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Equivalent Black volatilities

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  • Patrick Hagan
  • Diana Woodward

Abstract

We consider European calls and puts on an asset whose forward price F(t) obeys dF(t)=α(t)A(F)dW(t,) under the forward measure. By using singular perturbation techniques, we obtain explicit algebraic formulas for the implied volatility σB in terms of today's forward price F0 ≡ F(0), the strike K of the option, and the time to expiry tex. The price of any call or put can then be calculated simply by substituting this implied volatility into Black's formula. For example, for a power law (constant elasticity of variance) model dF(t)=aFβdW(t) we obtain σB = a/faυ1-β {1 + (1-β)(2+β)/24 (F0 - K/faυ)2 + (1 - β)2/24 a2tex/faυ2-2β +…} where faυ = ½(F0 + K). Our formula for the implied volatility is not exact. However, we show that the error is insignificant, rarely approaching 1/1000 of the time value of the option. We also present more accurate (albeit more complicated) formulas which can be used for the implied volatility.

Suggested Citation

  • Patrick Hagan & Diana Woodward, 1999. "Equivalent Black volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 147-157.
    • Handle: RePEc:taf:apmtfi:v:6:y:1999:i:3:p:147-157
    DOI: 10.1080/135048699334500
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    References listed on IDEAS

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    1. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    2. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    3. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
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