black Scholes formula as the number of paths increases, and the Binomial OPM is a discrete time approximation to the continuous Black Scholes formula.

If that is the case, why do we still use both of these models to value options when we know Black Scholes formula can give us the option price? And also, are the assumptions for all three models the same?

Thanks.
 
If that is the case, why do we still use both of these models to value options when we know Black Scholes formula can give us the option price? And also, are the assumptions for all three models the same?
Click to expand...

BS prices only European options while binomial can price Americans. Assumptions are the same. Binomial in infinity converges to BS also.
 
karafrylee said:
I have a couple of questions to ask regarding the above and I hope you can help me out.

If that is the case, why do we still use both of these models to value options when we know Black Scholes formula can give us the option price? And also, are the assumptions for all three models the same?

Thanks.
Click to expand...

Like mentioned above - American options - Cannot be priced using BS. Can be priced using Binomial / Trinomial and Finite Difference Methods.

Also for exotics options (Asian, Lookback, etc) numerical methods such as Monte Carlo, Finite Differences etc are required.

There is an advantage to explaining numerical methods using European options although we know the closed form solution. You can compare the number of iterations / paths / mesh refiness you need to get within a few cents of the closed form solution. Also you have something to compare your answer to and you will know if you have coded the method correctly. Once you get the hang of the numerical method being implemented, you can move on to more complex options and / or make refinements in the methods (for faster convergence etc).
 
Thanks for the info. I'm interested in pricing European options using the two aforementioned models. But it seems impractical to use these two models to price European option when there is a closed form solution, as said above.

One more question, is there a good method to calculate the rate of convergence for any of these models? Thanks.
 
For Binomial, as the number of paths increases, the accuracy of the solution increases,
i.e. as n goes to infinity, the solution converges to Black Scholes formula.

But how fast is this convergence? Is there any formula/method that can calculate the rate of this convergence with increasing number of paths? I hope that makes sense.
 
karafrylee said:
For Binomial, as the number of paths increases, the accuracy of the solution increases,
i.e. as n goes to infinity, the solution converges to Black Scholes formula.

But how fast is this convergence? Is there any formula/method that can calculate the rate of this convergence with increasing number of paths? I hope that makes sense.
Click to expand...

If you had an analytical approximation to Black-Sholes model then you could simply differentiate that function with respect to n and generally find the shape of the function to see the rate at each point - that is, find the rate at which the approximation is getting better as n increases. But since you are now dealing with numerical method (binomial) then you can simulate n-s and find the rate at which each following corresponding result is getting close to BS outcome. Once you have gotten the rates for each n, then you can fit in some model like regression to have a clear analytical view of the dependence of model on n (how n values affect it).
 
Running approximations of black scholes helps us flush out the amount of error in our valuation of more complex options which do not necessarily have closed form solutions (American options were mentioned), and thus decrease it.

http://en.wikipedia.org/wiki/Control_variates
 
depends how correlated or similar the two options are. you are not going to use a black scholes put as a control variate for an asian call option
 
@ <karafrylee>: You have mentioned: "But it seems impractical to use these two models to price European option when there is a closed form solution, as said above." However, as <enthusiast> has pointed out, using your Binomial tree code for simple European options help you check your method against the closed-form "BS" solutions so that you can confidently use your Binomial code for more complicated options.

When it comes to American options (which are exercisable at any time in a given interval) or Bermudan options (which are exercisable at specific instances of time), Binomial pricing is the choice as it is based on the description of an underlying instrument over a period of time rather than a single point. Furthermore, Binomial pricing is very simple to implement.

For more complicated options (such as Asian), Monte Carlo techniques are commonly used as Binomial pricing becomes less practical. The problem with Monte Carlo is that it is computationally time-consuming. Hence for less complicated simplistic American options, Binomial pricing wins the favor! :)

Good Luck!
 
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