Why do many traders use Black-Scholes formula to price vanillas?

[A modest quant]

Dear all readers,

Before i raise my questions, let me provide background: From my observations and experience, almost surely prices of the underlyings (may it be stock, bonds, forex, etc...) are not log-normal, that is the distribution of log-returns is not of a normal distribution with a constant variance. In terms of stochastic calculus, as a process the return is not likely a Brownian motion with a constant volatility ( the diffusion term). Hence, the well-known Black-Scholes model (or formula), as everybody knows, is not a correct model/formula to price vanillas and of course not for exotics. Besides, in practice traders do not have control of many prices from OTC markets as well as Exchange centers. Therefore my first question: Why do we use the Black-Scholes formula to vanillas knowing that it is not correct? Second question: why do we so trust and use the value of a so-called implied volatility, which is simply an inverted price thru an incorrect formula, to price exotics?

I would be greatly appreciate good explanations and only good explanations...

 

[diegosanaz]

Black-Scholes formula is not used to price options. Is used as a benchmark. When an option is traded in the market, in the BS formula the only unknown variable is the implied volatility. This is why it

The fact that the implied volatility for a series of strikes (with same maturity) is a smile instead of a constant, proves that the BS formula is wrong. But this is what makes it a useful benchmark.

If in any way a person uses the BS formula to price an option, what such person is really doing is pricing the volatility for that particular strike and maturity.

Consider this: imagine a person believes that the Heston model correctly reflects the dynamics of an underlying asset, then it goes Heston Model -> Option Prices -> Implied Volatility by those option prices (via BS formula).

Hope you consider this a good explanation

 

[A modest quant]

Black-Scholes formula is not used to price options. Is used as a benchmark. When an option is traded in the market, in the BS formula the only unknown variable is the implied volatility. This is why it

The fact that the implied volatility for a series of strikes (with same maturity) is a smile instead of a constant, proves that the BS formula is wrong. But this is what makes it a useful benchmark.

If in any way a person uses the BS formula to price an option, what such person is really doing is pricing the volatility for that particular strike and maturity.

Consider this: imagine a person believes that the Heston model correctly reflects the dynamics of an underlying asset, then it goes Heston Model -> Option Prices -> Implied Volatility by those option prices (via BS formula).

Hope you consider this a good explanation

Thanks for your explanation which i consider very good!

Agree with you on the BS formula as benchmark for pricing a vanilla with different strike and maturity. Believing a model or a benchmark (eg BlackSchole, Heston, local vol, stoch local vol, etc...) and using it to price derivatives is quite a subjective approach. Don't you agree? Nothing wrong in having a subjective approach because a trader like myself donot know the future and i must weigh the costs and risks involved, therefore i must take a view (or expectation that quants often talk about technically) on the market over a time horizon.

In relation to implied volatility surface, in the rates world, traders often quote Sabr prices; traders and quants believe in a local vol or Heston or SLV, etc... for pricing derivatives in equities, forex and commodities markets. Therefore, am i right in saying: Derivatives prices are subjective? If you agree, i will post this question as a separate thread.

Thanks again.

 

[diegosanaz]

In a way I would say yes, derivatives prices are subjective. The thing is that because volatility is stochastic and there is not an efficient way to trade it, then the markets are incomplete, meaning that doing delta hedging still carries risk are the derivative position cannot be replicated.

This is just my view, but valuating a derivative in a way is not that different as value investing is for equity. You have to assume a model, chosen by whatever reason, and then try to use market data somehow to calibrate your model and then get a "fair" price. You buy or sell the security comparing if such fair price is above or below market price.

For sell side firms, the fact that they cannot replicate the security perfectly comes into play. They will not sell at a "fair" price since at that price there is similar likelihood of making either profit or loss. So what they would do is sell at a price high enough to justify the risk they are incurring, but low enough so that other people are willing to buy at that price.

As for risk neutrality (don't take my word for this) but I think risk neutral <-> arbitrage free price. So if the price does not generate an arbitrage opportunity, then it is a risk neutral price.

