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The Buffett Critique: Volatility and Long-dated Options
Neeraj J. Gupta, Ph.D., CFAa,*
, Mark Kurt, Ph.D.b, Reilly White, Ph.D.
c
a Associate Professor of Finance, Elon University, Martha and Spencer Love School of Business,
2075 Campus Box, Elon, NC 27244, e-mail [email protected] b
Assistant Professor of Economics, Elon University, Martha and Spencer Love School of
Business, 2075 Campus Box, Elon, NC 27244, e-mail [email protected] c Assistant Professor of Finance, University of New Mexico, Anderson School of Business, 1
University of New Mexico, Albuquerque, NM 87131, e-mail [email protected]
*Corresponding Author: telephone 336-278-5962
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The Buffett Critique: Volatility and Long-dated Options
Abstract
In his 2008 letter to shareholders, Warren Buffett, Chairman and CEO of Berkshire Hathaway,
criticizes the ability of the Black-Scholes model to accurately price long-dated options. He
discusses how the model leads to over-pricing of put options with long maturities using
examples of Berkshire’s investments in derivatives contracts. We confirm that traditional
implied volatility estimates indeed overstate long-term volatility. As an alternative, we
propose a maturity-matching technique for estimating long-term volatility using historical
holding period returns. We focus on three large asset classes, large cap stocks, long-term
bonds, and treasury bills, to demonstrate how volatility evolves over different holding periods.
We apply this rolling-period simulation method to estimate volatility for annual maturities
ranging from 1-year to 30-years within the Black-Scholes model. This method generates
superior long-dated option values, and effectively answers Buffett’s critique. This is of
particular importance for firms with significant investment holdings, as the issuance and
valuation of these derivatives can have a substantial effect on firm capital.
JEL Classifications: G11, G12, G32, M48
Keywords: Volatility, Risk Management, Black-Scholes, Option Pricing, Investments
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“If the [Black-Scholes] formula is applied to extended time periods….it can produce absurd
results.”
-Warren Buffett, 2008 Letter to Berkshire Hathaway Shareholders
In 2008, Warren Buffett penned what we in the paper refer to as the ‘Buffett Critique’: he
observed that when the Black-Scholes option pricing model is applied to longer time horizons,
the valuations produced are irrational. However, despite its flaws, the Black-Scholes model
remains the most popular option valuation technique among large firms.1 In this paper, we focus
on the issue of using implied volatility estimates to calculate long-term option values from the
Black-Scholes model. We find that using a matched-maturity volatility input, in lieu of an
implied volatility figure, more accurately values long-dated options, which provides an effective
answer to the Buffett critique.
In his 2008 letter to shareholders, Buffett uses the hypothetical example of a 100-year $1
billion put option on the S&P 500 index. Using the traditional implied volatility assumption, he
applies the Black-Scholes model to price this premium at $2.5 million. The question then
becomes: is this premium rational? The S&P 500 has a very small chance of being worth less in
100 years than it does currently. Inflation will likely decrease the value of the dollar
significantly, driving up the index factor. Retained earnings will boost the index levels further.
Thus, even by assuming a conservative 1% likelihood of a 50% market decline, Buffett assesses
the maximum expectation of loss at $5 million.2 To mitigate this potential loss, Buffett argues
that the issuer would only have to reinvest at 0.7% annually to cover this loss expectancy at a
potential borrowing cost of only 6.2%. Thus, while 99% of the time creating such a put option
1 A 2005 report by ComplianceWeek found that 80% of companies used the Black-Scholes option pricing
framework to determine the value of their option holdings; the remainder generally used a binomial (lattice-form)
model. 2 $1 billion put contract x 1% likelihood x 50% anticipated loss = $5 million.
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would be risk-free and offer an immediate benefit, the 1% worse-case scenario can be easily
mitigated by conservative investments.
