The value of an option can be estimated using a variety of quantitative techniques based on the concept of risk neutral pricing and using stochastic calculus. The most basic model is the Black-Scholes model. More sophisticated models are used to model the volatility smile. These models are implemented using a variety of numerical techniques.

In general, standard option valuation models depend on the following factors:

• The current market price of the underlying security,
• the strike price of the option, particularly in relation to the current market price of the underlier (in the money vs. out of the money),
• the cost of holding a position in the underlying security, including interest and dividends,
• the time to expiration together with any restrictions on when exercise may occur, and
• an estimate of the future volatility of the underlying security's price over the life of the option.

More advanced models can require additional factors, such as an estimate of how volatility changes over time and for various underlying price levels, or the dynamics of stochastic interest rates.

The following are some of the principal valuation techniques used in practice to evaluate option contracts.

Black Scholes

In the early 1970s, Fischer Black and Myron Scholes made a major breakthrough by deriving a differential equation that must be satisfied by the price of any derivative dependent on a non-dividend-paying stock. By employing the technique of constructing a risk neutral portfolio that replicates the returns of holding an option, Black and Scholes produced a closed-form solution for a European option's theoretical price. At the same time, the model generates hedge parameters necessary for effective risk management of option holdings. While the ideas behind the Black-Scholes model were ground-breaking and eventually led to Scholes and Merton receiving the Swedish Central Bank's associated Prize for Achievement in Economics (often mistakenly referred to as the Nobel Prize),

the application of the model in actual options trading is clumsy because of the assumptions of continuous (or no) dividend payment, constant volatility, and a constant interest rate. Nevertheless, the Black-Scholes model is still one of the most important methods and foundations for the existing financial market in which the result is within the reasonable range.

Stochastic volatility models

Since the market crash of 1987, it has been observed that market implied volatility for options of lower strike prices are typically higher than for higher strike prices, suggesting that volatility is stochastic, varying both for time and for the price level of the underlying security. Stochastic volatility models have been developed including one developed by S.L. Heston. One principal advantage of the Heston model is that it can be solved in closed-form, while other stochastic volatility models require complex numerical methods.

Options valuation models Topic - Investing

In finance, investment is the commitment of funds by buying securities or other monetary or paper (financial) assets in the money markets or capital markets, or in fairly liquid real assets, such as gold, real estate, or collectibles. Valuation is the method for assessing whether a potential investment is worth its price. Returns on investments will follow the risk-return spectrum.