Explaining the volatility smile (very mathematical)

In today’s article, I wanted to do a more advanced review about how I think about option pricing, investing in options and we’ll review my thesis on why the volatility smile exists.

Memo to: Vega Capital clients
From: Scott Shuttleworth


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The below assumes some basic proficiency in option pricing theory and exchange margin requirements. If you are not familiar with options I’d recommend spending some time understanding the basics at Investopedia or any other of the free online resources.

Today's post also includes pictures which may not show up in some email browsers. Click 'View this email in your browser' and they should show up.

 

Intuition of the Option Pricing problem

The basic problem in option pricing is as follows. Let’s say I have some money to invest and wish to buy Commonwealth Bank of Australia stock (ASX: CBA). If I buy, I could potentially increase my wealth…but of course, something could go wrong and I might lose money instead.

However if I bought an option, it wouldn’t matter if the price fell and I’d still enjoy the upside if it eventuated. My only loss would be the amount which I paid for the option. So how much should an investor pay for such a luxury?

This is a hard question because before we even consider any formulas, there are clearly many answers. The value of such a commodity will vary between different people given their risk tastes and idiosyncrasies. Hence in a way, asking 'what is the value of an option?’ is a poor question. A better question is perhaps ‘what is the highest price any investor should be willing to pay for such an option?’

But we’ll try to answer the former anyway.

 

Finding a solution to a poor question.

Let’s start by breaking this problem down in a simplified manner – even if you've never bought an option on market, you’ve probably bought car insurance. The insurer naturally wants to sell the insurance for an amount which covers her cost – but what is the cost? Clearly that of paying for everyones damaged cars (let’s leave out operating expenses for the time being).

If the insurer believes that car collisions would be more frequent in future she would price policies at a higher cost and vice versa. From this, it’s easy to see why the price of insurance would be linked to how volatile car drivers are (since these drivers would crash more and incur losses for the insurer).

Going back to the markets, let’s say we’re a market maker and provide a sale price for every option security on the exchange. It’s difficult to forecast stock prices precisely so we can’t know which ones will cause us big losses but we want to charge enough to cover any that may arise.

As a market maker, we also need to consider how much margin capital the exchange will require us to deposit for every option trade and what sort of return we might need on that capital.

Hence the total amount we wish to charge is equal to our required return on capital plus the summed amount of probability weighted losses.

This is an important concept because it allows us to;

  1. Explain the volatility smile.
  2. Focus on option prices being compensation (a risk-free return plus a risk premium) for a market participant putting up margin capital and taking on risk in order to sell another participant insurance.

 

Explaining the volatility smile

Before we continue on point 2, I wish to show how my thinking above explains the volatility smile. And we don’t need much math to do it.

Let’s consider a 1 year far out-the-money option. We whip out the Black Scholes Merton model to calculate its value and find it’s worth 1 cent and yet we noticed its market price is $1. This is reflective of the well-known volatility smile phenomena - but why?

Put yourself in the shoes of the person selling this option. If they transact with you, they’ll have to put up margin capital. On the SPY, say this was $2,900 per option contract sold less the premium recieved. The option seller will get $100 for selling this contract, generating a 3.57% (100/(2900-100)) return on investment should the option expire worthless. The return is reflective of the risk-free rate as well as some compensation for the risk the market maker is bearing (since the option is far out of the money, the compensation is quite low).

As you go further out-the-money in the option chain you notice the prices fall but very slowly - 95 cents, 90 cents etc etc. This is despite the Black Scholes Merton model valuing these options at less than 1 cent. So how are they being held at such a premium? Surely investors would sell the option en masse to restore an efficient market.

Well, all of these options still require $2,900 in margin per contract for a market maker to sell and so the far out-the-money option prices will stay high enough to provide adequate compensation – otherwise, why would the market maker provide liquidity? If the Black Scholes model says the option is valued at 1 cent but the market maker's minimum return requires $1, the option will trade at $1.

That option is very unlikely to fall to 1 cent unless the market maker's required return on capital can justify such a price. This creates the volatility smile.

Now it’s true that market makers may have access to cheap funding and special margin arrangements with their bankers/the exchange which reduce the margin requirements somewhat, but all participants are still trying to generate an acceptable return on capital and so the logic holds.

In my view, this is a better explanation for why the smile exists than many of the competing theories. I could be wrong, but I am keen for anyone to refute my thinking.

 

Back to the poor question

So what is the main risk option sellers bare? The answer is adverse changes in our volatility expectations. After all, if we sell say a put option and the stock price moves a little or volatility falls/stays the same, that’s no problem. Our selling price would have us covered in all scenarios.

But with higher volatility, there’s a higher chance of us enduring a larger loss than we may have provisioned in our selling price.

