1

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 ngupta@elon.edu b

Assistant Professor of Economics, Elon University, Martha and Spencer Love School of

Business, 2075 Campus Box, Elon, NC 27244, e-mail mkurt@elon.edu c Assistant Professor of Finance, University of New Mexico, Anderson School of Business, 1

University of New Mexico, Albuquerque, NM 87131, e-mail rwhite17@elon.edu

*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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