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Estimating volatility in the Merton model: The KMV estimate is not maximum likelihood

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  • Benjamin Christoffersen
  • David Lando
  • Søren Feodor Nielsen

Abstract

We compare two methods for estimating the asset volatility in the Merton model using observed equity prices: maximum likelihood and an iterative method commonly referred to as the KMV method. The two methods often yield extremely similar estimates, which has led to the conjecture that the two methods are equivalent. We show that this is not true and we provide a necessary and sufficient condition that the inverse of the equity pricing function would have to satisfy for the two methods to be equivalent. Moreover, we show numerically that this condition is very close to being true for in‐the‐money options.

Suggested Citation

  • Benjamin Christoffersen & David Lando & Søren Feodor Nielsen, 2022. "Estimating volatility in the Merton model: The KMV estimate is not maximum likelihood," Mathematical Finance, Wiley Blackwell, vol. 32(4), pages 1214-1230, October.
    • Handle: RePEc:bla:mathfi:v:32:y:2022:i:4:p:1214-1230
    DOI: 10.1111/mafi.12362
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    References listed on IDEAS

    as
    1. Merton, Robert C, 1974. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 29(2), pages 449-470, May.
    2. repec:bla:jfinan:v:59:y:2004:i:2:p:831-868 is not listed on IDEAS
    3. J. T. Chang & D. Pollard, 1997. "Conditioning as disintegration," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 51(3), pages 287-317, November.
    4. Sreedhar T. Bharath & Tyler Shumway, 2008. "Forecasting Default with the Merton Distance to Default Model," The Review of Financial Studies, Society for Financial Studies, vol. 21(3), pages 1339-1369, May.
    5. Jin‐Chuan Duan, 1994. "Maximum Likelihood Estimation Using Price Data Of The Derivative Contract," Mathematical Finance, Wiley Blackwell, vol. 4(2), pages 155-167, April.
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