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An Option Pricing Model For The BuySide http://seekingalpha.com/p/1t1gn Jul 2, 2014
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An Options Model For The Fundamental Analyst
Despite a love of all things both nerdy and investing related, I find myself somewhat dispassionate about many of the modern quantitative methods in use today. Although awestruck by the mathematical revelations of Bachelier, Black, Merton, and Scholes, I find myself with a kind of subconscious resistance to much of their thinking. Truth be told, I'm probably not smart enough to fully grasp the differential and statistical equations that litter the pages of pricing theory textbooks. I am smart enough, however, to understand that many of the assumptions underlying the models can be less than perfect. Efficient markets, Brownian motion, riskneutral probabilities, normal returns, lognormal prices, homoskedasticity, and noarbitrage conditions are just a few that are often baked into an options pricing cake. Individually, each assumption might not be very far off the mark. Collectively however, they have potential to steer the models far from the realm of reality. Most importantly, there doesn't seem to be much mention of the company's underlying fundamentals anywhere in sight, and I'm not sure why.
For decades banks have hired armies of MBAs and CFAs who have spent countless hours digging through SEC documents, talking with company management, and working tirelessly on DCF models that incorporate factors like company market share, gross margins, tax rates, earnings estimates, free cash flow, and everything else an analyst might use to estimate the current value or the expected future price of a security. Reports are published with price targets or buy and sell recommendations. A typical conclusion may well read something like…
"After careful study of company SoandSo's current and projected financial condition, market position, strategic alternatives, and previous execution track record, we believe that the current stock price of $15.00/share does not reflect the true value of the company. We have set a price target of $20.00/share for company SoAndSo, with a time horizon of 9 months from today."
Yet at the same time on a different floor at the same bank, physics and applied science PhDs are busily pricing the options of these same stocks, seemingly with little regard for what their research analyst colleagues have to say about the company. These wizards of financial alchemy might care more about a security's historic or implied return volatility than the company's earnings or cash flow.
One can't help noticing just how divorced these two worlds appear. I find myself asking;
Many modern option pricing models start by assuming that the current market price is correct. The binomial model for instance, sends you on a trip into the future (a bunch of futures actually), tabulating option payouts along the way by constructing the probabilities of potential parallel future realities. Oddly, these socalled "riskneutral" probabilities envisioned by the model have more to do with ensuring consistency with current bond returns than with trying to predict future events. When the trip is over, the models come back to the present in a DeLorean with Marty McFly carrying an option price that's in every way consistent with today's stock price, current bond yields, and all possible future scenarios imagined. Like the flux capacitor, the model truly is a stroke of genius, but without all the fiction.
Unfortunately, many models such as the binomial model share similar drawbacks. Mainly, the future realities projected are rigidly constrained by what is happening today or in the recent past. These expectations are hinged to current stock or interest rate levels, and assume them to be "correct." If they weren't, then armies of arbitragers would make a bunch of money until prices came back into line, so the story goes. The models also assume that risk is defined by stock price fluctuations, or the volatility of stock price returns. In other words, how vigorously any random day's return bounces around its mean return.
But what if you were certain that the market is wrong, and you want to make an unhedged bet to capitalize on it? For some time this idea was considered a heresy in the academic community. It's clear now that the academics had latched onto unfounded memes like the efficient market hypotheses for way too long, despite the indisputable track record of so many successful value investors. It takes a certain collective tenacity by the academic community to ignore the significance of writings like "Security Analysis" or "The Super Investors of Graham and Doddsville". I believe that this can only be justified by a fatal flaw in the constitutions of some of our oldest and most respected institutions. Ivory tower hubris, perhaps. Only now has the consensus finally started to drift toward ideas that have been common understanding since the days of Graham himself. Mainly that, markets can be very wrong, and a keen observer can profit from it.
A Model for the BuySide
This option pricing model is geared towards research analysts with a view of the future. As such, it starts from the future, and works its way back to the present. Not only does the model price an option based on ones assumptions of the future, but it also provides an implied current fair value of the underlying based on that future.
WARNING: Option critics would often profess that it's hard enough to value a stock, let alone get the timing right on an option. They would argue that since the expiration of an option acts like a fuse on a ticking bomb, options can be much more risky than being long or short the underlying. I believe the critics are absolutely right and their warnings should be heeded. Models such as these probably work best for long dated options, where prices have ample time to correct to ones expectations and the correction time window can be predicted with some level of certainty. If the timing aspect of your position is too uncertain, you're probably better off just buying the underlying.
