How Stochastic Volatility and Other Extensions Relax Black-Scholes Assumptions

Summary

This discussion explains why the constant-volatility assumption in Black-Scholes can be restrictive and outlines broader ways to extend the model. Stochastic-volatility models make volatility vary over time, with model dynamics linked to quantities such as variance and the behavior of returns. The answer also cautions that familiar stochastic-volatility specifications generally do not incorporate company fundamentals such as debt load.

Other listed extensions relax assumptions about return distributions, continuous trading, continuous price paths, fixed interest rates, constant variance, and the absence of dividends. The examples connect these departures to established model families and papers, but the exchange does not derive the models, compare their pricing performance, or recommend one for a particular option. It is therefore a conceptual survey of dimensions in which Black-Scholes can be generalized, rather than a complete modeling guide. Any extension still requires assumptions and calibration, and adding parameters does not by itself ensure a better fit or more reliable prices.

Key ideas

• Black-Scholes assumes constant volatility over the derivative's life, an assumption stochastic-volatility models relax.
• Common stochastic-volatility models use volatility dynamics rather than company fundamentals such as debt.
• Extensions can also relax assumptions about return distributions, trading continuity, price paths, rates, variance, and dividends.
• The examples identify modeling directions but provide no performance comparison or selection rule.

Tags

# Extensions of Black-Scholes model


# Extensions of Black-Scholes model












For the Black-Scholes model my feeling is that the volatility parameter is like sweeping stuff under the rug.

Are there models which improve on the volatility aspect of Black-Scholes by adding other parameters (I'm guessing things like the distribution of past returns, or perhaps some measure of debt load held by the company).

## Answer by Joseph Tanenbaum (score 5)

https://quant.stackexchange.com/a/268

The Black-Scholes model assumes that the underlying volatility is constant over the life of the derivative, which is indeed a gross oversimplification. Stochastic Volatility models improve on that assumption by making volatility dependent on additional parameters such as distribution of returns and variance itself. However, the well known stochastic volatility models do not include company fundamentals among their parameters.

## Answer by JohnAndrews (score 1)

https://quant.stackexchange.com/a/8608

Extensions of the Black-Scholes model usually focus on relaxing one or more assumptions. Some of these generalizations include:

- Log normal distribution of returns (e.g., Corrado and Su, 1996)

- Continuous trading (e.g., relaxed by Merton, 1976)

- Continuous evolution of the share price (e.g., relaxed by Cox-Ross-Rubinstein, 1979)

- Constant interest rates (e.g., relaxed by Baksi et al, 1997)

- Constant variance on the underlying returns (e.g., relaxed by Heston, 1993)

- No dividends (e.g., relaxed by Merton, 1973)

- Contentious diffusion of the underlying (e.g., relaxed by Merton, 1976)"

Etc. Etc.

Note: I give example of relative `simple' and extensions.

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