Today I hope to address this question from one angle. What does standard finance theory actually have to say about the matter?

Now of course standard finance theory may very well be wrong. Some of the assumptions that will be used today are problematic.

As it turns out, this presentation will be a sort of a proof by contradiction. I believe it will show that standard finance theory is

*inconsistent*. Or at a minimum, certain propositions that are often assumed in various financial models are

*incompatible*depending upon interpretation.

While my interest in this is somewhat academic (aren't all of my interests?) there may be a practical application to this which has to do with estimating the cost of equity. It may be possible to use this device as an alternative method of determining cost of equity (say, versus the lousy CAPM model.)

But that will have to be worked out some other time. Today I just want to show how one can calculate the expected return on a call option.

#### The Problem

What got me to thinking about this is related to the problem of constructing a call option index. How would one construct an index of call options and what returns could one get from such an index?

As far as I can tell, there's no good way to construct such an index. At some point the options will expire worthless. So won't the index eventually go to $0?

After doing a preliminary search, I haven't found anyone that has attempted to construct such an index. (Does anyone happen to know of anything?) About the closest thing I can find are protective put indices. A protective put index involves buying an index and also buying a protective put option on the portfolio. As an example of this, see the DAX plus Protective Put Index which buys the DAX and buys a 3 month, 5% OTM (out-of-the-money) put option on the DAX.

According to put-call parity, this is equivalent to buying a call option and putting cash in a cash account earning "risk-free" interest. This addresses the problem of having the options expire as a large percentage of the assets will be held as cash which can then be redeployed.

Simply buying calls, however, is like taking a levered position in a protective put strategy. That leverage can result in substantial losses.

So instead of looking at an index, I'm going to model the expected returns of a call option based on

*theory*. This has obvious drawbacks of course. Part of my goal, however, is to show problems with traditional finance theory.

#### The Model

I will assume, as the Black-Scholes equation does, that stock prices follow a Geometric Brownian motion with mean $\mu$ and standard deviation $\sigma$. This results in returns following a normal distribution. For example, if annual returns follow a normal distribution where the mean is $\mu = 8\%$ and volatility is $\sigma = 16\%$, then the distribution of possible returns for the stock will look like this:

We can then calculate the probability that the stock will be between $x\%$ and $y\%$ of the current price by measuring the area under the curve between those two points (the total area is equal to 100%).

We can then multiple the probability by the returns in that little block and add them all up. This will give us the expected return. If you were to do this procedure, you would get 8% which is about what you'd expect (considering that this distribution has a mean $\mu = 8\%$.)

But this same procedure can be done for the call option. We can calculate the expected value of the call option by multiplying the value of the call option by the corresponding probabilities and then adding them all up.

__Example Calculation__

Allow me to illustrate with a simple example that should show how this works. Suppose there is a stock that's currently trading at \$100 whose price returns follows a normal distribution with $\mu = 8\%$ and $\sigma = 16\%$ (so it looks like our graph above).

Suppose we purchase a call option with a \$108 strike price that expires in one year. If you look at the normal distribution, you'll see that half of the area under the curve falls below the 8% return mark (\$108). So there's a 50% chance this option will expire worthless. So we can ignore it if the returns are anywhere between -100% and 8%.

Since the normal distribution has thin tails, we can also ignore returns in excess of about 72% as those amount to about a 0% chance of occurring. The result is a table that looks like this:

The total probability sums up to 50% which is fine because when we add that to the other 50% (for returns between -100% and 8%) it will sum up to 100%.

I implicitly assumed the stock would be about halfway between the return values. So between 8% and 16%, I assumed the stock would go up 12% to \$112.

The estimate for the expected value here is \$6.51. The present value of the call option according to Black Scholes (with a dividend of 0%) comes out to \$3.46. This amounts to an expected return of 88.15%.

Of course some context is required here. There's a 50% chance that you'll lose 100% in this scenario.

__A More Precise Calculation__

Now the calculation I did here is not terribly precise. Our intervals were 0.5 standard deviation intervals. Fortunately, we can utilize the calculating power of computers to give a more precise calculation. We can reduce the intervals to, say, 0.01 intervals and that will make the calculations better.

We can also calculate higher moments as well to give a more general characterization of what the theory says option returns look like. So here's what that looks like.

#### Call Option Returns

Like the previous example, I'll assume we're dealing with a stock that's currently trading for \$100 whose price returns will follow a normal distribution with $\mu=8\%$ and $\sigma=16\%$.

Here's what returns look like against volatility:

Here's what they look like against strike price:

As one might expect, as you move to higher strike prices, volatility increases. A higher strike price is, after all, riskier than a lower strike price.

