3.6 Financial derivatives and option pricing
9 min read•august 20, 2024
Financial derivatives are contracts that derive value from underlying assets like stocks, bonds, or commodities. , , and are common types, each offering unique ways to manage risk or speculate on price movements.
Derivative pricing models, like Black-Scholes and binomial trees, help determine fair values. Understanding option sensitivities (, , , ) is crucial for risk management. Various trading strategies, from covered calls to straddles, cater to different market views and risk appetites.
Types of financial derivatives
- Financial derivatives are contracts that derive their value from an , such as stocks, bonds, commodities, currencies, interest rates or market indexes
- Common types of derivatives include options, which give the holder the right to buy () or sell () an asset at a predetermined price and date
- Other derivatives are futures contracts that obligate parties to transact an asset at a future date and price, forward contracts similar to futures but traded over-the-counter (OTC), and swaps that exchange cash flows between parties based on a notional principal amount
Underlying assets of derivatives
- Derivatives can be based on a wide variety of underlying assets, each with its own risk and return characteristics that impact the derivative's value
- Equity derivatives have stocks or stock indexes as the underlying, such as stock options, futures, and equity swaps (total return swaps)
- Fixed income derivatives are based on bonds, interest rates, and credit products, including interest rate swaps, bond futures, and credit default swaps (CDS)
- Commodity derivatives use physical commodities (crude oil, precious metals) or commodity indexes as the underlying asset
- Foreign exchange derivatives are based on currency exchange rates and include FX forwards, futures, options and swaps
- Other underlying assets include volatility indexes, inflation rates, weather conditions, and even cryptocurrency prices in some newer derivatives markets
Derivative markets and exchanges
- Derivatives can be traded on regulated exchanges or over-the-counter (OTC) between counterparties
- Exchange-traded derivatives like futures and options provide standardization, price transparency and reduced counterparty risk through central clearing
- Popular derivatives exchanges include the Chicago Mercantile Exchange (CME), Intercontinental Exchange (ICE), Eurex, and the Japan Exchange Group (JPX)
- OTC derivatives are customized contracts negotiated bilaterally between counterparties, typically large financial institutions, corporations or sophisticated investors
- The OTC market is much larger than exchange-traded but faces additional counterparty and liquidity risks
Pricing models for derivatives
Black-Scholes option pricing model
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- The is a widely used formula for pricing European-style options based on key inputs like the underlying price, , time to expiration, risk-free interest rate, and implied volatility
- It assumes the underlying asset follows a geometric Brownian motion with constant drift and volatility, and that there are no transaction costs or taxes
- The model calculates the theoretical price of call or put options, which can be compared to market prices to determine if options are over- or undervalued
Binomial option pricing model
- The uses a discrete-time lattice approach to model the varying price paths an underlying asset can take during the life of the option
- At each time step, the underlying price can move up or down by a specific factor, creating a binomial tree of potential price paths
- The option price is determined by working backwards through the tree, discounting the option payoffs at the risk-free rate
Monte Carlo simulation for option pricing
- Monte Carlo methods use repeated random sampling to simulate the price paths of an underlying asset and estimate the option price
- They are particularly useful for path-dependent exotic options or options with multiple underlying assets or complex payout structures
- The simulation generates thousands of possible price paths based on assumed parameters like volatility and drift, and the average discounted payoff across all paths is the estimated option price
Option price sensitivities
Delta vs gamma
- Delta measures an option's price sensitivity to changes in the underlying asset price, representing the slope of the option price curve
- Call options have positive delta between 0 and 1, while put options have negative delta between -1 and 0
- Gamma measures the rate of change of delta with respect to the underlying price, or the convexity of the option price curve
- Options with high gamma are more sensitive to underlying price moves and can be profitable for option sellers
Vega vs theta
- Vega represents the sensitivity of an option's price to changes in the implied volatility of the underlying asset
- Options with higher vega are more sensitive to volatility changes, and long options have positive vega while short options have negative vega
- Theta measures the time decay of an option's value as it approaches expiration, all else equal
- Long options have negative theta as they lose value over time, while short options have positive theta
Rho sensitivity
- Rho measures the sensitivity of an option's price to changes in the risk-free interest rate used for discounting
- Call options have positive rho as they become more valuable when interest rates rise (lower present value of strike price)
- Put options have negative rho as they lose value when rates rise
- Rho is generally small unless options have long times to expiration
