Options, Futures and Other Derivatives

1. Derivatives: Essential Tools for Risk Management and Market Participation

Whether you like or hate derivatives, you cannot ignore them!

Derivatives defined. Derivatives are financial instruments whose value is derived from an underlying asset or variable. These instruments, including options, futures, forwards, and swaps, play a crucial role in modern finance, enabling risk management, speculation, and arbitrage.

Risk management. Derivatives allow businesses to hedge against various risks, such as fluctuations in commodity prices, interest rates, and exchange rates. By using derivatives, companies can stabilize their cash flows and make more informed decisions. For example, an airline can use fuel futures to protect itself from rising jet fuel costs, or a manufacturer can use currency forwards to hedge against exchange rate risk when exporting goods.

Speculation and arbitrage. Derivatives also provide opportunities for speculation, allowing investors to bet on the future direction of markets. Additionally, arbitrageurs use derivatives to exploit price discrepancies in different markets, ensuring market efficiency. However, it's important to note that derivatives can be complex and carry significant risks if not used properly.

2. Futures Markets: Standardized Contracts and Clearinghouse Guarantees

Once two traders have agreed on a trade, the exchange's clearing house steps in to act as an intermediary and to guarantee the performance of the trade.

Standardized contracts. Futures contracts are agreements to buy or sell an asset at a predetermined price and date in the future. These contracts are standardized by exchanges, specifying the quantity, quality, and delivery location of the underlying asset. Standardization facilitates trading and ensures liquidity.

Clearinghouse role. A key feature of futures markets is the role of the clearinghouse, which acts as an intermediary between buyers and sellers. The clearinghouse guarantees the performance of the contract, mitigating counterparty risk. This is achieved through margin requirements, where both parties must deposit funds to cover potential losses.

Electronic trading. Modern futures exchanges have largely transitioned to electronic trading platforms, replacing traditional open outcry systems. Electronic trading has increased efficiency, transparency, and accessibility, allowing for high-frequency trading and algorithmic strategies.

3. Hedging Strategies: Mitigating Risk with Futures and Options

Hedging is designed to reduce risk, it does not necessarily improve the outcome.

Hedging defined. Hedging involves using derivatives to reduce or eliminate the risk associated with an existing position or future transaction. Hedgers seek to protect themselves from adverse price movements by taking an offsetting position in the derivatives market.

Short hedge. A short hedge is used when a company or individual owns an asset and expects to sell it in the future. By selling a futures contract, the hedger locks in a price for the asset, protecting against potential price declines. For example, a farmer can use a short hedge to guarantee a price for their crop before harvest.

Long hedge. A long hedge is used when a company or individual expects to purchase an asset in the future. By buying a futures contract, the hedger locks in a price for the asset, protecting against potential price increases. For example, a bakery can use a long hedge to secure a price for wheat they will need in the future.

4. Interest Rate Dynamics: Understanding and Measuring Interest Rate Risk

The derivatives markets have been so successful partly because they attract many different types of traders and are usually very liquid.

Interest rate importance. Interest rates are fundamental to the pricing of many derivatives, particularly those related to fixed income securities. Understanding interest rate dynamics, including term structure, compounding, and zero rates, is crucial for effective risk management.

Measuring interest rates. Interest rates can be measured using different compounding frequencies, such as annually, semi-annually, or continuously. Continuous compounding is often used in derivative pricing due to its mathematical convenience. The equivalent annual interest rate is the rate that, when compounded annually, yields the same return as the given rate with its compounding frequency.

Duration and convexity. Duration measures the sensitivity of a bond's price to changes in interest rates. Convexity measures the curvature of the price-yield relationship, providing a more accurate estimate of price changes for larger interest rate movements. These measures are essential tools for managing interest rate risk in fixed income portfolios.

5. Forward and Futures Pricing: Arbitrage and Market Equilibrium

Arbitrageurs are constantly watching the markets for mispricing.

Forward price determination. The forward price of an asset is the price agreed upon today for delivery of the asset at a specified future date. The forward price is determined by arbitrage arguments, ensuring that no risk-free profit opportunities exist.

Cost of carry. The forward price is typically related to the spot price through the cost of carry, which includes storage costs, insurance, and financing costs, less any income generated by the asset. For example, the forward price of gold will reflect the cost of storing the gold over the life of the contract, as well as the interest rate used to finance the purchase.

Futures vs. forwards. While forward and futures contracts are similar, they differ in their standardization, trading location, and settlement procedures. Futures contracts are traded on exchanges and are subject to daily settlement, while forward contracts are customized and traded over-the-counter. However, in most cases, the forward price and futures price for the same asset and delivery date are very close.

6. Options Pricing: The Black-Scholes-Merton Model and Beyond

The key to the Black-Scholes-Merton analysis is to set up a riskless portfolio consisting of the stock and the option.

Black-Scholes-Merton model. The Black-Scholes-Merton model is a cornerstone of options pricing theory. It provides a mathematical formula for calculating the theoretical price of European-style options based on factors such as the underlying asset price, strike price, time to expiration, risk-free interest rate, and volatility.

Model assumptions. The model relies on several key assumptions, including constant volatility, a risk-free interest rate, and efficient markets. While these assumptions are not always met in practice, the model provides a valuable framework for understanding option pricing.