To start, let's assume that, upon completion of extensive due diligence, the analyst can predict with some level of certainty where the stock price will be on the expiration date of the option. Certainly, this is no easy task. It's never possible to predict the future with exacting precision. What we can attempt to do is form an expectation of the future by our knowledge of the present, then estimate how uncertain we are about it. If we can be honest with ourselves about our uncertainty, we can form a conservative probabilistic view of future events. Once a reasonable expectation of the future is established, it's a simple task to find the expected option payout and discount that back to the present.
One solution is to assume that the future price of a stock will fall within a range of equally probable prices, with the average of that range being the expected value. For instance, if a European option expires in a year from now and we expect the stock price to be $10.00 at expiration, we can make an assumption on the range of possible prices at that date based on our uncertainty with our own analysis and timing expectations. Let's define the range as $10.00 ± $5.00 and assume that all prices in the expected range are equally probable. We can also assume that the exercise price of the European option is $12.00. What we end up with is a probability density function that looks like Figure 1.
(click to enlarge)
Figure 1Probability Density Function
The function shows an equally likely chance that the future stock price will fall between $5.00 and $15.00, with an expected price of $10.00. The area under the curve must always sum to one, thus explaining all possible future events.
If we then multiply the probability density function, P(S_{T}), by the call (put) option pay0ff function f_{c}_{(}S_{T)} (f_{p}_{(}S_{T)}) at each interval , we get;
The results of our example are shown graphically in Figure 2.
(click to enlarge)
Figure 2  Payoff Distribution of a Call and Put Option with Strike Price of $12.00.
To find the option value at expiration, simply integrate (sum) the area under both the green and red curves from 0 to +∞ to get;
The expected payoff in our example yields;
c_{T}= $0.45
p_{T}= $2.45
Finally, we can discount the expected option values to the present using the current risk free rate.
In our example, the current European option values assuming a $12.00 strike price, a 3.0% risk free rate, and one year until expiration are as follows;
c_{0}= $0.43
p_{0}= $2.38
Notice that these values were found without knowledge of the current stock price. When pricing American options, one can simplify the problem by taking the higher of the European option value, or the difference between the current market price and the strike price. This assumption implies a smooth price convergence toward the expected future price over the time of the option.
Honing the Model
The previous example assumed that there were equal probabilities for each potential future outcome inside the range chosen. Since the probability density function represents the analyst's subjective or a posteriori probabilities, there are no restrictions on the shape of the function used in the model, as long as the total area under the curve sums to 1. Since the analyst typically expects a higher probability for a certain expected future value, and a lower probability for values farther from the expected value, a bell shaped curve may better reflect the analyst's expectations. Instead of equal probabilities, we can use a normal distribution, where the analyst's expected future stock price at expiration is equal to mean µ, and the uncertainty inherent in the analyst's valuation and timing expectations is equal to standard deviation σ. Notice here that σ has less to do with historic or implied return volatility, and more to do with uncertainty of future prices. The model assumptions are as follows;
As in the previous example we have;
But in this case;
P(S_{T})= One or more normal distributions defined by the expected future stock price at expiration µ, and the uncertainty inherent in the analyst's valuation and timing expectations σ.
To proceed with our example, if we define the expected return µ=$10.00, with a standard deviation σ= ±$3.00, we get the following probability and payout distribution shown in Figure 3. (Remember here that σ represents our uncertainty of the future, not return volatility.)
(click to enlarge)
Figure 3 Expiration Payoff of European Option with $12.00 Strike Price and Expectation Defined By Normal DIstribution of µ =$10.00, and σ=$3.00.
When using the normal distribution, we can define a confidence interval to be used as a mental litmus test. If a 90% confidence interval is defined, the analyst would expect that 9 out of 10 expected future price scenarios would lie between µ ± 1.64 σ, or in our case between $5.07 and $14.93. With the future probabilities defined as they are, and a risk free rate of 3.0%, a one year European call and put option with a strike price of $12.00 is worth;
c_{0}= $0.44
p_{0}= $2.37
By varying the exercise price, a model generated option pricing table is produced as follows;
(click to enlarge)
(Here we assume a current stock price of $11.50 to determine American option values and ITM status.)
