[R-SIG-Finance] [R-sig-finance] How good is Black-Scholes vs actual option prices

[Luwingo]

Hi Peter- the Black-Scholes-Merton closed-form solution is not a particularly
good approximation to real option prices, because of its inherent
assumptions. The BSM formulation assumes lognormally distributed asset
prices, constant dividend yield/risk-free rate/implied volatility, and
frictionless complete continuous markets. Not one of these assumptions holds
in practice.

However, the beauty of the BSM technique is that it is to a very large
extent self-correcting; that is, one can use the correct inputs to "tweak"
the option price until it most closely approximates the real world, as long
as you never violate risk-neutrality and no-arbitrage principles. In
practice, this means using bootstrapping techniques to build a zero rate
curve from market data, using volatility term structures to find the right
implied volatility, applying credit spreads to the risk-free rate to
preserve risk neutrality using CDS spreads, and possibly adjusting the BSM
formula for non-lognormal prices. I've done the first three in VBA using
FINCAD for mark-to-market accounting purposes.

Regarding the MSCI paper- if they're using options on a monthly basis to
hedge portfolios, the use of the 3M T-bill is not appropriate. I wouldn't
even use Treasury rates for risk-free rates anyway- they're too susceptible
to manipulation and are not "true" market rates. I would use the 1M LIBOR
rate for a monthly hedge, not a 3M rate. It is true that the LIBOR rate is
more of a swap rate, but this is conventional practice in the markets and
those are the rates used for bootstrapping zero curves.

Also, CBOE VIX data are "blended" across a variety of strikes, which means
that the implied volatility for the options used is not the "true" implied
volatility for each option. If you wanted to be really rigourous, you would
use the market implied volatility for a particular strike and, if you have
to smooth things, use a Kalman filter or some other kernel-smoothing
approach.

In answer to your question about the price: the VIX volatility is very
likely to be lower than the true volatility for an out-of-the-money option,
and very likely to be higher for an in-the-money option. So if the option is
far OTM, the price given by the BSM framework will be too low, and if the
option is far ITM, the price will be too high. This isn't a big deal in some
illiquid commodity markets but it's a VERY big deal in highly liquid equity
or rates markets.

I hope all of that helps to answer your question(s).


Peter Mennie wrote:
> 
> MSCI published this report recently:
> http://www.mscibarra.com/resources/pdfs/research/Portfolio_BCP_Nov_2009.pdf
> which basically looks at various methods of mitigating extreme event risk
> for equity portfolios.
> 
> One method they test is to buy options when their indicators suggest
> downside risk.  On pg 13 they mention they they use Black-Scholes to
> estimate the price of these options, using the VIX index as volatility and
> US 3m T-bills for the risk free rate
> 
> I was wondering if anyone had any experience of how accurate this
> assumption is likely to be in practice, and whether in practice the price
> would be likely to be greater or less than this estimate
> 
> Peter Mennie
> 
> 
> 
>       
> 	[[alternative HTML version deleted]]
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