Option Pricing Theory

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Option Pricing Theory

What is option pricing theory?

Option pricing theory estimates the fair value of an options contract by quantifying the probability it will finish in the money (ITM) at expiration. Traders, market makers, and risk managers use mathematical models to convert market variables into a theoretical option price, which helps assess potential profitability and manage risk.

Core inputs and the Greeks

Common inputs used by pricing models:
* Current price of the underlying asset
* Strike (exercise) price
* Time to expiration
* Volatility (expected or implied)
* Risk-free interest rate
* Dividends (when applicable)

From these inputs models also derive the “Greeks” — sensitivity measures that show how option value changes with small moves in underlying variables:
* Delta (price sensitivity)
* Gamma (rate of change of delta)
* Vega (sensitivity to volatility)
* Theta (time decay)
* Rho (sensitivity to interest rates)

Primary pricing models

  1. Black–Scholes model
  2. Widely used to price European-style options with a known expiration.
  3. Requires inputs above, including an estimate of volatility (often implied volatility).
  4. Assumes log-normal price distribution, constant volatility and interest rates, frictionless markets, and no arbitrage.

  5. Dividends can be added as an extra input for many stock options.

  6. Binomial (and trinomial) models

  7. Discrete-time models that build a recombining tree of possible underlying prices.
  8. Can handle both European and American-style options because they evaluate early exercise at every node.

  9. Flexible for varying assumptions about volatility, dividends, and exercise features.

  10. Monte Carlo simulation

  11. Simulates many random price paths for the underlying to estimate expected option payoff.
  12. Useful for complex, path-dependent, or multi-dimensional payoffs (e.g., Asian options, certain exotics).
  13. Computationally intensive but flexible in modeling dynamics beyond Black–Scholes assumptions.

Model assumptions and real-world limitations

Standard models make simplifying assumptions that often differ from market reality:
* Constant volatility and interest rates — in practice both can change.
* No transaction costs, taxes, or liquidity constraints.
* Ability to continuously hedge without friction.
* Black–Scholes assumes European exercise only.

Because of these gaps:
* Implied volatility (derived from market prices) can differ from historical or realized volatility and is often used instead of a direct volatility estimate.
* Volatility skew or smile appears in markets — implied volatility varies by strike and maturity, often higher for far OTM options.
* Market prices can deviate from theoretical values due to supply/demand, liquidity, and differing model choices.

Practical insights for traders

  • The more likely an option will finish ITM, the more valuable it is; longer time to expiration and higher volatility generally increase option value.
  • Use theoretical prices to identify mispricings and to size or hedge positions, but account for execution costs and model risk.
  • For American options or options with early-exercise considerations, prefer tree models or numerical methods that check exercise at each step.
  • Calibrate models with market-implied parameters where possible (e.g., implied volatility surface) to reflect current sentiment and skew.

Conclusion

Option pricing theory provides structured methods to estimate an option’s fair value and sensitivities using a set of market inputs. Black–Scholes, binomial/trinomial trees, and Monte Carlo simulation are the main approaches, each with strengths and limits. Understanding their assumptions and how market realities (implied volatility, skew, liquidity, early exercise) affect prices is essential for effective trading and risk management.