Monday, April 25, 2005

What is the expected return on options? Zero? RF rate? Or something else?

FEN has a great article on some of the limitations of Black-Scholes World! It is by Ross Miller and it is definitely worth your time!!! It is both thought provoking and funny at once. A great combo!!!

Be Careful What You Model

A few quick highlights:

* "You have in your hand (or on your screen) an at-the-money call option with a year until it expires. Because you are in BSW, you know exactly what that option is worth at the present moment in time. Consider this: What is your expected rate of return on the option between now and the option’s expiration in one year?

An easy question, right? Think some more.

The typical profit-and-loss diagram for options, popularly known as the “hockey-stick,” assumes that the funds invested in options earn a zero return regardless of the time until expiration. According to this diagram, the absolute return from an option is simply the terminal payoff minus the current cost. The possibility that one might require a positive return to compensate for the opportunity cost of funds used to finance the option is either ignored for the sake of pedagogical simplicity or relegated to a footnote.

Zero is clearly the wrong answer, so what about the risk-free rate? That was the nearly unanimous answer to my informal, nonscientific survey and it is what Paul Wilmott appears to be saying (if I understand his notation) on the top of page 34 of the first volume of his magnum opus on quantitative finance.

This answer might be defensible, but it is not what BSW’s creators had in mind"

*The strange world in which every asset earns the risk-free rate of return for the life of the option is not Black-Scholes World, but a universe that I will dub Cox-Ross World (CRW) after the two economists, John Cox and Stephen Ross, who colonized this world in their 1976 Journal of Financial Economics article. (Cox and Ross explicitly refer to their theoretical construct as a “world.”) CRW is a degenerate neighborhood of Black-Scholes World in which risk-neutrality rules. What Cox and Ross recommend (and what Black and Scholes allude to in an unpublished early draft of their famous article) is that when you have a messy option it usually pays to visit CRW to find its value."

*"Take the collapse in spreads on risky debt. In a risk-neutral world, yield spreads are just wide enough to cover the expected capital losses from adverse credit events. While unquestionably much of the tightening over the past few years has come from good news on the credit front, there appears to be more going on – vanishing risk premia."

* "It is natural to wonder whether all of this is just another recipe for disaster whipped up in the financial engineers’ kitchens. Unfounded assumptions of option replicability (portfolio insurance in 1987) and market liquidity (LTCM in 1998) turned out to have a destabilizing effect on financial markets"


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