Question on Black-Scholes model

Dazarath · #1 08-02-2006, 04:40 AM

I'm taking a finance class right now, and we just learned about the Black-Scholes model. We were taught that it is meant to price European options. Is there a model for pricing American options? I mean, everything else equal, an option that allows the owner to exercise it anytime up until the expiration date, should have more value than an option that only allows the owner exercise it on the expiration date. Sorry if this is a really beginner's question.

Evan · #2 08-02-2006, 11:47 AM

Try googling the binomial model for an intro on something more applicable to American options.

edtost · #3 08-02-2006, 09:21 PM

There are lots of models for pricing American options but few (perhaps none) will give a closed form solution. As Evan mentioned, somehting involving binomial trees is probably the best easy-to-understand model you'll find.

It is probably somewhere between absurdly hard and completely impossible to model American Options in the Black-Scholes (continuous-time) framework.

Dazarath · #4 08-02-2006, 09:24 PM

We learned about the binomial model as well, but only with two branches. The book touched on the fact that you can have more, but we didn't really get into it. I was just wondering if such a model existed. Thanks for the response guys.

joeynav · #5 08-03-2006, 12:39 AM

I recently took a mathamatics of finance class, and to my understanding American and Eurpoean "call" options are priced the same. Mathematically, the difference in the exercise time does not affect the price...don't ask me to prove this. American and Euro "put" option prices, however, are affected by the exercise date diference.

I think when pricing the American put option we were taught a reverse tree diagram method, in which you start with the branches(various stock prices and corresponding option values) and work your way towards the base. So when S'(t)>S(t) the put option value is zero. So one only focuses on the S'(t)<S(t) braches. You can extend the diagram for any given amount of periods and you assign probabilities to each outcome. I think we price Euro put options with the formula. Hope this makes sense, it kinda does to me.

But, I don't even know how the market actually prices these options. Do they actually use black-scholes to calculate option prices. I didn't think so, because the formula includes certain probabality assumptions, such as distributions and variance of stock prices. Anyways how does the market price options???

Felt like I had to chime in since I've recently studied it

edtost · #6 08-03-2006, 07:41 AM

Dividends can make it optimal to exercise american calls early.

That sounds like a description of a binomial tree model, though the option may have some value at S(t) > K for t < T. (For a put).

As for how the market actually prices options, I don't really know, but I have a sneaking suspicion that there really isn't a consensus, and that models are constantly evolving.

Evan · #7 08-03-2006, 09:41 AM

[ QUOTE ]
We learned about the binomial model as well, but only with two branches. The book touched on the fact that you can have more, but we didn't really get into it. I was just wondering if such a model existed. Thanks for the response guys.

[/ QUOTE ]
You just keep cutting the time periods of each branch down and making more branches. The problem is that you can never reach continuous time with this model. Well, "problem" is maybe not the best word, but that is why you can't find a black-scholes style solution to this.

RiverDancer · #8 08-03-2006, 03:17 PM

To partially answer this question you could check out the book "Option Volatility and Pricing" by Natenburg (Natenberg maybe). It deals in both European and American style options, I believe, as well as commodity and equity options. Personally, I trade commodity options for a living and we use the Black Model. This is the same model as Black-Scholes model expcept there are no dividends. In order to price these options most Market Makers use a computer program running the Black model to create pricing sheets. These sheets are basically a matrix of possible underlying prices vs. the theoretical option value at that given price.

Hope this helps a little,
RiverDancer

crazy canuck · #9 08-04-2006, 12:09 AM

[ QUOTE ]
You just keep cutting the time periods of each branch down and making more branches. The problem is that you can never reach continuous time with this model. Well, "problem" is maybe not the best word, but that is why you can't find a black-scholes style solution to this.

[/ QUOTE ]

Actually, the binomial model becomes the Black-Scholes model in the limit (as tree length go to 0). See Cox, Ross and Rubinstein; Option Pricing: A Simplified Approach.

Evan · #10 08-04-2006, 11:15 AM

[ QUOTE ]
[ QUOTE ]
You just keep cutting the time periods of each branch down and making more branches. The problem is that you can never reach continuous time with this model. Well, "problem" is maybe not the best word, but that is why you can't find a black-scholes style solution to this.

[/ QUOTE ]

Actually, the binomial model becomes the Black-Scholes model in the limit (as tree length go to 0). See Cox, Ross and Rubinstein; Option Pricing: A Simplified Approach.

[/ QUOTE ]
I'm confused by your use of the word "actually". When modeling American options the binomial model will not lead you to a Black-Scholes style solution.