Physical Review Letters · 2002 · 275 citations · 11 references
BusinessEconomicsFinanceFinancial EconomicsQuantitative FinanceThermodynamicsAsset Pricing TheoryImplied VolatilityOption PricingOption Pricing ModelsStock OptionsVolatility ModelingBlack-scholes Implied VolatilityBlack-scholes ModelImplied Volatility SmileOption Pricing TheoryOption Pricing ValuationVolatilityVolatility Smile Modeling
Options derive their value from the underlying stock, yet the classic Black‑Scholes model (q = 1) requires a volatility smile, i.e., multiple volatility inputs. The study aims to explain option price dynamics by modeling non‑Gaussian fluctuations with the nonextensive thermodynamics parameter q. The authors derive a generalized Black‑Scholes partial differential equation incorporating q and provide closed‑form solutions. With q = 1.5, the model fits option prices using a single volatility, eliminating the need for a volatility smile.
Options are financial instruments that depend on the underlying stock. We explain their non-Gaussian fluctuations using the nonextensive thermodynamics parameter q. A generalized form of the Black-Scholes (BS) partial differential equation and some closed-form solutions are obtained. The standard BS equation (q=1) which is used by economists to calculate option prices requires multiple values of the stock volatility (known as the volatility smile). Using q=1.5 which well models the empirical distribution of returns, we get a good description of option prices using a single volatility.
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The Pricing of Options and Corporate Liabilities
Fischer Black, Myron S. Scholes · Journal of Political Economy · 1973 · 29.4K citations