Risk-neutral valuation. The Black-Scholes-Merton model is based on the principle of risk-neutral valuation, which states that the price of an option is equal to the expected payoff of the option in a risk-neutral world, discounted at the risk-free interest rate. This principle simplifies the pricing process by eliminating the need to consider investors' risk preferences.

7. Volatility Smiles and Skews: Capturing Market Sentiment

The volatility smile shows that the implied volatility is not constant.

Volatility smile defined. The volatility smile is a pattern observed in the implied volatilities of options with the same expiration date but different strike prices. It reflects the market's perception of the relative likelihood of different price movements.

Smile vs. skew. In equity markets, the volatility smile is often more of a skew, with implied volatilities generally increasing as strike prices decrease. This skew reflects a greater demand for downside protection, as investors are more concerned about potential market crashes than large gains.

Implied distribution. The shape of the volatility smile or skew provides insights into the market's implied probability distribution of the underlying asset's price. A volatility smile suggests that the market believes the distribution has fatter tails than a normal distribution, while a skew indicates asymmetry in the perceived probabilities of upside and downside movements.

8. Numerical Methods: Approximating Derivative Prices

The purpose of numerical procedures is to value derivatives when analytic results are not available.

Numerical methods defined. Numerical methods are techniques used to approximate the prices of derivatives when analytical solutions are not available. These methods are particularly useful for valuing complex options with features such as early exercise or path-dependent payoffs.

Binomial trees. Binomial trees are a popular numerical method for option pricing. They involve constructing a tree-like structure that represents the possible price paths of the underlying asset over time. By working backward through the tree, the option's value can be determined at each node, ultimately leading to an estimate of the option's current price.

Monte Carlo simulation. Monte Carlo simulation is another widely used numerical method. It involves generating a large number of random price paths for the underlying asset and calculating the option's payoff for each path. The average payoff, discounted to the present, provides an estimate of the option's price.

9. Credit Risk: Assessing and Managing Default Probabilities

The credit rating of a company is an important determinant of the interest rate it has to pay when borrowing money.

Credit risk defined. Credit risk is the risk that a borrower will default on their debt obligations, resulting in a loss for the lender. Assessing and managing credit risk is crucial for financial institutions, as it directly impacts their profitability and solvency.

Credit ratings. Credit rating agencies, such as Moody's, Standard & Poor's, and Fitch, assign credit ratings to companies and their debt instruments. These ratings provide an assessment of the borrower's creditworthiness, helping investors evaluate the risk of default.

Default probabilities. Default probabilities are estimates of the likelihood that a borrower will default on their debt obligations within a specified time horizon. These probabilities can be derived from historical data, bond yields, or stock prices.

10. Credit Derivatives: Transferring Credit Risk in the Financial System

Credit derivatives allow companies to trade credit risk in much the same way as they trade market risk.

Credit derivatives defined. Credit derivatives are financial instruments that allow investors to transfer credit risk from one party to another. These instruments, such as credit default swaps (CDS) and collateralized debt obligations (CDOs), have become increasingly important in the financial system.

Credit default swaps. A CDS is a contract that provides insurance against the default of a specific entity or asset. The buyer of protection pays a premium to the seller, who agrees to compensate the buyer if a credit event occurs. CDS are widely used to hedge credit risk or to speculate on the creditworthiness of borrowers.

Collateralized debt obligations. CDOs are structured finance products that pool together a portfolio of debt instruments, such as bonds or loans, and then divide the cash flows into different tranches with varying levels of credit risk. CDOs allow investors to gain exposure to a diversified portfolio of debt, while also tailoring their risk and return profile.

11. Exotic Options: Tailoring Derivatives to Specific Needs

Exotic options are designed to meet the particular needs of a corporate treasurer or fund manager.

Exotic options defined. Exotic options are options with non-standard features that make them more complex than plain vanilla options. These options are often customized to meet the specific needs of corporate treasurers, fund managers, or other sophisticated investors.

Types of exotic options:

• Barrier options: Options that are activated or terminated when the underlying asset reaches a certain price level.
• Asian options: Options whose payoff depends on the average price of the underlying asset over a specified period.
• Lookback options: Options whose payoff depends on the maximum or minimum price of the underlying asset over a specified period.
• Chooser options: Options that allow the holder to choose whether the option is a call or a put at a later date.

Static option replication. Static option replication involves creating a portfolio of standard options that replicates the payoff of an exotic option. This technique can be used to hedge the risk of exotic options or to create synthetic exotic options.

12. Advanced Models: Martingales, Measures, and Interest Rate Derivatives

The key result in this chapter is the equivalent martingale measure result.

Martingales and measures. Martingales are stochastic processes with a zero drift, meaning that their expected future value is equal to their current value. Measures are probability distributions used to price derivatives. The equivalent martingale measure result states that there exists a measure under which all asset prices, when discounted by the risk-free rate, are martingales.

HJM and LMM models. The Heath-Jarrow-Morton (HJM) and LIBOR Market Model (LMM) are advanced models used to price interest rate derivatives. These models allow for a more flexible and realistic representation of interest rate dynamics than simpler models.

OIS discounting. OIS discounting involves using overnight indexed swap (OIS) rates to discount cash flows in derivative pricing. This approach has become increasingly popular since the 2008 financial crisis, as OIS rates are considered to be a more reliable measure of the risk-free rate than LIBOR.

Last updated: March 11, 2025