__Contradiction in Traditional Finance?__

But something strange happens. Expected returns actually start to decline as you move further out. Typically, traditional finance assumes that the relationship between risk and returns is a monotonically increasing function which is just a fancy way of saying more risk = more returns.

One possibility here is that other characterizations of risk (such as skewness and kurtosis) may be affecting risk in different ways.

When the strike price is low, the option will behave more like the stock (which we assumed to follow a normal distribution.) So it will not exhibit any skewness or excess kurtosis.

But as you move to higher strike prices (especially out of the money options), we start to see both skewness and kurtosis.

The positive skewness comes from the fact that, a good portion of the time, you'll end up losing money (often 100%) but every now and then you'll hit it big. The positive excess kurtosis comes from the fact that "tail" events are going to be more common than the normal distribution.

How to translate that into risk and returns is a subject for another blog (and I'm not sure if I have an answer for that.) Suffice to say, the material taught in traditional finance courses is inconsistent with its own theory.

#### Final Notes

**1) Calculations in Excel**Most of the calculations were done in Excel using some custom functions. I already have two of the functions posted here:

Financial Excel Add-In

I may later add the option expected value function which calculated the raw moments for a call or put option. This enabled me to do all of the analysis above fairly easily.

Feel free to try out the add-ins and leave feedback on any problems you may have as it was programmed by a moron (yours truly.)

**2) Normal Distribution Assumption**It's been noted that stock returns don't follow a Geometric Brownian Motion as the theory assumes. But there's another interesting consequence here. If you assume that they do (as the theory does), then call options will not as they will exhibit nonzero skewness and excess kurtosis (especially as you move to higher strike prices.

What makes this interesting is that if you assume that the

*firm*follows a Geometric Brownian Motion then by the Merton Model

^{1}it follows that stocks will not. In other words, the Merton Model is inconsistent with the Black-Scholes model which I find quite ironic.

The defense here would be that skewness and kurtosis are near zero provided the strike price is low (leverage is relatively low.) You really don't see the optionality take effect until a strike price of around \$60 which corresponds to a leverage of 60% (debt to equity would be 1.5). So the normal distribution may still be a reasonable assumption provided leverage is low.

Granted, given the empirical literature on stock returns, it seems likely that the firm's returns do not follow a normal distribution.

__3) Cost of Equity - Alternative Method__One potential application of my derivations above is that it may be possible to derive an alternative cost of equity for firms. If it's possible to use market based credit spreads and plug them into the Black-Scholes + Merton Model to derive an implied volatility for the

*firm*, we could then use that information (along with the leverage ratio of the firm) to derive a cost of equity for the firm based upon the expected returns of the call option.

I suspect David Merkel has something along this in mind when he discusses his idea regarding cost of equity. See the following blogs:

Toward a New Theory of the Cost of Equity Capital

Toward a New Theory of the Cost of Equity Capital, Part 2

My Theory of Asset Pricing

I suppose the main drawback is that I would need to assume a rate of return for firm. In other words, I would need to know the unlevered cost of capital which is roughly WACC. Since cost of debt is also known it would seem trivial to simply calculate the cost of equity.

At the very least, it would provide an alternative to standard CAPM based on bond yields for the firm.

__4) Expected Returns versus CAGR__The model I use is a calculation of arithmetic returns, not compounded returns. Compounded returns will, of course, be lower than the arithmetic returns. I only note this for clarification.

**5) Risk versus Return**As I noted above, the relationship between risk and return is not monotonically increasing as theory suggests. There are several possible defenses of standard finance theory that might save it from this criticism.

- As the strike price gets higher, we start to see nonzero skewness and kurtosis. Our theory of risk and return would need to factor in these as possible risk factors. At one point in the progression, the increase in volatility does not result in an increase in expected returns but that may be offset by the skewness and kurtosis. When those get factored in, it may be possible to maintain the monotonically increasing relationship between risk and return.
- The relationship fails when the strike price is higher than the asset price. We might, then, question whether or not the risk/return relationship even needs to hold when the strike price is beyond the value of the asset.
- As it turns out, some assets have a negative expected return. Put options, for example, (which act like insurance) have negative expected returns in spite of being quite risky (in terms of volatility). So there might be other theoretical grounds for believing the standard risk/return relationship need not hold.

^{1}The Merton Model begins by supposing that the firm is a tradable asset and divides up the firm into (1) bondholders (which are considered a homogenous class) and (2) stockholders which have an equity or residual claim on the firm.

If the firm issues options on it, it follows that the equity claim is akin to a call option and that cash covered put options are akin to zero-coupon bonds on the firm. This allows one to analyze corporate debt using the Black-Scholes equation.

For the original paper, see On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.

## No comments:

## Post a Comment

Some common OpenID URLs (no change to URL required):

Google: https://www.google.com/accounts/o8/id

Yahoo: http://me.yahoo.com/