However, as Cornell (2009) observes, to price a European-style option contract over long
periods requires three inputs: the dividend yield, the risk-free rate, and the volatility. In Buffett’s
case, he accesses a fairly standard projection for these three values: a dividend yield of 3% that
approximates the 50-year average market payout; a risk-free rate of 4.8% that is in line (if not
slightly higher) than historical estimates; and a volatility measure of 18% that approximates the
long-term S&P volatility.3 Altering the dividend yield and risk-free rate values generates swings
in the calculated option premia.4 However, the necessary deviations are implausible and not
justifiable based on historical evidence.
In this paper, we propose a readily applicable alternative to the Buffett Critique: long-
term volatility matching. We adopt the rolling-period methodology developed in Gupta, Pavlik,
and Synn (2012) to calculate that the observed volatility of the S&P 500 for long-term holding-
period returns (15 to 30 years) is as low as 4%; this is significantly lower than the 18% historical
volatility of one-year returns on the S&P 500, and consistent with the notion that volatility
decreases in the long-run. We measure this long-term holding-period volatility since it matches
the maturity of Berkshire Hathaway’s long-term options. During periods of market stress, these
historical volatilities can be much lower than the short-term volatility implied by options prices.
The recalculated put premium was essentially $0; in other words, an option with no value. This
matches the expectations of the rational investor for a security with these specifications. When
3 Cornell (2009) states that the values are reasonable estimates based on the Buffett letter.
4 Cornell (2009) notes that raising the risk free rate to 5.8% lowers the put value to $0.45 million, but we argue that
in an era of low interest levels, this adjustment is unreasonable. Dividends are likewise deemed appropriate,
matching both the current and long-term S&P yields.
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employed by the issuer, our calculation generates a value that successfully answers Buffett’s
critique.
The balance of our paper is structured as follows: we review the relevant options
literature in Section 2. In Section 3, we outline our methodology and technique. Section 4
presents our Results, and Section 5 concludes.
Literature Review
In the nearly forty years since the Black-Scholes Model was first published, substantial research
has been generated to improve the model’s usage. Black (1979) adapted the model to forwards
and futures; Garman and Kohlhagen (1983) modified the model to price European currency
options. Haug (1998) generates an extensive review of the prevailing option pricing methods,
and provides considerable references on the pricing of exotic options. Alternative option-pricing
models also exist: modern versions of the Bachelier (1900), Sprenkle (1964), and the continuous-
time, positive-drift Brownian model proposed by Samuelson (1965) are also used widely. Cox,
Ross, and Rubinstein (1979) created a binomial option pricing model that is particularly useful in
pricing American options. As noted by Buffett, the Black-Scholes model, certainly the most
famous option pricing model, is capable of tremendous error in regards to long maturities.
In the sections that follow, we discuss known issues in the use of implied volatilities to
value options, and the use of asset volatility to predict default and bankruptcy.
Implied Volatility to Value Options
Significant attention has been given to volatility as a source of poor option pricing. The
stochastic nature of volatility has led many researchers to conclude that the implied volatility is
independent of option maturity. Applying this empirically, market uncertainty over different
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time horizons generates a ‘term structure’ of implied volatilities. One character of this structure
is the ‘volatility smile’ (Dupire 1994 and Rubinstein 1994). There has been significant research
on forecasting volatility, beginning with Poterba and Summers (1986) who find that long-term
volatility is not significantly affected in response to changes in short-term volatility. Campa and
Chang (1995) test the expectations hypothesis with foreign currency options, and find no
evidence for overreaction in the long-term market. Smoothing volatility levels across strike
prices and maturities have resulted in significant empirical improvement to the Black-Scholes
model. The ‘ad hoc Black-Scholes’ outperformed both the Black-Scholes and deterministic
volatility functions (Dumas, Fleming, and Whaley 1998), the DVF model in valuing FTSE 100
index options (Brant and Wu 2002), and S&P 100 index options (Berkowitz 2009). Davis
(2001) noted that this empirical improvement is likely a factor of the option being ‘mapped onto’
itself, a notion proven asymptotically by Berkowitz (2009). As a general rule, while research has
focused on the prediction of long-term volatility in regards to long-dated options, the
inconsistency of the results compiled with uncertainty as to the predictive power of long-range
maturities limit its practical application. Our volatility-matching technique improves the existing
results considerably for long-dated options, enabling firms to make better and more practical
determinations of option value.
