💹Financial Mathematics Unit 5 – Derivatives and Options Pricing

Key Concepts and Definitions

• Derivatives financial instruments that derive their value from an underlying asset, such as stocks, bonds, commodities, or currencies
• Options contracts that give the holder the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a predetermined price (strike price) on or before a specific date (expiration date)
• Futures contracts that obligate the buyer and seller to exchange an asset at a predetermined price on a specific future date
  • Futures are standardized contracts traded on exchanges, while forwards are customized contracts traded over-the-counter (OTC)
• Swaps agreements between two parties to exchange cash flows or liabilities from two different financial instruments, often used to hedge against interest rate risk or currency fluctuations
• Black-Scholes model a widely used mathematical model for pricing European-style options, which takes into account factors such as the current price of the underlying asset, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset
• Greeks a set of risk measures that quantify the sensitivity of an option's price to changes in various parameters, such as the price of the underlying asset (delta), the passage of time (theta), and the volatility of the underlying asset (vega)

Types of Derivatives

• Options financial derivatives that provide the holder with the right, but not the obligation, to buy or sell an underlying asset at a predetermined price on or before a specific date
  • Call options give the holder the right to buy the underlying asset
  • Put options give the holder the right to sell the underlying asset
• Futures contracts that obligate the buyer and seller to exchange an asset at a predetermined price on a specific future date, typically used for hedging or speculating on the price movements of commodities, currencies, or financial instruments
• Forwards customized contracts similar to futures, but traded over-the-counter (OTC) and settled at the end of the contract
• Swaps agreements to exchange cash flows or liabilities from two different financial instruments, such as exchanging fixed-rate payments for floating-rate payments (interest rate swaps) or exchanging cash flows in different currencies (currency swaps)
• Credit derivatives instruments that allow the transfer of credit risk from one party to another, such as credit default swaps (CDS), which provide insurance against the default of a borrower
• Exotic derivatives complex or non-standard derivatives that are often tailored to specific client needs, such as barrier options, lookback options, or Asian options

Underlying Assets and Markets

• Stocks equity securities that represent ownership in a company, with derivatives such as stock options and single-stock futures based on these underlying assets
• Bonds debt securities issued by corporations or governments, with derivatives such as bond futures and interest rate swaps based on these underlying assets
• Commodities physical goods such as agricultural products (corn, wheat), energy (oil, natural gas), and metals (gold, silver), with derivatives such as commodity futures and options based on these underlying assets
• Currencies foreign exchange markets, with derivatives such as currency futures, options, and swaps based on these underlying assets
• Indices market benchmarks that represent a basket of securities, such as the S&P 500 or the Dow Jones Industrial Average, with derivatives such as index futures and options based on these underlying assets
  • Indices can be equity indices, bond indices, or commodity indices
• Over-the-counter (OTC) markets decentralized markets where derivatives are traded directly between two parties, often customized to meet specific needs and subject to counterparty risk
• Exchange-traded markets centralized markets where standardized derivative contracts are traded on regulated exchanges, such as the Chicago Mercantile Exchange (CME) or the Intercontinental Exchange (ICE), which provide transparency and reduce counterparty risk

Pricing Models and Techniques

• Black-Scholes model a widely used mathematical model for pricing European-style options, which assumes that the underlying asset follows a geometric Brownian motion and that the market is frictionless
  • Inputs include the current price of the underlying asset, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset
• Binomial option pricing model a discrete-time model that uses a binomial tree to represent the possible paths the price of the underlying asset can take over the life of the option
  • The model assumes that the price of the underlying asset can only move up or down by a certain amount at each time step
• Monte Carlo simulation a technique that uses random sampling to simulate the possible paths the price of the underlying asset can take, and then calculates the average payoff of the derivative across all simulated paths
  • Particularly useful for pricing path-dependent or exotic derivatives
• Finite difference methods numerical methods that solve the partial differential equations (PDEs) governing the price of a derivative by discretizing the equations and solving them iteratively
• Volatility estimation techniques methods for estimating the volatility parameter used in pricing models, such as historical volatility (based on past price data), implied volatility (derived from option prices), or GARCH models (which capture time-varying volatility)
• Risk-neutral pricing a fundamental principle in derivatives pricing that states that the price of a derivative is the expected value of its future payoff, discounted at the risk-free rate, assuming that investors are risk-neutral
  • Allows for pricing derivatives without needing to estimate the actual risk preferences of investors

