How is the price of a derivative determined?

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.

Futures Pricing Basics

Futures contracts are standardized financial contracts that allow holders to buy or sell an underlying asset or commodity at a certain price in the future, which is locked in today. Therefore, the futures contract's value is based on the commodity's cash price.

Futures prices will often deviate somewhat from the cash, or spot price, of the underlying. The difference between the cash price of the commodity and the futures price is the basis. It is a crucial concept for portfolio managers and traders because this relationship between cash and futures prices affects the value of the contracts used in hedging. As there are gaps between spot and relative price until expiry of the nearest contract, the basis is not necessarily accurate.

In addition to the deviations created because of the time gap between expiry of the futures contract and the spot commodity, product quality, location of delivery and the actuals may also vary. In general, the basis is used by investors to gauge the profitability of delivery of cash or the actual, and is also used to search for arbitrage opportunities.

For example, consider a corn futures contract that represents 5,000 bushels of corn. If corn is trading at $5 per bushel, the value of the contract is $25,000. Futures contracts are standardized to include a certain amount and quality of the underlying commodity, so they can be traded on a centralized exchange. The futures price moves in relation to the spot price for the commodity based on supply and demand for that commodity.

Forwards are priced similarly to futures, but forwards are non-standardized contracts that arranged instead between two counterparties and transacted over-the-counter with more flexibility of terms.

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.

The value of the swap at the initiation date will be zero to both parties. For this statement to be true, the values of the cash flow streams that the swap parties are going to exchange should be equal. This concept is illustrated with a hypothetical example in which the value of the fixed leg and floating leg of the swap will be V_fix and V_fl respectively. Thus, at initiation:

 $V_{fix} = V_{fl}$Vfix​=Vfl​

Notional amounts are not exchanged in interest rate swaps because these amounts are equal and it does not make sense to exchange them. If it is assumed that parties also decide to exchange the notional amount at the end of the period, the process will be similar to an exchange of a fixed rate bond to a floating rate bond with the same notional amount. Therefore such swap contracts can be valued in terms of fixed and floating-rate bonds.