Asset Volatility to Predict Default and Bankruptcy
These issues with volatility duration also extend into the default literature. Merton (1974)
extends the Black-Scholes model to evaluate the risk structure of corporate debt by modeling
equity as a call option on the assets of the firm with a strike price equal to the firm’s liabilities.
The appeal of adopting this structural model is that it connects the theory of how firms’ assets
and profits relate to equity pricing and the risk of firms’ debt obligations. The default probability
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can be calculated from the derivate pricing equations. This model has been extensively modified
since its introduction (Lando 2005) and a version has been employed by Moody’s KMV section
(Dwyer, Kocagil, and Stein 2005). However, the ability of this class of models to accurately
predict default has been called into question. Eom, Helwege, and Huang (2005) test five
different structural models based on the Black Scholes framework. They find that most models
perform poorly by predicting implausible yield spreads.
In light of these shortcomings, reduced form hazard models have received more attention.
However, these models do not have strong theoretical underpinnings as compared to structural
models. It should also be noted that, while this approach provides flexibility, it often takes the
form of a duration model with limited information regarding a firm’s default, it may not capture
the true underlying process which governs the change in the probability of default. Moreover, it
does not explicitly consider the relationship between default and firm value (Jarrow and Turnbull
1995, Duffie and Lando 2001, and Jeanblanc and Le Cam 2007).
Walker (2005) argues that both classes of models over-predict default and bankruptcy.
Duan and Fulop (2009) point out that using daily equity data could introduce substantial noise.
Walker (2005) employs a hybrid model which relaxes the assumption of parameterizing the
model with daily equity data. Using the monthly average of daily equity data (Walker 2005) or a
maturity matching approach mitigates market noise from short term market fluctuations.
Defaults and bankruptcies are rare events which culminate from systemic declines in firms’
value relative to their debt obligations. Using daily equity data introduces short-term volatility
which degrades models ability to predict default and bankruptcy.
Methodology
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Black-Scholes Option Pricing Model
In its original form, Black and Scholes (1973) presented the following formula for a call
valuation C or a put valuation P:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
(
) (
) ( )
√
(
) (
) ( )
√ √
(1)
where, S is the spot price of the underlying financial asset; K is the strike price; r is the risk-free
rate, σ is the volatility; T - t is the time to maturity; and N(d) is the cumulative normal density
function.
The standard form of the Black-Scholes model assumes that stock returns follow a
lognormal diffusion where the stock price distribution at time T is:
(
) (2)
where, S0 is the current stock price, ST is the stock price at expiration, and μ is the drift (expected
return on the index minus the dividend yield).
In practice, these inputs are generally based on observed data – with the exception of
volatility.
Estimating Long-term Volatility
Variations in volatility over the life of the option will have profound effects on its long-term
value and require some element of volatility ‘forecasting’.
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Traditional models use historical market volatility. Following Figlewski (2004), one first
estimates the mean price change as the simple average of a series of past returns Rt:
∑
(3)
The variance of past returns is then:
∑( )
( ) (4)
To compute volatility, one multiplies this value by the number of price observations in a year N
and takes the square root:
√ (5)
This value in turn becomes the volatility forecast, and can be applied over any time horizon.
We know from Figlewski (2004) that traders and researchers alike favor this estimation
technique, despite the fact it may generate suboptimal values when compared to more advanced
methods. Figlewski (2004) suggests that these commonly-used stochastic techniques have failed
to generate widely accepted or generally superior volatility forecasts, so common usage has
adopted the use of either short-term volatility estimates or the application of the lognormal-
diffusion assumption on long-dated options.
However, long-dated options are particularly vulnerable to significant changes in
volatility. Calculating the value of a long-term option portfolio using short-term volatility
estimates can generate tremendous errors. The lognormal diffusion assumption suggests a linear
relationship between volatility and the horizon over which it is measured. But, as Cornell (2009)
notes, empirical research has demonstrated that this fails to hold over long horizons. Using the
lognormal-diffusion estimation for the calculation of long-term option values incorrectly
estimates volatility and skews the option values, which is the primary motivator of the Buffett
Critique.