Risk Management Strategies

• Hedging the practice of taking an offsetting position in a derivative to reduce or eliminate the risk associated with an existing position in the underlying asset
  • For example, a company can use currency forwards to hedge against the risk of unfavorable exchange rate movements
• Diversification the strategy of investing in a variety of assets or derivatives to spread risk across different markets, sectors, or asset classes
  • Helps to mitigate the impact of any single adverse event on the overall portfolio
• Value at Risk (VaR) a statistical measure that quantifies the potential loss a portfolio or derivative position could incur over a given time horizon, at a specified confidence level
  • Commonly used by financial institutions to assess and manage market risk
• Stress testing a risk management technique that involves simulating how a portfolio or derivative position would perform under various adverse market scenarios, such as a significant stock market decline or a sudden increase in interest rates
  • Helps to identify potential vulnerabilities and inform risk mitigation strategies
• Counterparty risk management the process of assessing and mitigating the risk that a counterparty in a derivative transaction may fail to fulfill their obligations
  • Involves setting exposure limits, requiring collateral, and using credit derivatives such as credit default swaps (CDS) to transfer credit risk
• Margin requirements the amount of collateral that must be posted by parties in a derivative transaction to cover potential losses, as required by exchanges or clearinghouses to mitigate counterparty risk
• Risk governance the framework of policies, procedures, and oversight mechanisms that an organization puts in place to identify, measure, monitor, and control the risks associated with its derivative activities
  • Includes establishing risk limits, defining roles and responsibilities, and ensuring adequate risk reporting and escalation processes

Trading and Hedging Strategies

• Covered call writing a strategy where an investor holds a long position in the underlying asset and sells (writes) call options on that asset to generate additional income
  • The investor's potential upside is limited by the strike price of the call options, but they can keep the premium received from selling the options
• Protective put buying a strategy where an investor holds a long position in the underlying asset and buys put options on that asset to protect against potential downside risk
  • The put options act as insurance, providing the right to sell the underlying asset at a predetermined price
• Bull spread a strategy that involves buying a call (or put) option with a lower strike price and selling a call (or put) option with a higher strike price, both with the same expiration date
  • Profits when the underlying asset price increases (for call spreads) or decreases (for put spreads), but has limited upside and downside potential
• Bear spread a strategy that involves selling a call (or put) option with a lower strike price and buying a call (or put) option with a higher strike price, both with the same expiration date
  • Profits when the underlying asset price decreases (for call spreads) or increases (for put spreads), but has limited upside and downside potential
• Straddle a strategy that involves simultaneously buying a call and a put option with the same strike price and expiration date
  • Profits when the underlying asset price moves significantly in either direction, but loses money if the price remains stable
• Delta hedging a dynamic hedging strategy that involves continuously adjusting the position in the underlying asset to maintain a target delta (sensitivity to price changes) for the overall derivative position
  • Aims to minimize the risk associated with changes in the underlying asset price
• Volatility trading strategies that aim to profit from changes in the implied volatility of options, such as buying options when implied volatility is low (expecting it to increase) or selling options when implied volatility is high (expecting it to decrease)
  • Requires a good understanding of the factors that drive implied volatility, such as market sentiment and economic events