There are a number of observable differences between this model, which I'll call the expected value model, and others such as BlackScholesMerton. In the BSM model, call values increase and put values decrease as the risk free rate increases. This is due mainly to the expectation that the stock prices drift upward from the current price, allowing the holder of the underling to earn a return that is at least equivalent to that of a risk free bond. The expected value model on the other hand starts at expiration and works back to the present. Thus, any return expectations of the underlying are already implicit in the probability density function defined by the analyst. If the analyst believes that a stock price will grow at the risk free rate (or more likely at the company's sustainable ROE in the event that no dividends are paid out), she will reflect that in her expectations of the future. The only explicit effect of increasing the risk free rate assumption is to decrease both call and put values as a result of the discounting process.
On the surface, it may also appear that the expected value model is much less sensitive to the timetoexpiration than BSM. As this time is varied, the risk free rate appears to be the only influencing variable affecting the option value, which may have relatively little impact when rates are low. However, a proper evaluation made by the analyst of the uncertainty inherent in greater lengths of time would likely result in the modeling of a larger expected standard deviation as time to expiration increases. This results in a larger confidence interval, and acts as an additional indirect factor affecting option value.
Finally there is volatility. Unlike the BSM, the expected value model cares nothing about the volatility of returns. The "σ" of the expected value model embodies the risks one inherits each time one tries to predict the future (without a DeLorian and a flux capacitor). The information available to us is imperfect, the past is only a soso predictor of the future, and our understanding of markets is nothing like our understanding of our planetary orbits. We just can never know the future for certain, but we might be able to estimate the bounds of our uncertainty.
Stretching the Model
Occasions often arise where two distinct future outcomes are possible. This might happen following a merger announcement where the closing is less than certain, prior to a material litigation ruling, or before a critical earnings announcement. The expected return model can account for this by defining a probability density function with two separately defined probability curves. The following example will explore this concept.
Example Post Merger Announcement
A tender offer has just been announced for the XYZ company at $15.00/share. It is being bought by a private equity firm with a checkered history of closing deals. After extensive due diligence, an analyst determines that the probability for the deal closing is about 70%. For simplicity, we can assume that the deal is expected to close on the option expiration date. He believes that it is reasonable to expect a $15.00 future price within a narrow band of expectations. However it is also expected that the stock price might fall to $9.00 if the deal falls through. The analyst is less certain about the magnitude of the price drop if this happens. He constructs the following probability distribution based on his expectation;
(click to enlarge)Figure 4  XYZ Company Expected Probability Density Function
The distribution shows two curves, one with an expected price of $15.00 and standard deviation of $0.50, and the other with an expected price of $9.00 and standard deviation of $2.00. The area under the left curve is 7/3 times the area of the left, reflecting the analyst's expectation of the deal closing. The cumulative probability of the entire function must always equal one. If the analyst is interested in pricing a call and put option with a strike price of $15.00, the following payoff distribution will result;
(click to enlarge)
Figure 5  XYZ Merger Payoff Distribution with $15.00 Strike Price
The modelgenerated European option values assuming 6 months until expiration, a 3% risk free rate, and a $15.00 strike price are as follows;
c_{0}= $0.14
p_{0}= $1.91
With the current price at $14.20, the modelgenerated pricing table is;
(click to enlarge)
PutCall Parity
PutCall parity states that a fiduciary call (C_{0} +PV(Exercise Price)) should remain equal to a protective put (S_{0}+P_{0}), or an arbitrage opportunity is possible. Typically it assumes that today's current market price S_{0} is as it should be, and the option prices must adjust to maintain the balance. However, the expected return model assumes that S_{0} may be incorrect. In our case, putcall parity is maintained if it assumed that one's expectation about the future is correct, and the implied spot price that results from this view is substituted for S_{0}. The implied spot price is simply the present value of the future expected stock price. Revisiting the previous example, the putcall parity equation is as follows;
If anyone is interested in the copy of Excel file, I expect to post a link in the comments sometime in the near future. Here is a screen shot of what it looks like.
(click to enlarge)
Joe Gradzki
The Economics Of Share Repurchases
One of the many great challenges of investing is properly evaluating the business economics of a share repurchase plan. The subject has left many of us scratching our heads. The intent of this article is to dispel the misconceptions and halftruths commonly associated with repurchases. The basic mechanics of repurchases are illustrated, as well as the economic consequences to the continuing longterm shareholder. In addition, the increasing marginal benefit of share repurchases will be discussed, as well as the benefit that arises when new investors acquire stock during a period of both undervaluation and ongoing share repurchases. Finally, the responsibility of a company's management to properly evaluate a repurchase opportunity is addressed.