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Estimating Volatility using Historical Data
Traditionally, the volatility input of the Black Scholes model even for valuing long-term options
is computed using a short-term implied volatility estimate. This is typically done by observing
the movement of short-term (two- to three-year maturity) contracts on a given day.
We, instead, use rolling period methods from past studies such as Cooley, Hubbard, and
Walz (2003) and Gupta, Pavlik, and Synn (2012) to simulate observed short-term and long-term
volatilities of portfolios of stocks (using the S&P 500 index). These studies note that rolling
period simulations better replicate historical performance since they retain the serial correlation
observed in stock returns – they inherently assume that stocks returns in a particular year are
dependent on stocks returns in past years.
We calculate the volatility of short-term and long-term stock portfolio returns for
maturities of 1 year to 30 years in annual intervals, that is we calculate the volatility of the
portfolio for maturities T taking values 1, 2, …30 as described below.
The value of the portfolio at maturity T, is calculated as:
T
t
tstT rPP1
,1 )1( (6)
where, t is each year in the life of the put option (1, 2, … T), rs,t is stock returns (S&P 500) in
year t, and P0 is the initial portfolio value ($1).
The annual buy-and-hold return of the portfolio at time T is then:
1
1
0
TT
TP
Pr (7)
For each maturity T, we generate multiple iterations of T-year portfolios created at the
start of 1926 through the maximum possible iterations possible using data ending in 2008. For
example, for portfolios with T=30 (30-year maturities), we generate N=53 iterations where
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starting year = [1926 through 1978]. Stock returns correspond to the return t years after date the
retirement portfolio is created. For example, for portfolios created at the start of 1926, stock
returns in the 20th year rs,20, correspond to the stock returns in 1945.
We measure volatility σT as the sample standard deviation of annual returns rT for all
simulated portfolios with maturity T.
1
)( 2
N
rr TT
T (8)
Hypotheses Development
In this paper, we test Buffett’s critique of the S&P option directly, using both a traditional short-
term implied volatility approach and a long term observed volatility estimate. For our long-term
estimate, we use observed historical data to compute volatility over the same horizon of the
average option maturity. Thus, an option portfolio with N years to maturity should use a
historical volatility estimate of N prior years.
We present the following hypothesis:
H1: Estimating the volatility input for the Black-Scholes model using the matched-
volatility approach will generate improved long-term option valuations when compared
to the traditional method of implied volatility
If an underlying option averages N years until maturity, we propose that the volatility
input to the Black-Scholes model should be based on the calculated holding period value over N
years. The resulting estimates should provide a result more consistent with logic and common
sense than the $2.5 million put premium described by Buffet. This has considerable advantages
over the traditional implied volatility estimates; namely, they incorporate the long-term leveling
effects of volatility of the market, and do not capture the short-term volatility fluctuations that
make long-term option pricing unreliable.
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Results
We find the value for all forward contracts written by Berkshire Hathaway at the start of 2008 by
aggregating the notional value of all forward contracts issued. This amount generates a proxy
value for the total value of the underlying claims on which these options are traded, and presents
an opportunity to accurately price the ‘market’. Initially, we value this at $37.10 billion in 2008;
at year end, this value drops to $27.08 billion and at the end of the first quarter of 2009, $18.55
billion. Furthermore, we find that contracts are generally issued at the money with an average
maturity of 17 years. Traditionally, the volatility input to the Black-Scholes model is the implied
volatility taken from long-dated options. Our annualized risk-free rate, taken directly from mid-
term treasury-bonds, was adjusted from 4.3% in 2008 to 3.70% at year-end 2008 and 3.4% in
2009. To provide a more conservative estimate of the option value, we set the dividend rate at
0% since there is no exchange of funds on these forward contracts prior to maturity, whether it
be margin moneys or dividends.
