# Theoretical Pricing of Derivatives

08/09/2022

Different types of derivatives have different pricing mechanisms. A derivative is simply a financial contract with a value that is based on some underlying asset (e.g. the price of a stock, bond, or commodity). The most common derivative types are futures contracts, forward contracts, options and swaps. More exotic derivatives can be based on factors such as weather or carbon emissions.

Options Pricing Basics

Options are also common derivative contracts. Options give the buyer the right, but not the obligation, to buy or sell a set amount of the underlying asset at a pre-determined price, known as the strike price, before the contract expires.

The primary goal of option pricing theory is to calculate the probability that an option will be exercised, or be in-the-money (ITM), at expiration. Underlying asset price (stock price), exercise price, volatility, interest rate, and time to expiration, which is the number of days between the calculation date and the option’s exercise date, are commonly used variables that are input into mathematical models to derive an option’s theoretical fair value.

Aside from a company’s stock and strike prices, time, volatility, and interest rates are also quite integral in accurately pricing an option. The longer that an investor has to exercise the option, the greater the likelihood that it will be ITM at expiration. Similarly, the more volatile the underlying asset, the greater the odds that it will expire ITM. Higher interest rates should translate into higher option prices.

The best-known pricing model for options is the Black-Scholes method. This method considers the underlying stock price, option strike price, time until the option expires, underlying stock volatility and risk-free interest rate to provide a value for the option. Other popular models exist such as the binomial tree and trinomial tree pricing models.

Swaps Pricing Basics

Swaps are derivative instruments that represent an agreement between two parties to exchange a series of cash flows over a specific period of time. Swaps offer great flexibility in designing and structuring contracts based on mutual agreement. This flexibility generates many swap variations, with each serving a specific purpose. For instance, one party may swap a fixed cash flow to receive a variables cash flow that fluctuates as interest rates change. Others may swap cash flows associated with the interest rates in one country for that of another.

The most basic type of swap is a plain vanilla interest rate swap. In this type of swap, parties agree to exchange interest payments. For example, assume Bank A agrees to make payments to Bank B based on a fixed interest rate while Bank B agrees to make payments to Bank A based on a floating interest rate.

### Models:

The Expectancy Model

The Expectancy Model of futures pricing states that the futures price of an asset is basically what the spot price of the asset is expected to be in the future.

This means, if the overall market sentiment leans towards a higher price for an asset in the future, the futures price of the asset will be positive.

In the exact same way, a rise in bearish sentiments in the market would lead to a fall in the futures price of the asset.

Unlike the Cost of Carry model, this model believes that there is no relationship between the present spot price of the asset and its futures price. What matters is only what the future spot price of the asset is expected to be.

This is also why many stock market participants look to the trends in futures prices to anticipate the price fluctuation in the cash segment.

The Cost of Carry Model

The Cost of Carry Model assumes that markets tend to be perfectly efficient. This means there are no differences in the cash and futures price. This, thereby, eliminates any opportunity for arbitrage the phenomenon where traders take advantage of price differences in two or more markets.

When there is no opportunity for arbitrage, investors are indifferent to the spot and futures market prices while they trade in the underlying asset. This is because their final earnings are eventually the same.

The model also assumes, for simplicity sake, that the contract is held till maturity, so that a fair price can be arrived at.

In short, the price of a futures contract (FP) will be equal to the spot price (SP) plus the net cost incurred in carrying the asset till the maturity date of the futures contract.

FP = SP + (Carry Cost – Carry Return)

Here Carry Cost refers to the cost of holding the asset till the futures contract matures. This could include storage cost, interest paid to acquire and hold the asset, financing costs etc. Carry Return refers to any income derived from the asset while holding it like dividends, bonuses etc. While calculating the futures price of an index, the Carry Return refers to the average returns given by the index during the holding period in the cash market. A net of these two is called the net cost of carry.

The bottom line of this pricing model is that keeping a position open in the cash market can have benefits or costs. The price of a futures contract basically reflects these costs or benefits to charge or reward you accordingly.