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. Buffet 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 do 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 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.
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Notes
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.
$1 billion put contract × 1% likelihood × 50% anticipated loss = $5 million.
Cornell (2010) states that the values are reasonable estimates based on the Buffett letter.
Cornell (2010) 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.
See Sears, Steven “Here come the Super Options.” Barron’s, September, 7, 2012. http://online.barrons.com/news/articles/SB50001424053111904294104577631491840049660
Last Accessed 7/17/2014.
See SEC Release No. 34-68164: http://www.sec.gov/rules/sro/cboe/2012/34-68164.pdf
The small discrepancies result from fluctuations in the market between Buffett’s contract initiation date and our assumed date.
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Gupta, N.J., Kurt, M. & White, R. The Buffett critique: volatility and long-dated options. J Econ Finan 40, 524–537 (2016). https://doi.org/10.1007/s12197-015-9319-5
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DOI: https://doi.org/10.1007/s12197-015-9319-5