We calculated the value of the option using Black-Scholes, and have presented our work
in Exhibit 1. Our traditional inputs generate a contracted put value of $4.92 billion, $10.06
billion, and $12.61 billion based on high market volatility. In the far right column, we create
Buffet’s original critique: using an example option worth $1 billion, we match Buffett’s put
estimate at $2.5 million at origination. This is based on assumptions that closely match
Buffett’s.5
5 The small discrepancies result from fluctuations in the market between Buffett’s contract initiation date and our
assumed date.
Exhibit 1: Option Valuation using the Traditional Implied Volatility Assumption In this table, we present the results of the Black-Scholes option pricing model on the S&P 500 market, to provide
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In Exhibits 2 and 3, we present our calculations of long-term holding period volatilities
across maturity levels of 1 to 30 years for stocks. For comparison, we also present the same
calculations for bonds and short-term treasury bills. From the data, we observe that stock returns
have been positive in all long-term holding periods (15-years and up). That is, writers of put
option on the S&P 500 with these maturities have never needed to pay out on settlement. We
observe also that longer holding periods have lower volatility of returns, and the volatility
estimate of over 19% used in the traditional implied model significantly overstates volatility for
long-dated options.
6 Uses implied volatility from traded options - this is the most commonly used volatility measure in the Black-
Scholes valuation of options.
a representative S&P 500 option. Our stock price is the notional value (in billions of dollars) of all forward
contracts issued by Berkshire Hathaway using data from their SEC filings; dividend yield is set at 0.00% since
there is no exchange of funds prior to maturity; the annualized standard-deviation is computed from the implied
volatility of long dated options; the risk free rate is taken from the 10-year treasury bonds; and time to maturity
was taken as the average maturity on the outstanding contracts. We provide three columns representing three
periods in the financial crisis: contract initiation (mid 2008), year-end 2008, and the end of first-quarter 2009. The
‘example’ column aligns our results with Warren Buffet’s S&P example.
Initiation 2008YE 2009Q1E Example
Model Inputs
Stock Price (S0) $37.10 $27.08 $18.55 $1,000.00
Dividend Yield - Annualized (δ) 0.00% 0.00% 0.00% 0.00%
Standard Deviation - Annualized (σ) 6 30.00% 38.00% 38.00% 19.70%
Riskfree Rate - Annualized (rf) 4.30% 3.70% 3.40% 3.70%
Exercise Price (X) $37.10 $37.10 $37.10 $1,000.00
Time to Maturity in Years (T) 17 16 16 100
Model Outputs
d1 1.2094 0.9424 0.6619 2.8632
d2 (0.0275) (0.5776) (0.8581) 0.8932
N(d1) 0.8868 0.8270 0.7460 0.9979
N(d2) 0.4890 0.2818 0.1954 0.8141
Call Price (Co) $24.16 $16.61 $9.63 $977.78
Put Price (Po) $4.92 $10.06 $12.61 $2.50
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Exhibit 2: Holding Period Volatility, Mean Returns, and the Sharpe Ratio In this table, we present the results of the calculated annual volatility; mean holding period return, and the Sharpe
Ratio across Large Cap Stocks (defined as the S&P 500), Long-Term Treasury Bonds, and US T-Bills. Holding
period in years is calculated over the sample period 1926-2008, and ranges from 1 (N=73) to 30 years (N=44).
Annual volatility is the standard-deviation of annual returns computed for each holding period; mean holding period
return is the average annual holding period return; and the Sharpe Ratio is computed by dividing the difference
between the holding period returns of large cap stocks (or long term T-Bonds) and US-T-Bills by the calculated
volatility.
Annual Volatility Mean Holding Period Return Sharpe Ratio
Holding Period (Years)
Large Cap Stocks
(S&P 500)
Long Term
T-Bonds U.S.
T-Bills
Large Cap Stocks
(S&P 500)
Long Term
T-Bonds U.S.