Hence understanding how volatility expectations can change is fertile ground for our study into option pricing. The famed Black Scholes Merton model would assume that volatility is a constant over time but as many would know, this isn’t true.  So what model should we consider?

 

Introducing Gosset

There are several models which break this assumption – our favoured being the Gosset model (a link to its paper can be found here by Cassidy et al, 2008).

Below is the probability density function of the Student’s T distribution versus the Normal Distribution. Note the presence of fat tails.

The Black Scholes Merton model employs the normal distribution whereas the Gosset model employs the Student’s T distribution and there’s significant evidence (Platen & Rendek, 2007 for example) that this is a more apt distribution for option pricing on stock indices due to its fat tails.

The Gosset model is shown below for a vanilla European put option over a non-dividend paying security. For those not accustomed to advanced calculus it's a bit daunting, but the expression's logic is generally the same as that of the Black Scholes model. Only here you need to set a number for a concept known as the 'degrees of freedom' (DF).

The papers of Cassidy et al 2008 & 2010 review formulating expectations for future volatility whilst employing the Student's T distribution. Ignoring the complicated math of each paper, their model generally states that if you want to form future expectations about volatility, you need only the following;

  • The stock’s current volatility (as measured by the standard deviation of historical stock price returns).
  • The time horizon over which you wish to forecast.
  • The ‘degrees of freedom’.

To put it simply, the DF controls how fat-tailed we make our distribution and thus how much we expect volatility to change in the future. Put in a low number for the DF and you’ll get something very fat-tailed (e.g. the red line on the density functions above) and be assuming lots of future change in volatility. Put in a high number (30) and you’ll get something which looks akin to the normal distribution which is not fat-tailed (the blue line) and is assuming constant volatility.

Jumping back to our prior example, the red line would be used by a car insurer expecting highly volatile drivers (low DF & expensive insurance) since she’s expecting more crashes. The blue line would be used when they’re anticipating less volatile drivers and fewer crashes (high DF & cheap insurance).

Now let’s recap, we’ve established that we’ll likely get a better sense of an option’s value by considering:

  1. The margin capital required.
  2. How volatility expectations of the stock we’re writing the option over might change.
  3. Given (1) and (2), what’s the return rate for taking risk?

And using the Gosset model, we set our expectations for volatility for (2) by assuming a certain DF and knowing our forecast timeframe.

So what DF do we use?

 

The DF problem

Generally speaking, Cassidy et al (2010) found that for short-dated options, higher DFs should be utilised and in 2007, Platen and Rendek found that for longer-dated securities, a lower DF should be used. Hence a fairly logical relationship seems to exist in that for longer-term horizons we should assume fatter tails and for nearer term horizons less fat tails.

Platen, Rendek & Cassidy et al have thus given us a rather large range of between 4 and 30 to work with and this is a problem. To see this, let’s say we’re looking at a 1-year call option on a hypothetical stock. The stock is trading at $50, the strike is $50, the interest rate is 5% and volatility is 20%. If we value the option using a DF of 4, it’s theoretically worth $6.25 and yet with a DF of 25 it’s $4.38 – quite a gap. The answer is probably somewhere in the middle but where?

Working out the nature of this relationship and how it changes over time is difficult and requires some advanced statistics found in my own unpublished works developed in the latter half of 2017. These relationships drives the option pricing algorithms I employ at Vega Capital.

 

Using DF in practice

Without going into the advanced statistics, anyone can invest around this issue in the markets. Take the 1-year call option we considered above. Using a low DF of 4, we found that the value was $6.25. If we were in the fortunate circumstance that we believed the underlying stock was headed down and the market price of the option was say $6.60 (circa 5% above the value), then this would clearly be a very compelling option to sell.

Now perhaps the DF ends up being 30. That’s fine since we ended up selling something worth $4.38 for $6.60. If the DF ends up being 4, we’ve been well compensated for our risk. Hedging and other risk management techniques can be utilised to limit the impact of losses beyond the model bounds.

Vega’s advanced statistical understanding of how to set DF’s allows us to more precisely dig out such opportunities but the underlying logic is still the same – if we’re receiving more in compensation than the risk would justify, then clearly alpha has been found.

 

A freebie

As a final note, if you would like a copy of Cassidy et al’s 2008 Gosset formula, please feel free to contact me. I can provide it for personal use coded in either Maple, R or Excel. Users should be aware that Vega Capital nor I will be taking any responsibility for how you use the formula. The Vega Fund employs a similar but improved model so our valuations on various options may also differ.

 

 

Vega Capital is a wealth manager for high net worth individuals and financial institutions. Call us now on 1800 960 707 to find out more about how we can grow your wealth.

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