The mechanics of a share repurchase are straight forward. Typically, the management of a company uses excess cash on the company's balance sheet to purchase outstanding shares of its own company. Often times, the shares are bought directly from the secondary market at the prevailing market price. These shares are effectively taken out of circulation, creating a greater concentration of equity ownership in the hands of the remaining shareholders. Although it's natural to assume that this will benefit shareholders in all instances, it's not always the case. The following examples will illustrate why this is;
Example1: Share Repurchases During a Period of Undervaluation
Company XYZ is a hypothetical company with a total of only 5 common shares outstanding. This simple company does not own any property, plant, or equipment, doesn't provide a service, has no debt, doesn't pay any salaries, and doesn't do much of anything for that matter. It does, however, hold two types of assets:
Let's also assume that the tax rate on all company dividends and earnings is 0%, and the discount rate used to value the earnings is 10%. (These assumptions, although unrealistic, will affect only the company valuation, not the underlying principles discussed here).
Under these conditions, the $50 in the company's bank account should generate at least $50 dollar of intrinsic business value, or $10/share. The $10/year of perpetual interest payments would add a present value of $100, or $20/share (The value of a perpetuity = cashflow/discount rate). During "normal" times, the market value of the stock will most likely be aligned with the intrinsic value of the company. The shares will trade close to their fair value of $30/share. This scenario is illustrated below:
(click to enlarge)Each of the 5 shares of company XYZ have an equal claim on 1/5th of the $50 in the bank account, as well as 1/5th of the earnings value from the perpetuity. The different colors represent the two different types of assets, the green blocks being cash, and purple blocks, earnings power. The relative length of each block depicts the dollar amount of the asset's value.
Let's now assume that the entire stock market suddenly loses 1/3 of its value as the result of a large widespread market panic. The fundamentals of XYZ remain intact (and interest rates remain unchanged), but the share price of the stock plummets to $20/share from its original $30. Each share can now be acquired for a discount to its intrinsic value of $10/share. The situation is represented below;
(click to enlarge)
Aware of this large undervaluation, management decides to repurchase its own stock. Since the company does not need the cash to make operating or capital expenditures, the management considers all of the $50.00 as "excess" capital, and is free to spend it buying back stock. The next trading day the company buys back a single share of XYZ (20% of the outstanding shares) for $20/share. The result of the share repurchase is shown below.
(click to enlarge)
Since the company has one less shareholder after the repurchase, the $20 earnings value of the repurchased share has now been transferred equally to the remaining shareholders. This will materialize in an annual earnings increase of $0.50/share, or an increased present value of $5.00/share. It's critical to note that the EPS increase is only half of the story. Since the stock was repurchased with $20.00 from the company's balance sheet, the company's new cash position has decreased to $30.00, or $7.50/share. To properly evaluate the economic effect of the repurchase, both the benefit and the cost to remaining shareholders must be considered. In this example, each remaining shareholder paid 2.50/share in cash to receive $5.00/share in earnings value. The net effect of management's decision to buy back shares resulted in an increase pershare intrinsic value of $2.50, to $32.50, or 8.3% (relative to intrinsic value). Put another way, for every $1 the continuing shareholders gave up in cash, they received $2 in earnings value from the folks leaving. This example demonstrates a repurchase situation that created value for the remaining shareholders.
Example 2: Share Repurchases During a Period of Share Overvaluation
Now let's assume the share repurchase in Example 1 never happened and XYZ is back to 5 shares. This time, instead of a depression, the market surges into a period of euphoria. As a result, market values become detached from underlying business values and market prices skyrocket. Instead of the company selling in the market at a discount to intrinsic value, as in the previous example, it now sells for a $10 premium, or $40/share. XYZ's financial position is shown below;
(click to enlarge)
Despite the share premium, the management decides to repurchase one share of the company's stock. XYZ's financial position after the repurchase is as follows;
(click to enlarge)
This time the economic effect of the repurchase yields a quite different result. As in the last example, management's decision to repurchase shares has led to an increase in EPS of $0.50 per annum, or an added earnings value of $5/share. (The management team will always be quick to advertise this seemingly pleasant result). The unfortunate truth is that the economic benefit to the remaining shareholder did not outweigh the cost. As one can see, the company spent $7.50/share for only $5 of increased earnings value. The net result of the repurchase is a total pershare intrinsic value decrease of $2.50. In this case, the management effectively took $10.00 of cash from the balance sheet and donated it to the shareholders dumping the stock. So much for fiduciary duty!