T-Bills
Large Cap Stocks
(S&P 500)
Long Term
T-Bonds
1 20.57% 9.36% 3.08% 11.67% 6.08% 3.75% 0.38 0.25
2 14.88% 6.19% 3.02% 11.05% 5.74% 3.76% 0.48 0.32
3 14.68% 6.08% 3.03% 11.49% 5.60% 3.77% 0.52 0.30
4 9.98% 4.78% 2.93% 10.40% 5.59% 3.77% 0.66 0.38
5 8.61% 4.53% 2.90% 10.28% 5.58% 3.76% 0.75 0.40
6 7.54% 4.31% 2.88% 10.29% 5.56% 3.77% 0.86 0.42
7 6.78% 4.16% 2.86% 10.42% 5.58% 3.79% 0.97 0.43
8 6.20% 4.08% 2.84% 10.61% 5.56% 3.81% 1.09 0.43
9 5.80% 3.97% 2.83% 10.75% 5.55% 3.83% 1.19 0.43
10 5.49% 3.96% 2.85% 10.92% 5.48% 3.84% 1.28 0.42
11 5.32% 3.85% 2.80% 11.01% 5.51% 3.87% 1.34 0.43
12 5.16% 3.83% 2.79% 11.05% 5.49% 3.89% 1.38 0.42
13 4.96% 3.80% 2.77% 11.12% 5.47% 3.91% 1.45 0.41
14 4.75% 3.78% 2.75% 11.15% 5.46% 3.92% 1.52 0.41
15 4.53% 3.74% 2.73% 11.17% 5.43% 3.94% 1.59 0.40
16 4.26% 3.72% 2.71% 11.21% 5.41% 3.97% 1.70 0.39
17 4.03% 3.69% 2.69% 11.28% 5.39% 3.99% 1.80 0.38
18 3.83% 3.66% 2.67% 11.33% 5.37% 4.02% 1.91 0.37
19 3.66% 3.61% 2.64% 11.36% 5.34% 4.04% 2.00 0.36
20 3.44% 3.57% 2.62% 11.38% 5.32% 4.06% 2.12 0.35
21 3.27% 3.52% 2.59% 11.38% 5.28% 4.08% 2.23 0.34
22 3.09% 3.47% 2.56% 11.39% 5.25% 4.11% 2.36 0.33
23 2.89% 3.43% 2.53% 11.40% 5.23% 4.13% 2.52 0.32
24 2.65% 3.38% 2.49% 11.42% 5.20% 4.15% 2.74 0.31
25 2.38% 3.33% 2.51% 11.41% 5.10% 4.14% 3.05 0.29
26 2.14% 3.23% 2.42% 11.40% 5.10% 4.18% 3.36 0.29
27 1.93% 3.16% 2.38% 11.37% 5.07% 4.20% 3.71 0.28
28 1.73% 3.06% 2.34% 11.33% 5.02% 4.21% 4.11 0.26
29 1.57% 2.96% 2.30% 11.32% 4.97% 4.22% 4.51 0.25
30 1.41% 2.88% 2.25% 11.27% 4.93% 4.22% 5.00 0.25
Exhibit 3: Volatility by Holding Period across different Securities This figure demonstrates the relationship between volatility (y-axis) and holding period in years (x-axis). Holding
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In Exhibit 4, we present our results using our matched-maturity model. In our model, the
assumptions, other than for volatility, are the same as Exhibit 1. However, we calculate that the
average maturity on outstanding options in the market was about 17 years. Our volatility
estimate therefore reflects the 17-year holding period return volatility – the matched-maturity
volatility estimate. Using rolling holding period methods for return data from 1926-2008, we
compute the 17-year match-maturity volatility to be roughly 4%. Then, when we calculate the
put values in the Buffet example, we find that the put’s true value is established at virtually $0 –
period volatility is defined by equation (5). We include data for S&P 500 firms (large-cap stocks), long-term treasury
bonds (30-year maturity), and short-term (<1 year maturity) treasury bills. The sample includes 44 thirty-year holding
periods and 73 one-year holding periods across all variables from 1926-2008.
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similar to the intrinsic value at origination of forward put contracts issued by Berkshire
Hathaway.
