How to correct the Black-Scholes-Merton Formula
One thing I often read on LinkedIn is that a formula is only as good as the assumptions underlying it. This normally comes up in connection with the Black-Scholes-Merton formula for option prices. A simplified version of the formula might look like this:
(1) option_price = f(volatility, other_stuff)
This formula is based on a big list of assumptions, including the assumption that the price of the underlying asset follows geometric Brownian motion. It's not an assumption that I can say is definitely wrong but it's certainly an assumption that nobody believes reflects reality. And if the formula's based on an assumption that we don't believe, it must be rubbish, right?
Well, I'm claiming today that I can replace this formula with a better one. Better in that there are no underlying assumptions and that it's always 100% accurate. Don't believe me? Read on.
I'm going to start by moving away from derivative pricing to simpler stuff. There was a Life Convention in Birmingham last week and many life actuaries would have been faced with the problem of working how long it would take to drive there. Is there a formula for that? How about this?
(2) time = distance / speed
It's a formula I came up with just now, based on a simple model where I assumed that a car drives at the same speed for the whole journey. And that's a pretty naff assumption. Nobody's going to drive at the same pace for the entire journey. That makes this a pretty useless formula.
Now consider this formula.
(3) time = distance / average_speed
This formula serves two purposes. First, and most importantly, it's the definition (after a slight rearrangement) of average speed. You might think there's only one possible definition of average speed but that's because you've been conditioned into only using a time-weighted average. There's an alternative distance-weighted average; if you drive the kids to Grandma's at 50mph then bring them home in the evening at 70mph and think your average speed is 60mph, then you're using the distance-weighted version. But we all tend to settle on the time-weighted version and I think it's because (3) is so similar to (2).
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Because the second purpose of (3) is to work out travelling times. It's only a slightly amended version of (2) but that amendment is enough to result in a formula with no underlying assumptions and that is 100% accurate. The 100% accuracy and lack of underlying assumptions are due to the formula for travel time being a rearrangement of the definition of average speed. You're not making any assumptions if all you're doing is rearranging the formula in a definition!
Obviously to work out how long it's going to take to drive to Birmingham, you're going to have to estimate your average speed, so your final result is going to include some assumptions but they're not assumptions within the formula: they're assumptions within one of the parameters. A formula's only as good as the numbers you plug into it. That won't change but at least it's only the inputs and not the formula that are up for debate.
I think people will have guessed by now where this is going but I'll carry on. I'm heading back to the world of derivatives and would like to present to you this alternative to the Black-Scholes-Merton formula:
(4) option_price = f(implied_volatility, other_stuff)
And guess what? This formula serves two purposes. First, and most importantly, it serves as the definition of implied volatility, albeit subject to a backsolving exercise. Because option prices are monotonic functions of volatility, there's only one volatility that will satisfy this equation, so, yes, it's an unambiguous definition.
And the second purpose of (4) is to work out option prices. It's only a slightly amended version of (1) but that amendment is enough to result in a formula with no underlying assumptions and that is 100% accurate. The 100% accuracy and lack of underlying assumptions are due to the formula for the option price being identical to the formula that defined implied volatility. Once again, no assumptions are involved in regurgitating a definition!
Obviously to work out an option price you're going to have to come up with an implied volatility, so that constitutes an assumption that underlies your final result but that's not an assumption within the formula: it's just a parameter. I can't remember where I first heard this piece of wisdom but a formula's only as good as the numbers you plug into it.
And that's where my story ends. If you replace volatility in the Black-Scholes-Merton formula with implied volatility, you get a formula that is not based on any underlying assumptions and that is as accurate as the parameters that are plugged into it.
Repeat after me
THERE ARE NO ASSUMOTIONS UNDERLYING FORMULA (4)
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1 年The BS formula is based on risk-neutral probabilities. Implied volatility is generally higher than historical volatility to take into account that the expected return in the BS formula is the risk free rate minus the volatility squared over two rather than a separate parameter. To properly set parameters for option pricing, short-term or long-dated, both a process mean and process volatility should be used without risk-neutral properties. The values will differ from the market price and that’s your opportunity to arbitrage.
Partner at Barnett Waddingham
1 年Hi Steve - Your note reminded me of the writings of elie ayache, who has written some deep philosophical pieces that describe a system where prices are exogenous and probabilistic models' only role is as instruments to express those prices. I think you'd find his book, The Medium of Contingency, interesting (if it times a bit hard going). I wrote a review of it a few years ago that you can find here: https://craigturnbullfia.com/reflections-prompted-by-elie-ayaches-the-medium-of-contingency/
Actuarial Contractor - Actuarial Systems Expert
1 年Implied volatility is defined by Black-Scholes, so it seems a bit circular…
Actuarial Contractor - Actuarial Systems Expert
1 年Hans Vaihinger:- "An idea whose theoretical untruth or incorrectness, and therewith its falsity, is admitted, is not for that reason practically valueless and useless; for such an idea, in spite of its theoretical nullity[,] may have great practical importance."