# Insights A Simplified Synthesis of Financial Options Pricing - Comments

Related Set Theory, Logic, Probability, Statistics News on Phys.org

#### andrewkirk

Homework Helper
Gold Member
I like the approach of providing intuition by describing how the model works for a very short-term option, so that discounting can be ignored. That makes it a lot easier to grasp.

The para that deals with interest rates (it starts with 'The prevailing interest rate should figure in these computations') is not quite right. The interest rate is the most complex part of the calculation. It is where the concept of changing the measure using the Cameron-Martin-Girsanov theorem comes in. It involves advanced concepts like equivalent martingale measures. So I would not say it is easily taken care of. The way to choose the interest rate is quite nuanced, and it affects not only the discounting of cash flows but also the assumed growth rate for the underlying stock, which determines how the mean of the bell curve shifts rightwards based on the time to option expiry. In the short-term example the bell curve does not shift at all, because the term to expiry is short enough to ignore expected growth.

You could deal with that by writing a para something like:

'In the above examples we have ignored interest rates and expected growth of the stock, because they will not be significant over a short period. For most options these need to be taken into account, and that is where the additional complexity comes in. But the principle is the same as in the above examples: measuring the expected payoff over a bell curve based on the final stock price.'

#### rude man

Homework Helper
Gold Member
EDIT occured shortly after publication (2018-08-18 19:15 GMT).

#### rude man

Homework Helper
Gold Member
I like the approach of providing intuition by describing how the model works for a very short-term option, so that discounting can be ignored. That makes it a lot easier to grasp.

The para that deals with interest rates (it starts with 'The prevailing interest rate should figure in these computations') is not quite right. The interest rate is the most complex part of the calculation. It is where the concept of changing the measure using the Cameron-Martin-Girsanov theorem comes in. It involves advanced concepts like equivalent martingale measures. So I would not say it is easily taken care of. The way to choose the interest rate is quite nuanced, and it affects not only the discounting of cash flows but also the assumed growth rate for the underlying stock, which determines how the mean of the bell curve shifts rightwards based on the time to option expiry. In the short-term example the bell curve does not shift at all, because the term to expiry is short enough to ignore expected growth.

You could deal with that by writing a para something like:

'In the above examples we have ignored interest rates and expected growth of the stock, because they will not be significant over a short period. For most options these need to be taken into account, and that is where the additional complexity comes in. But the principle is the same as in the above examples: measuring the expected payoff over a bell curve based on the final stock price.'
The Black-Scholes model also ignores expected growth, at least the on-line calculators do. I suppose that is one reason why the full Blk-Sch model is so much more complex. Barring growth I see nothing wrong with using present value to take care of interest. Also, no one else seems to take inflation into account which nowadays is a much greater cost determinant than the short-term, safe interest rate.

The other thing I omitted was of course dividends. This is a real problem since ex-dividend stock prices typically plunge. I don't know if Black-Scholes deals with that. Anyway, as I tried to emphasize, I was not trying to out-gun Blk.-Sch., just trying to help people see the big picture.

Thanks for your expert comment!

#### alan2

But of course, we know that option pricing doesn't actually work. Just like MPT and CAPM it is fun math but it doesn't reflect reality.

#### andrewkirk

Homework Helper
Gold Member
The Black-Scholes model also ignores expected growth, at least the on-line calculators do.
It doesn't ignore expected growth. It assumes that the stock will grow at a rate equal to the risk-free interest rate minus the dividend rate (or other net benefits of holding the assets). The reasoning behind that assumption is the main insight that the original Black-Scholes paper introduced. Discussing it is beyond the scope and purpose of your note, but I think it is important to acknowledge that growth is not ignored but rather assumed to be at that rate.

The expected growth cancels out against the discounting of the payout on maturity, leaving a discount factor on the first term in the formula that is $e^{-uT}$ where $T$ is time to option maturity and $u$ is the dividend rate. In examples where the underlying asset has negligible dividends or storage costs (eg some tech stocks, and commodities like gold with low ratios of storage cost to value) that discount factor is approximately 1 and is omitted. That's why it can look like growth is being ignored even though it isn't.

Inflation doesn't need to be explicitly accounted for in pricing options because it is automatically incorporated into the interest rate and hence also the expected growth rate.
But of course, we know that option pricing doesn't actually work. Just like MPT and CAPM it is fun math but it doesn't reflect reality.
It depends what is meant by 'work'. Banks around the world use option pricing techniques like Black-Scholes to set prices for the hedging deals they offer their business customers, and they make consistent big profits with it. Usually it's a good predictor. But like all models, there are rare circumstances where it breaks down, and that's where one can lose big money. Long-Term Capital Management was a famous case of that.

#### rude man

Homework Helper
Gold Member
"Inflation doesn't need to be explicitly accounted for in pricing options because it is automatically incorporated into the interest rate and hence also the expected growth rate."

Not so. As I said, current safe short-term interest rates are far below inflation. The former about 1.5%, the latter close to 4%.

#### rude man

Homework Helper
Gold Member
But of course, we know that option pricing doesn't actually work. Just like MPT and CAPM it is fun math but it doesn't reflect reality.
Absolutely correct! Couldn't agree more. It's fun math, is all. But it is fun!

#### andrewkirk

Homework Helper
Gold Member
current safe short-term interest rates are far below inflation. The former about 1.5%, the latter close to 4%.
I don't know which economy you're referring to, or which inflation measure, but all the inflation measures I know of* are about what has happened in the recent past, whereas interest rates relate to what is expected in the future. So the two are not comparable.

The Black-Scholes option pricing formula is based on the construction of an arbitrage with a dynamic hedging portfolio that replicates the performance of the option. Inflation doesn't need to be considered because, regardless of what inflation does, the hedging portfolio will replicate the value of the option, and the value of a particular hedging portfolio can be explicitly observed based only on current stock prices and interest rates.

* Other than the expected-inflation component of the yield on an inflation-linked bond. But those bonds are unusual and I doubt that is what the 4% relates to.

#### rude man

Homework Helper
Gold Member
current safe short-term interest rates are far below inflation. The former about 1.5%, the latter close to 4%.
I don't know which economy you're referring to, or which inflation measure, but all the inflation measures I know of* are about what has happened in the recent past, whereas interest rates relate to what is expected in the future. So the two are not comparable.

The Black-Scholes option pricing formula is based on the construction of an arbitrage with a dynamic hedging portfolio that replicates the performance of the option. Inflation doesn't need to be considered because, regardless of what inflation does, the hedging portfolio will replicate the value of the option, and the value of a particular hedging portfolio can be explicitly observed based only on current stock prices and interest rates.

* Other than the expected-inflation component of the yield on an inflation-linked bond. But those bonds are unusual and I doubt that is what the 4% relates to.
"I don't know which economy you're referring to, or which inflation measure, but all the inflation measures I know of* are about what has happened in the recent past, whereas interest rates relate to what is expected in the future. So the two are not comparable."
??? They don't change that fast!

Thanks
Bill

### Want to reply to this thread?

"A Simplified Synthesis of Financial Options Pricing - Comments"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving