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Comment by ncclporterror | Hacker News Reader

In modern finance the Black-Scholes formula is not used to "price" options in any meaningful sense. The price of options is given by supply and demand. Black-Scholes is used in the opposite way: traders deduce the implied volatility from the observed option prices. This volatility is a representation of the risk-neutral probability distribution that the markets puts on the underlying returns. From that distribution we can price other financial products for which prices are not directly observable.

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I have seen the insides of an options market maker, and can say this is not really true (at least for some regions of the market). Black-Scholes is used to derive theoretical prices for options. Good option traders will have an opinion on volatility and won't just take whatever the market says.

However, one of the interesting aspects of serious option trading is that Black-Scholes is merely your bread and butter. There is a lot of information that goes into option pricing, including supply/demand signals. The mix of signals also depends on the time scale on which you are trading.

What rings true to me with this comment is the correlation between products. Option traders are often concerned with many relationships between product pricing: between underlying and option, across expiries, across strikes, between products in indices, between products in sectors, etc .

It’s still used as an input into illiquid 409a valuations.

It’s also frequently used to price stock options given to employees at publicly traded companies.

Black-Scholes assumes constant volatility and cannot compute option prices without a volatility input.

This volatility is backed out of nearby options prices, often using the formula for European options.

There isn’t any purely theoretical option price because an assumption depends on observed prices.

>In modern finance the Black-Scholes formula is not used to "price" options in any meaningful sense. The price of options is given by supply and demand.

I'm not sure what your point is. Yes, actual market prices are determined by...the market. The Black-Scholes formula is widely used in modern finance to MODEL the price of an option given different sets of inputs in theoretical situations.

And it's a cycle. Supply and demand are partially driven by pricing models used by hedge funds, and variants of Black Scholes is one of those.

The way the article is written, it appears that the formula is used as: 1. Observe market parameters (volatility of the underlying and risk free rate) 2. Plug into formula 3. Deduce a price for the option.

My point is that it is used in the opposite way: observe prices to deduce market parameters. You claim my point is obvious, but I'm not sure it would be obvious to a reader unfamiliar with modern finance reading this article, which is the target audience.

> 1. Observe market parameters (volatility of the underlying and risk free rate) 2. Plug into formula 3. Deduce a price for the option.

In the FX market (interbank), the quoted and "traded" number is Implied Vol - the price of the option then follows from there (via the Black–Scholes model).

Sure, but isn't most of supply and demand in the market driven by large investors who use such formulas to derive the fair price of the option?

That is, if the real price ever differred significantly from what Black-Scholes predicts, wouldn't algorithmic trading very quickly correct this deviation?

If there was a way to directly formulate every parameter of the black Scholes formula you would be correct. The problem that you run into is how to calculate volatility itself? Without the volatility value, your algorithm cannot trade on it.

Using history of volatility is insufficient, because volatility is a forward looking measure. Just because the stock was volatile in the past does not mean it will be in the future, and vice versa. There are even more nuances with this, as volatility is a smile (or a surface), not a singular number https://en.wikipedia.org/wiki/Volatility_smile.

TLDR Trading in volatility is a very complicated topic. However, volatility is a useful parameter, and black Scholes is typically used to deduce the forward looking volatility from option price, in addition to volatility -> option price.

> There are even more nuances with this, as volatility is a smile (or a surface), not a singular number https://en.wikipedia.org/wiki/Volatility_smile.

That article says that implied volatility is inconsistent, with options at strike prices that are very far from the current market price having costs that imply a different level of volatility than options at strike prices that are close to the current price. The cute question here is "should an option be priced according to the actual level of volatility in the price of the underlying asset, or should it be priced according to the level of volatility that exercising the option would require?"

Volatility is just a quantity.

In a sense, BS and the option market enable trading in volatility itself.

Specifically, you trade in the estimate of the stock's volatility over the time from now to expiry of the option.

If you don't want to trade options directly to do that (it is cumbersome, as it involves "continuous" delta hedging), you can trade in VIX futures for the same purpose. Or variance swaps.

> traders deduce the implied volatility from the observed option prices.

I've only ever seen one thing:

Black-Scholes models say IV should be less but your broker/brokerage/the market are overpaying for it.

I always figured it was closer to a Vegas juice/vig.

I never understood the benefit really.

Complicated math to tell you lots of people want to play roulette on NVDA earnings and whatever you are going to pay for it is going to be "overpriced/overvalued" in at least one way.

I've never seen the opposite where it helps you find an edge and something was undervalued.

I’ve seen it used for OTC option pricing - there’s no liquid market, so you are more of a market maker than a market taker.

Black Scholes is not used for any otc option pricing, except perhaps to provide an instantaneous estimate to get in the ballpark, but no one would use it for the final price.

The real purpose of models is risk anyway e.g. implied vol is handy, delta is essential.

Also, isn't it only used for European style options, not American?

European and American calls cost the same on non-dividend paying stocks (on dividend paying stocks, it might make sense to exercise an American just before the ex-date).

Either way, as was pointed out, in reality BS is used as a deterministic one-to-one mapping between option prices and BS vols. Then, from market quotes (either as prices or as BS vols) a vol-surface is fitted (as a function of strike and expiry time), from which a stochastic process is fitted that correctly re-prices all these points (using a model such as "local vol" or "stochastic vol" or a combination of those two, or others), and then everything is priced of that.

> European and American calls cost the same on non-dividend paying stocks

All else being equal, I would prefer to buy an option contract I can exercise at any time vs one I can only exercise on a certain date. It doesn’t make intuitive sense they would be priced the same, can you please elaborate?

The parent is assuming that you can always sell your option to someone else for its fair value. If that's the case, there would never be a time where it's optimal to exercise a call option, because the optionality will always make the option value higher that the value of owning the stock.

This is shown in the article: the curved lines representing the option value are always above the straight lines of the final option payoff (the value if exercised).

This is not necessarily true for put options or for call options if the stock pays dividends. In those cases the option value can be below the payoff line and early exercise would be better than selling the option.

American style options are inherently more valuable. Imagine you had options on a stock that experienced a sharp but possibly temporary move. As a holder of an American style option, you could benefit from that temporary move, making it more valuable.

The way the market is typically modeled, temporary moves are not a thing.

The way the market actually exists, temporary moves are definitely a thing.

By put-call parity, C - P = S - df K, thus C = S - df K + P > S - K.

This is contingent only on the discount factor df being <= 1, and P >= 0, which is basically always the case. Thus, the value of the call exceeds the exercise value, making exercise never optimal.

Exercise for the reader: Understand why the same argument doesn't work for puts (or calls on dividend paying stocks).

For valuing financial products with no directly observable price, BS or its descendants matter quite a lot. For actually pricing a transaction on those, it becomes more complex but model value is typically an important input.

Kinda, but it’s not great because of the volatility smile

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