It becomes clear that the increase in pershare intrinsic value resulting from share repurchases originates exclusively from the undervaluation of the repurchased stock. In Example 1, the $10.00/share discount was effectively transferred prorata from the single exiting shareholder to the four remaining shareholders. During a well executed share repurchase, capital is not so much returned to shareholders as it is transferred from active ones to the less active ones. (More on this later). In Example 2 however, that same $10.00 flowed out of the company to the selling shareholder.
A Bit of Math
We can mathematically express the value added (or lost) as a result of the repurchases. The following formula shows the economic pershare value gain or loss (in dollars) as a result of repurchases in a given period;
(click to enlarge)
The formula simply expresses what was already observed in the previous examples. It shows that the pershare value created (or destroyed) as a result of share repurchases is simply the average pershare discount to intrinsic value (V_{i}P_{avg}), times the number of shares repurchased S_{rp}, distributed evenly across the remaining shareholder base S_{o}.
It's just as useful to express this result as a percentage of intrinsic value;
(click to enlarge)
Close investigation into these formulas yields the following conclusions;
(click to enlarge)Figure 1 Economic Benefit of Repurchases at Different Market Discounts
The graph uses Equation 1c to plot the value added pershare as a result of repurchases, V_{c}/V_{i}, against the percentage of the company repurchased in a given period, S_{rp}/S_{tot}. Each curve represents a different discount to intrinsic value. As discussed above, a greater discount will result in a proportionally greater added value. In this undervalued condition, the value added to each share increases exponentially as more if the company is repurchased.
In practice, the amount of shares repurchased will be limited by the amount of excess cash on a company's balance sheet. A greater market discount will not only magnify the repurchase benefit, but will also allow for a greater number of shares to be repurchased in a given period. This will thus maximize the increasing marginal benefit effect of the repurchase. The benefit to shareholders becomes selfreinforcing the more the stock price falls relative to value. The dotted lines in Figure 1 represent the maximum amount of shares available for repurchase with a given amount of excess cash. The excess cash is represented as a proportion of intrinsic value. It's clear that the optimum benefit to shareholders will result from the repurchase of the maximum amount of shares possible during a period of maximum undervaluation. In the case of share repurchases, an opportunistic management is the friend of the long term investor.
Example 3: Entering Shareholders
So far, the discussion has focused on the benefits of share repurchases to existing shareholders. However, shareholders buying into a company during a period of both undervaluation and share repurchases can receive substantially more value than the existing shareholders of that same company. Both groups receive the benefit of the share repurchase previously discussed, but the new shareholder also receives the value of the current market discount. This is because the initial investment is made at the depressed market price. The outcome can be illustrated using the postrepurchase results of Example 1 presented earlier;
(click to enlarge)
Since a new shareholder can buy a share at the depressed market price of $20.00, the total benefit from this well timed share acquisition is $2.50/share due to the repurchase, plus $10/share from the market discount. The savvy investor can buy $32.50 of value for only $20.00. This "dual benefit" for the new shareholder is expressed mathematically below;
(click to enlarge)
Equation 2b is also presented below as a percent of intrinsic value;
(click to enlarge)
The expression sums both the value created from the ongoing repurchase and value obtained from the initial purchase of the undervalued share. It also assumes that the average purchase price of the entering investor is the same price the management is paying for their shares. If the company is indeed undervalued, the total economic benefit received by the new shareholder will be, at a minimum, equal to the discount to intrinsic value. The benefit will multiply as the company repurchases greater amounts of stock.
Management Responsibility and the Return of Capital Myth
Managements that choose to repurchase shares must first acknowledge that, unlike a dividend, their judgment alone determines the benefit to shareholders. As shown above, the optimum benefit arises when repurchases are executed shrewdly. Thus, it is managements' responsibility to perform the following tasks when confronted with the decision of share repurchases;
Incidentally, these guidelines are nearly identical to the due diligence performed before any typical business investment or acquisition. The main difference for a repurchase evaluation is that management already knows everything there is to know about their potential target. Although share repurchases are often labeled as a "return" of capital, and can be shown as mathematically equivalent under certain circumstances, it may be more appropriate to classify them as a reinvestment of capital. Beware of managements that hide behind the pretext of the former definition to avoid the accountability inherent in the latter. Whether investing in whole companies, real estate, oil wells, or the earnings of exiting shareholders, the benefit to long term shareholders originates from management paying less for an asset than what that asset is intrinsically worth. As in any investment, it's management's judgment and skill, or lack of, that will be the deciding factor. When evaluating companies that regularly repurchases their own stock, choosing a company with the right management will be just as important as choosing the right company. Be sure to choose wisely.