Limitation of black scholes

[Lun]

I want to overcome 2 limitations of BS, they are
1) can't price American options
2) assume no dividend
In the real case, we have dividends, how can we price the option ? use another model or modify the BS ?
Similarly, for American options, use another model or modify the BS ?

 

[Sarang]

I think Merton did write a paper on Dividend Paying form for BS.
An adaptation here -

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node9.html

 

[Lun]

what I want to ask most is what the practice in the industry is ? it's very common to price American options, and to price options with dividend.

 

[Uncle Max]

You can price American call option (no dividends) using BS formula. To price puts - there are many different methods - trees, finate differences etc

 

[Joy Pathak]

You can use Crank-Nicholson to price American options through finite difference.

Binomial Trees will work just as fine as Max mentioned for American Options and for European with dividends.
For trees you use the same method as you would for european options, just remember to take into consideration early exercise. So, in the case of an American Option, the value of the option at any node in the tree is the greater of the value calculated by backward induction, and the payoff from early exercise of option.

You can use least square monte carlo method to estimate the conditional expected payoff of American options to value them. It has been proved to even work for assets following jump diffusion process. Monte carlo would work for most path dependent options.

The modification to BS provided by Sarang's link should help for dividends.

 

[Yike Lu]

There's a nice no-arb proof that an American call on a non-dividend paying asset must have the same value as the corresponding European call, thus allowing for Black-Scholes valuation.

 

[Lun]

Thanks for all the replies !

the resources are useful, you all are helpful

 

[latency]

in practice, most exchange traded options in the US with dividends are american options. software for option market making typically use the binomial model.

 

[HyperVolatility]

you can still use the BSM model even if I'd recommend you to use an estimation of volatility which is not the VIX because that would lead you to overprice the options itself.

However, I have to agree with Joy Pathak both the binomial (or trinomial) tree and Monte Carlo simualtion work quite well for dividens paying options.

In my opinon,the key problem is volatility. that's the crucial factor for an accurate pricing

 

[HyperVolatility]

you can still use the BSM model even if I'd recommend you to use an estimation of volatility which is not the VIX because that would lead you to overprice the options itself.

However, I have to agree with Joy Pathak both the binomial (or trinomial) tree and Monte Carlo simualtion work quite well for dividens paying options.

In my opinon,the key problem is volatility. that's the crucial factor for an accurate pricing

I almost forgot that if you want to price correctly options using volatility you can always have a look at my website www.hypervolatility.com