The Black-Scholes model does an admirable job at valuing short-term options. If an option expires in a few weeks, the current price of the underlying stock and its recent volatility have a good deal of influence on the outcome of the option investment. A simple Black-Scholes calculation has a lot of flaws (none of which I’ll go over), but in my opinion it does alright on the short-term options. However, the further away the expiration date, the worse it gets.

Value investors know that the historic volatility of a stock has nothing to do with its long-term value, and therefore should never be used when making a purchase. However, when purchasing equities, value investors have the luxury of waiting however long they need until price eventually reaches fair value.

If a stock is worth $30, that doesn’t mean a call option with a strike of $20 is worth $10. The option value must also depend on the **duration **of the option: the further out the expiration, the greater the underlying valuation should affect the option price (and the less volatility should matter). A lot of value investors purchase LEAPs, or options a year or more out, for this very reason.

## The *Graham-Olson Option Valuation Model *

In honor of Benjamin Graham, I put forth the following equation as the value of a call option:

* Where*:

IV = Intrinsic value of underlying stock*

SP = Strike price of option

BS = Black-Scholes valuation of option

x = Time to expiration (in years)

The Black-Scholes value can be calculated using a spreadsheet model or from websites like this.

**Here’s a practical example**: Let’s say that you think Burlington Northern (BNI) is worth $90-110. The *Graham-Olson* model values the $80 January 2011 calls at **$8.3-$19.5**. Time to expiration is 1.75 years and the Black-Scholes model uses volatility of 25%, risk free rate of 3% and current price of $68.

*The graph to the right represents both valuation models of the BNI options with an IV of $100. The x-axis is the time to expiration in years. *

The formula isn’t very precise, but then again, neither is value investing. The *Intrinsic Value* input is obviously very subjective (that’s why I’d probably use a “range” of valuations like in the example above).

The numbers 3 and 5 in the exponent adjust the”shape” of the graph so that on average, a stock should reach its intrinsic value in around 4 years. As you can see from the graph above, the equation puts BNI at approximately intrinsic value in 3 years. These numbers can be adjusted based on how long you think the average stock takes to reach fair value.

## Conclusion

I think that most value investors who purchase options already intuitively use the above method when making a purchase. But the *Graham-Olson* model can be used to check your assumptions using a variety of different inputs.

If a stock is *overvalued*, it shows that the Black-Scholes formula can overprice even very short-term options. There are also many occasions when an option reaches its value even though the underlying stock hasn’t — in this scenario the *Graham-Olson* model could be a useful guide of when to sell.

If you have any suggestions or criticisms please feel free to comment below.

* In reality, IV should be the present value of your estimate of IV at the time of expiration.

What is ‘e’ in the denominator of your equation.

Also, I am a little unclear here – if you believe intrinsic value is reached in 4 years you use e^(-3x + 5) as the exponent.

If you believe it takes two years to reach intrinsic value would you then use (-x + 3) for the exponent?

I.e. how do you adjust the exponent for your target time frame for intrinsic value to be reached?

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Jamie:

The “e” represents the natural logarithm. It should be

e^xon a calculator or EXP() in Excel.Unfortunately, it’s not as simple as picking two numbers in the exponent that average out to the optimal year. The fact that 4 is the average of 3 and 5 is just a coincidence. I think the best (and least precise) method is just to eyeball a graph of the equation and adjust the numbers until it reaches the limit at x number of years to intrinsic value. So for two years, it would be

[edit]e^(-4.5x+5). The 4.5 number is approximate.,[/edit (sorry, had the equation wrong)]On a related note, even though the equation still works using shorter time periods to IV, I personally would never use anything under 3 years. For example, if you assume the stock gets to IV in 2 years, a call option expiring in Jan-11 (1.75 years from now) would be valued very close to its “full” value. It very well might happen, but I think it is a little unrealistic and not conservative enough.

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Is there any motivation for this formula? I don’t want to put you off, but just proposing a formula without 1) a theoretical justification or 2) a motivation why the formula would work from a pragmatic point of view (I’m the first to accept that there is use in formulae that are not theoretically correct but for which there is lot of pragmatic evidence why it works).

Furthermore, your statements seem to imply that the B&S formula doesn’t give weight to either the intrinsic value of a call or to the time value of an option, because it does.

The point of B&S option valuation (or the basis of it) is risk-neutral valuation. This means nothing more or less than that we value options in such a way that (given a set of assumptions) the option value can be replicated with the underlying, so that no one can make risk-free profits. From a pragmatical point of view, the real value of it, comes from the partial derivatives it has, its “greeks”. More specifically (if volatility realizes the implied volatility) than with delta hedging you exactly (under some assumptions) realize the option premium. Tweaking the the B-S formula when you have a view in the end probably will entail nothing more than being more long in the underlying then standard B-S greeks would cause you to do so. However, you did already knew that! đź™‚

Also, from another angle I see a flaw in your reasoning. You seem to want to determine a “fair value” for an option based upon your idea of the fair value of the underlying. My point so-far was that the fair value of an option doesn’t have to do with the fair value of the underlying, but with the costs of replicating an option. My second point is that the fair value of an option by itself is useless… I assume that you consider options for this article as another way of investing in the underlying. You now have a true value of the option, if held to maturity. So, what does this tell you? That options are over or undervalued? Now what? My point is that it doesn’t help in telling you whether it is better to buy an option or another one or just shares. Furthermore, even if you are right about the value, then you also still have to be right about volatilities before you can make a profit on the option. And that is sort of complex.