 

[financeguy]

I agree there is a certain amount of subjectivity in pricing path dependent exotics. But I think there is a bit less subjectivity than we're giving sell side traders credit for. As diegosanaz has pointed out, nobody believes that the B-S formula actually describes reality. This is why a smile exists. If the volatility smile implied unreasonable levels of volatility, spot-vol correlation, or vol of vol, and traders were willing to deal options at those levels, the street could go to town trading risk reversals, flies, and just atm options to realize the right values of these parameters. Unfortunately, the street generally prices things just about fairly to how they've realized in the past, down to pretty fine detail - unless there is some reason to believe the future won't be similar to the past (like an event coming up), or unless there is a supply/demand imbalance created by a non-economic user of options (such as a corporate hedging balance sheet exposure in enormous size, without regard to implied parameters that options traders would care about). That's in vanillas. In exotics, the most used model for single factor stuff is a blended stochastic-local vol model. While again it doesn't necessarily reflect reality 100%, it allows the trader to control how he feels the smile evolves as market conditions change through the mixing blend parameter. So rather than totally guessing at what the value of an exotic should be by just blindly plugging it into the model, the trader can try to assess the sensitivity of the slope of the smile to changes in movements of the underlying by using past realized mechanics as well as a pinch of intuition, use the resulting mixing fraction, and price the exotic. If the dynamic ends up being different from his prediction, he'll either make or lose money dynamically hedging the vega and smile exposures of that exotic using vanillas. The less confident he is in his ability to predict the dynamics of the smile within the model, the wider the range for the mixing fraction he will incorporate into his bid/offer spread. Sometimes a liquid enough interbank market exists for exotics and traders can offset their books' mixing fraction exposure at some prevailing rate dealing exotics in the market. Not every model is the same, but you should be able to draw similar analogies between this and whatever model you are working with.

 

[A modest quant]

I agree there is a certain amount of subjectivity in pricing path dependent exotics. But I think there is a bit less subjectivity than we're giving sell side traders credit for. As diegosanaz has pointed out, nobody believes that the B-S formula actually describes reality. This is why a smile exists. If the volatility smile implied unreasonable levels of volatility, spot-vol correlation, or vol of vol, and traders were willing to deal options at those levels, the street could go to town trading risk reversals, flies, and just atm options to realize the right values of these parameters. Unfortunately, the street generally prices things just about fairly to how they've realized in the past, down to pretty fine detail - unless there is some reason to believe the future won't be similar to the past (like an event coming up), or unless there is a supply/demand imbalance created by a non-economic user of options (such as a corporate hedging balance sheet exposure in enormous size, without regard to implied parameters that options traders would care about). That's in vanillas. In exotics, the most used model for single factor stuff is a blended stochastic-local vol model. While again it doesn't necessarily reflect reality 100%, it allows the trader to control how he feels the smile evolves as market conditions change through the mixing blend parameter. So rather than totally guessing at what the value of an exotic should be by just blindly plugging it into the model, the trader can try to assess the sensitivity of the slope of the smile to changes in movements of the underlying by using past realized mechanics as well as a pinch of intuition, use the resulting mixing fraction, and price the exotic. If the dynamic ends up being different from his prediction, he'll either make or lose money dynamically hedging the vega and smile exposures of that exotic using vanillas. The less confident he is in his ability to predict the dynamics of the smile within the model, the wider the range for the mixing fraction he will incorporate into his bid/offer spread. Sometimes a liquid enough interbank market exists for exotics and traders can offset their books' mixing fraction exposure at some prevailing rate dealing exotics in the market. Not every model is the same, but you should be able to draw similar analogies between this and whatever model you are working with.

I generally agree with you and diegosanaz on most points both raised. There are some points rather technical that i believe it's not convenient to discuss here. However, there is a practical issue i want to raise as a separate thread: spot volatility vs implied volatilities.

 

[coolarbitrager]

Haha...here it comes again - the BSM (BullShitMurderer) formula! I can't help the urge to put a stop to this stupid formula. What is there about it that ppl have been bragging on for years!

See my other posts guys!