17
In Exhibit 5, we combine our findings from Exhibits 1 and 4 to contrast the Black-
Scholes value of a put option on the S&P 500 using the traditional implied volatility measure
against our matched-maturity volatility technique. We present results for Berkshire Hathaway’s
basket of put option contracts at three different times (at origination, at 2008 year end, and 2009
first-quarter end), and for the example that Buffett discusses in his letter. Our graph highlights
7 Uses the matched-maturity volatility from the rolling periods method – calculated to be 4.03% for 17-year holding-
period returns. Data Source: Ibbotson Associates data for the S&P 500 for 1926-2008.
Exhibit 4: Option Valuation using the Matched-Maturity Technique In this table, we present the results of the Black-Scholes option pricing model using our matched-maturity technique
on the S&P 500 market, to provide a representative S&P 500 option. Our stock price is the notional value (in billions
of dollars) of all forward contracts issued by Berkshire Hathaway using data from their SEC filings; dividend yield is
set at 0.00% since there is no exchange of funds prior to maturity; the annualized standard-deviation is based on the
17-year holding period volatility level; the risk free rate is taken from the 10-year treasury bonds; and time to
maturity was taken as the average maturity on the outstanding contracts. We provide three columns representing
three periods in the financial crisis: contract initiation (mid 2008), year-end 2008, and the end of first-quarter 2009.
The ‘example’ column aligns our results with Warren Buffet’s S&P example.
Initiation 2008YE 2009Q1E Example
Model Inputs
Stock Price (S0) $37.10 $27.08 $18.55 $1,000.00
Dividend Yield - Annualized (δ) 0.00% 0.00% 0.00% 0.00%
Standard Deviation - Annualized (σ) 7 4.00% 4.00% 4.00% 4.00%
Riskfree Rate - Annualized (rf) 4.30% 3.70% 3.40% 3.70%
Exercise Price (X) $37.10 $37.10 $37.10 $1,000.00
Time to Maturity in Years (T) 17 16 16 100
Model Outputs
d1 4.5148 1.8131 (0.8522) 9.4500
d2 4.3499 1.6531 (1.0122) 9.0500
N(d1) 1.0000 0.9651 0.1971 1.0000
N(d2) 1.0000 0.9508 0.1557 1.0000
Call Price (C0) $19.24 $6.62 $0.30 $975.28
Put Price (P0) $0.00 $0.06 $3.29 $0.00
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that the traditional implied volatility approach consistently overstates the value of long-dated put
options, and underscores our findings that a matched-maturity volatility approach better
represents actual market volatility and option value than the traditional implied volatility
estimates.
Exhibit 5: Value of the Put Option using Implied and Matched-Maturity Volatilities In this figure, we examine the Black-Scholes value of a put option on the S&P 500 using both the traditional implied
volatility measure and the matched-maturity volatility technique employed by this paper. For each approach, we
graph the estimated premia in billions of dollars for Berkshire Hathaway’s basket of long-dated put option contracts
at three different times (at origination, at 2008 year end, and 2009 first-quarter end), and for the example that Buffett
discusses in his newsletter.
$4.92
$10.06
$12.61
$2.50
$0.00 $0.06
$3.29
$0.00$0.00
$2.00
$4.00
$6.00
$8.00
$10.00
$12.00
$14.00
At Contract
Origination
At 2008YE At 2009Q1E Buffett's Example
Implied Volatility Matched-Maturity Volatility
19
Conclusion
We find that using matched-maturity volatilities in the Black-Scholes model generate improved
values on long-dated options. This is of particular importance for firms with significant
investment holdings, as the issuance and valuation of these derivatives can have a substantial
effect on firm capital. Mispriced options can overstate expenses or profits at the expense of
shareholder information. Overstating volatility levels inflate the price of both calls and puts to
unreasonable levels, a fact that degrades the quality and viability of the option market. Further,
the proper valuation of long-term options should take into account long-term volatility levels that
better reflect market expectations.
We expect that our results can be practically used by traders, investors, and those
involved throughout corporate finance. We conclude that our approach provides guidance for
firms, policy makers and researchers for improving the performance of the Black-Scholes option
pricing model when applied to long-term securities.
20
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