Regulatory Framework

• Dodd-Frank Wall Street Reform and Consumer Protection Act (2010) a comprehensive reform of the U.S. financial regulatory system in response to the 2008 financial crisis, which introduced new regulations for the derivatives market
  • Established the Commodity Futures Trading Commission (CFTC) and the Securities and Exchange Commission (SEC) as the primary regulators of the derivatives market
• European Market Infrastructure Regulation (EMIR) a regulation introduced by the European Union in 2012 to improve transparency and reduce systemic risk in the over-the-counter (OTC) derivatives market
  • Requires the reporting of OTC derivative contracts to trade repositories and the clearing of certain OTC derivatives through central counterparties (CCPs)
• Basel III an international regulatory framework for banks, developed by the Basel Committee on Banking Supervision, which includes guidelines for the capital treatment of derivatives exposures
  • Introduces the Standardized Approach for Counterparty Credit Risk (SA-CCR) for calculating derivative exposures and the Credit Valuation Adjustment (CVA) capital charge for counterparty credit risk
• Margin requirements regulations that require market participants to post collateral (initial margin and variation margin) for non-centrally cleared OTC derivatives
  • Aims to reduce counterparty credit risk and promote central clearing of standardized OTC derivatives
• Position limits rules that restrict the number of derivative contracts a single market participant can hold, to prevent excessive speculation and market manipulation
  • Applies to certain physically-settled commodity derivatives, such as agricultural and energy futures and options
• Volcker Rule a provision of the Dodd-Frank Act that prohibits banks from engaging in proprietary trading and limits their investments in hedge funds and private equity funds
  • Aims to reduce risk-taking by banks and prevent potential conflicts of interest between banks and their clients
• Swap Execution Facilities (SEFs) regulated trading platforms introduced by the Dodd-Frank Act for the trading of OTC derivatives
  • Designed to increase transparency, competition, and efficiency in the OTC derivatives market by promoting electronic trading and central clearing

Real-World Applications

• Interest rate risk management companies and financial institutions use interest rate derivatives, such as interest rate swaps and futures, to hedge against the risk of changes in interest rates affecting their borrowing costs or investment returns
  • For example, a company with floating-rate debt can enter into an interest rate swap to pay a fixed rate and receive a floating rate, effectively converting its floating-rate exposure to a fixed-rate exposure
• Currency risk management multinational corporations use currency derivatives, such as currency forwards and options, to hedge against the risk of unfavorable exchange rate movements affecting their foreign currency cash flows or assets
  • For example, a U.S. company expecting to receive a payment in euros in six months can enter into a currency forward contract to sell euros and buy U.S. dollars at a predetermined exchange rate, locking in the future value of the payment
• Commodity price risk management producers and consumers of commodities, such as airlines (jet fuel), agricultural companies (crops), and mining companies (metals), use commodity derivatives to hedge against the risk of price fluctuations affecting their revenues or costs
  • For example, an airline can purchase crude oil futures to lock in the price of jet fuel for a certain period, protecting against the risk of rising fuel prices
• Portfolio insurance asset managers and institutional investors use put options or dynamic hedging strategies to protect their portfolios against potential market downturns
  • For example, a fund manager can buy put options on a stock market index to limit the downside risk of their equity portfolio, while still participating in the potential upside
• Speculation and arbitrage traders and hedge funds use derivatives to speculate on the direction of market prices or to exploit price discrepancies between related assets
  • For example, a trader can buy call options on a stock they believe will increase in price, or sell put options on a stock they believe will remain stable or increase in price
• Structured products investment banks use derivatives to create complex financial instruments that offer customized risk-return profiles for specific investor needs
  • For example, a bank can create a principal-protected note that combines a zero-coupon bond with call options on a basket of stocks, offering investors the potential for equity-linked returns with downside protection
• Risk transfer and securitization derivatives play a key role in the transfer and repackaging of risk through securitization, such as the creation of mortgage-backed securities (MBS) or collateralized debt obligations (CDOs)
  • Credit derivatives, such as credit default swaps (CDS), allow investors to transfer the credit risk of bonds or loans to other parties, enabling the creation of synthetic credit products