Equivalent Black volatilities

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 σ B in terms of today's forward price F 0 ≡ 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)=aFβ dW(t) we obtain σ B = a/f aυ 1− β {1 + (1−β)(2+β)/24 (F 0 − K/f aυ)2 + (1 − β)2/24 a 2 tex /f aυ 2−2β +…} where f aυ = ½(F 0 + 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.

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 133 citations exist upstream.

Papers that cite the same references — the relational fallback while embeddings are still backfilling.