Option Pricing Theory

The theory of options pricing uses variables such as stock price, strike price, interest rate, volatility, and time to expiration for valuing an option theoretically.

It gives an estimation of the fair value of the option that traders include in their strategies for maximizing profits.

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Although options valuation may look simple, it calls for some mathematical calculations. The following are the commonly used methods we will look into in detail.

• Risk neutral model

• Binomial model

• Black-Scholes model

• Monte-Carlo simulation

As these usually derive their values from the price of the stock, they have a high possibility for error.

Understanding Option Pricing Theory.

The prime goal of the theory for pricing options is to be able to calculate the probability of the option contract to be in-the-money at the time of expiry. The underlying stock price, exercise price, interest rate, volatility, and time to expiry are the variables commonly used as inputs for the mathematical models to arrive at the theoretical fair value of the option.

Apart from the stock and strike prices, interest rate, time, and volatility are also play a part in pricing an option accurately. The more time an investor has for exercising the option contract, the likelihood for it to be ITM at expiry is greater. Likewise, the more volatile an underlying asset is, the greater the chances that it will expire in-the-money. A higher interest rate means a higher option price.

Marketable, and non-marketable options both need a different method of valuation. The prices of those options that are traded are determined by the market. Like all other assets, the value can be different from the theoretical value. In any case, by having a theoretical value, the traders can assess the chances of getting a profit by trading in these options.

The pricing model published by Fischer Black and Myron Scholes in 1973, led to the development of the present-day option market. The Black-Scholes formula is used to arrive at the theoretical price for a financial instrument with a known expiry date. Apart from this, the models widely used are the Monte-Carlo simulation and Rubinstein binomial options pricing.

The Black-Scholes model also assumes that the stock prices go as per a log-normal distribution because the asset prices cannot be a negative value. The other assumptions are that there are no taxes or transaction costs. Also assumed is that the risk-free interest rate is the same for all maturities and that short selling of securities and the use of proceeds is permitted, and there are no risk-free arbitrage opportunities.

KEY TAKEAWAYS

• The option pricing theory uses stock price, strike price, interest rate, volatility, and time to expiry as variables.

• The prime goal of pricing theory is to calculate the probability of an option being exercised or to be in-the-money at the time of expiry.

• The commonly used models for valuing options are the Black-Scholes, binomial option pricing, and the Monte-Carlo simulation.

Using Black-Scholes Option Pricing Theory

The Black-Scholes model required six input variables namely strike price of the option, current market price, time to expiry, volatility, risk-free rate, and dividends. As observing volatility directly is not possible, it must either be implied or estimated. Implied volatility and historical or realized volatility is not the same.

The Black-Scholes model also assumes that the stock prices go as per a log-normal distribution as the asset cannot have a negative price. The other assumptions are that there are no taxes or transaction costs. It is also assumed that the risk-free interest rate is the same for all maturities and that short selling of securities and the use of proceeds is permitted, and there are no risk-free arbitrage opportunities. The Black-Scholes model assumes that all options are European style and can be executed only on maturity.

Some of the assumptions given above do not remain true all of the time. For example, this model assumes that volatility remains constant over the lifespan of the option. Normally, this is not the case because volatility keeps fluctuating depending on the level of demand and supply.