What I would like to propose is a slightly different approach (I’m willing to explain it more in depth than in this comment and I have some excel sheet in which I have developed this as well). If we assume that options will remain being valued with B-S formulae by the market than we can buy an option with say a 3 year maturity. Now if we at the same time assume a probability distribution for our share in an intermediate period (say, after two years) and we assume a volatility associated with it (not too difficult actually!) then we can calculate the distribution of option values and compare those with distributions of direct investements in the underlying and investigate which is the more interesting investment.

For example, suppose we are interested in Wells Fargo which we think should be valued at 45. What can we do.

1. We can buy the share now (say at $20) and hold it and keep earning returns on it or sell it when we feel it is overvalued.

2. We can try to make more money with a 2 years option if we have an opinion of where the underlying will be in 1 year. For example, suppose that we think that in 1 year the value of the Wells Fargo will be about $30 (half of where we think it should be), than we can assume it to be normally distributed with a mean of $30 and let’s say a standard deviation of $3 (any discrete distribution will work as well). Just investing in the the share will then give us an expected return of 50% with a standard deviation of 15%. However, what we also can do is take the $35 call. We know its price and volatility now. For each value of WF in 1 year time we also know the value if we know the volatility (for example, if WF is at $30 and we think volatility will then be xx% then we can plug in these numbers in a B-S calculator to get an option value). If we repeat this process over and over for a lot of values in the distribution of the share price of WF then can compute an expected option value that is associated with a particular distribution for WFC – for example our assumption of normally distributed with a certain standard deviation and mean). As a result of this, we can then compute an IRR of the option that we can compare with the IRR of buying the share.

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Sorry it took so long to respond. I appreciate the comments/criticisms, it was exactly what I was looking for. Before I respond keep in mind that I only know the very basics of the B-S model, and I am just a humble value investor not trained in financial math/statistics. Let me reply to your points:

Is there any motivation for this formula? â€¦ Furthermore, your statements seem to imply that the B&S formula doesnâ€™t give weight to either the intrinsic value of a call or to the time value of an option, because it does.I came up with this formula in order to have a better way of integrating an estimate of intrinsic value into an optionâ€™s price. It was a way of â€śputting on paperâ€ť my prior beliefs of how I thought about an optionâ€™s valuation. In other words, that it lays somewhere in between full value (i.e. $10 for your WFC example above and a $35 strike) and the short-term accurate Black-Scholes value. I didnâ€™t mean to imply the B-S formula doesnâ€™t take into account time value, as seen from the graph above it does very well. However, I wanted this formula to include

personalestimate of IV, and how long it takes to get there. The larger the spread between price/IV of the underlying stock, the more time will affect option value.Tweaking the the B-S formula when you have a view in the end probably will entail nothing more than being more long in the underlying then standard B-S greeks would cause you to do so.Like you said, I understand this, but when I purchase a call option thatâ€™s really my goal.

So, what does this tell you? That options are over or undervalued? Now what? My point is that it doesnâ€™t help in telling you whether it is better to buy an option or another one or just shares. â€¦ As a result of this, we can then compute an IRR of the option that we can compare with the IRR of buying the share.True for the most part. If you are looking at an option as it really isâ€”a contract to buy/sell at a certain priceâ€”and you actually exercise the option (buy or sell the underlying shares), then from my point of view valuing an option isnâ€™t too useful. If you think Wells Fargo is worth $45, and you plan on exercising those options, the way I use â€śtime valueâ€ť in my formula is of no use. But if you donâ€™t plan on exercising them (you plan on selling them on or before expiration), I think my formula (or just my way of thinking) can be helpful for knowing when to buy/sell.

The direct comparison between potential IRR from either an option or share purchase is an interesting idea that I hadnâ€™t thought of. Some may not have the option (i.e. they donâ€™t have enough capital to purchase the shares). But this is interesting; Iâ€™ll probably look into it further.

For example, suppose that we think that in 1 year the value of the Wells Fargo will be about $30 (half of where we think it should be), than we can assume it to be normally distributed with a mean of $30 and letâ€™s say a standard deviation of $3 (any discrete distribution will work as well).I did consider using a normalized distribution around the IV estimate. It makes the most sense from a mathematic stand point. However, I had two problems with this: (1) Even if I wanted to incorporate it, Iâ€™m not really sure how (again, Iâ€™m not too well versed in the math side); and (2) If you assumed that the â€śmeanâ€ť was $30, a normal dist. says that half of the time it would be above that. Of course that makes sense in reality, but as a value investor Iâ€™m uncomfortable with saying that it

mighttrade above intrinsic value at a certain time. Hence with my formula, I tried to make sure the option was never worth more than (IV â€“ Strike).I wrote the above fairly quickly, so I hope that some of it makes sense.

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