Choosing the Lognormal Mean for Black Scholes

[LM3886]

One thing I don't understand about Black Scholes is why the mean (drift) of the stock return lognormal distribution is set to the risk-free interest rate minus the dividend rate. Wouldn't it be more accurate to use historical mean return, if the history is sufficiently long?

According to https://www.investopedia.com/ask/answers/042415/what-average-annual-return-sp-500.asp, the average annualized return for S&P 500 is 10% within a ~100-year period. But if I use that as the lognormal mean, the option pricing for SPY will be quite different from the market (more expensive calls and cheaper puts), which is based on near-zero interest rate.

Does this mean the market doesn't really think S&P 500 will continue to yield 10% per year on average, even though it has performed like that reliably for almost 100 years?

 

[guru]

One thing I don't understand about Black Scholes is why the mean (drift) of the stock return lognormal distribution is set to the risk-free interest rate minus the dividend rate. Wouldn't it be more accurate to use historical mean return, if the history is sufficiently long?

According to https://www.investopedia.com/ask/answers/042415/what-average-annual-return-sp-500.asp, the average annualized return for S&P 500 is 10% within a ~100-year period. But if I use that as the lognormal mean, the option pricing for SPY will be quite different from the market (more expensive calls and cheaper puts), which is based on near-zero interest rate.

Does this mean the market doesn't really think S&P 500 will continue to yield 10% per year on average, even though it has performed like that reliably for almost 100 years?

I would love to hear your phone call to your broker while they’re liquidating your account and you’re trying to convince them that everything will be alright and your stock will recover and return 10% according to some data you’ve looked at. I also wonder how many people tried that trick last year when their accounts where being liquidated. Negative oil price must’ve been especially fun for brokers listening to their bankrupt customers/traders.

 

[Saltynuts]

I would love to hear your phone call to your broker while they’re liquidating your account and you’re trying to convince them that everything will be alright and your stock will recover and return 10% according to some data you’ve looked at. I also wonder how many people tried that trick last year when their accounts where being liquidated. Negative oil price must’ve been especially fun for brokers listening to their bankrupt customers/traders.

Now THERE is a helpful post lol....

 

[LM3886]

I would love to hear your phone call to your broker while they’re liquidating your account and you’re trying to convince them that everything will be alright and your stock will recover and return 10% according to some data you’ve looked at. I also wonder how many people tried that trick last year when their accounts where being liquidated. Negative oil price must’ve been especially fun for brokers listening to their bankrupt customers/traders.

Well, assuming the return is equal to risk-free interest rate would not prevent the broker from liquidating the account, either.

 

[DiceAreCast]

Because then the no arbitrage condition would not hold, aside the simple fact that you are pricing expectations, hence the future, not the past. Under the risk neutral measure the assumption of a constant rate is mathematically correct and matches with The Black model using a T forward measure. Problem is that when rates are stochastic it is very difficult to determine the value of B(T) because it's hard to calculate the entire integral within the expectation and numeric methods don't make it easier. Hence the trick via risk neutral measure and the assumption of a constant rate. My stochastic calculus knowledge is a bit rusty but I am sure some guys who currently price rates derivatives can chime in.

One thing I don't understand about Black Scholes is why the mean (drift) of the stock return lognormal distribution is set to the risk-free interest rate minus the dividend rate. Wouldn't it be more accurate to use historical mean return, if the history is sufficiently long?

According to https://www.investopedia.com/ask/answers/042415/what-average-annual-return-sp-500.asp, the average annualized return for S&P 500 is 10% within a ~100-year period. But if I use that as the lognormal mean, the option pricing for SPY will be quite different from the market (more expensive calls and cheaper puts), which is based on near-zero interest rate.

Does this mean the market doesn't really think S&P 500 will continue to yield 10% per year on average, even though it has performed like that reliably for almost 100 years?

 

[guru]

Well, assuming the return is equal to risk-free interest rate would not prevent the broker from liquidating the account, either.

Yes, but you’re discussing Black-Scholes vs market prices, as the market should be using BS.
Puts are expensive because we all know what can happen in an instant, regardless if the market will double again in 10 years. You’re always free to use any calculations you like and sell puts cheaper. The market isn’t based on BS but on supply and demand.
BS with its risk-free rate may be useful for estimating the current arbitrage-free pricing of options, and for hedging. Here are some interesting answers as well:
https://www.quora.com/Is-the-Black-Scholes-formula-useful-because-its-fundamentally-correct-or-is-it-useful-because-everyone-in-the-market-assumes-that-its-useful

 

[DiceAreCast]

Well, bs is still used every day by quants and everyone on wall street. But it should not be misunderstood. It's a translation device between prices and implied volatility. Not as a means to value an option.

Yes, but you’re discussing Black-Scholes vs market prices, as the market should be using BS.
Puts are expensive because we all know what can happen in an instant, regardless if the market will double again in 10 years. You’re always free to use any calculations you like and sell puts cheaper. The market isn’t based on BS but on supply and demand.
BS with its risk-free rate may be useful for estimating the current arbitrage-free pricing of options, and for hedging. Here are some interesting answers as well:
https://www.quora.com/Is-the-Black-Scholes-formula-useful-because-its-fundamentally-correct-or-is-it-useful-because-everyone-in-the-market-assumes-that-its-useful

 

[guru]

Well, bs is still used every day by quants and everyone on wall street. But it should not be misunderstood. It's a translation device between prices and implied volatility. Not as a means to value an option.

This is what I mean, maybe. Meaning working with the vol surface, where prices and IV are correlated, practically being one and the same. You mentioned “Not as a means to value an option.” but if you can estimate the IV then you can estimate the price. So it’s confusing when you say you can “translate IV to price” but you cannot price an option.

 

[guru]

Well, bs is still used every day by quants and everyone on wall street.

Btw, I use BS every day. But I still know that this doesn’t mean that the market cares. Tomorrow someone may buy 100K qty of some option without caring for BS. Then BS needs to adapt to market prices (for example IV and greeks can change), not market prices adapt to BS (except for keeping stuff arb-free).

 

[DiceAreCast]

For me valuing an option is valuing implieds

This is what I mean, maybe. Meaning working with the vol surface, where prices and IV are correlated, practically being one and the same. You mentioned “Not as a means to value an option.” but if you can estimate the IV then you can estimate the price. So it’s confusing when you say you can “translate IV to price” but you cannot price an option.