Option trading strategies
Covered calls vs protective puts
- strategy involves owning the underlying stock and selling call options on the same asset, generating premium income but limiting upside potential
- strategy involves owning the underlying stock and buying put options as downside protection, establishing a floor on losses but at the cost of the option premium
Bull spreads vs bear spreads
- Bull call spreads combine buying a lower strike call and selling a higher strike call with the same expiration, profiting from moderately rising underlying prices
- Bear put spreads involve buying a higher strike put and selling a lower strike put, profiting from moderately falling prices
- Both spreads limit risk and reward compared to outright option positions and benefit from declining implied volatility
Straddles vs strangles
- Long combines buying a call and put at the same strike and expiration, profiting from large price moves in either direction but losing money if prices remain stable
- Long involves buying a call and put with different strikes, typically out-of-the-money, which is cheaper than a straddle but requires larger price moves to profit
- Short straddles and strangles profit from stable prices and declining implied volatility but face unlimited risk if prices move sharply
Butterfly spreads vs condor spreads
- Butterfly spreads involve buying one call and put at a middle strike, and selling a call and put at an upper and lower equidistant strike (four options total)
- Condor spreads are similar to butterflies but use different strikes for the two sold options, creating a wider profit range
- Both strategies profit from stable underlying prices and declining volatility, with limited risk and reward
Hedging with options
Delta hedging techniques
- involves creating an option position with a delta that offsets the delta of an existing position, making the overall portfolio delta-neutral
- Delta-neutral portfolios are insensitive to small underlying price movements but require constant rebalancing as option deltas change
- Delta can be done by buying or selling the underlying asset in the right proportion to offset the option delta
Gamma hedging strategies
- aims to make a portfolio gamma-neutral by offsetting the gamma of existing positions
- This requires using options with opposite gamma, such as selling options to hedge a long option position
- Gamma hedging is more complex than delta hedging and requires monitoring the rate of change of delta, but can help maintain delta-neutrality over larger price moves
Exotic options and pricing
Asian options vs barrier options
- Asian options have a payout based on the average price of the underlying asset over a certain period, rather than the price at expiration
- Asian options are less sensitive to market manipulation or extreme price movements near expiration
- Barrier options have payouts that depend on whether the underlying price reaches a certain level (barrier) during the option's life
- Knock-in options only activate if the barrier is reached, while knock-out options become worthless if the barrier is reached
Lookback options vs forward-start options
- Lookback options allow the holder to buy (call) or sell (put) the underlying asset at the most favorable price observed during the life of the option
- Lookback options are path-dependent and more expensive than standard options due to the increased optionality
- Forward-start options have a strike price that is determined at a future date, typically based on the underlying price at that time
- Forward-start options allow investors to lock in future volatility exposure without specifying the strike in advance
Quanto options and foreign exchange risk
- Quanto options are cross-currency options where the payoff is determined in one currency but settled in another at a fixed exchange rate
- Quanto options allow investors to gain exposure to foreign assets without the foreign exchange risk
- The quanto feature adds complexity to pricing and hedging, as it introduces correlation between the underlying asset and exchange rate
Credit derivatives and swaps
Credit default swaps (CDS)
- A CDS is a contract where the buyer makes periodic payments to the seller in exchange for a contingent payment if a credit event occurs on a reference entity
- Credit events include bankruptcy, failure to pay, and restructuring, and the CDS payoff is typically the difference between the par value and market value of the reference debt
- CDS allow investors to hedge credit risk or speculate on changes in credit quality, and the CDS spread reflects the perceived creditworthiness of the reference entity
Total return swaps (TRS)
- In a TRS, one party makes payments based on a set rate (reference rate plus a spread) while the other party makes payments based on the total return (interest plus capital gains) of an underlying asset
- The underlying asset is often an equity index, loan, or bond, and the TRS allows the receiver to gain exposure to the economic performance without owning it directly
- TRS can be used for financing, leverage, or hedging purposes, but face counterparty and liquidity risks
Credit-linked notes (CLNs)
- CLNs are structured notes that combine a credit derivative with a fixed income security
- The CLN pays periodic coupons and a principal payment at maturity, but the payments are contingent on the performance of a reference credit or portfolio
- If a credit event occurs, the CLN may deliver the underlying debt or experience loss of principal, providing the investor with credit risk exposure
Risk management with derivatives
Value-at-Risk (VaR) for derivatives
- VaR measures the maximum potential loss on a derivatives portfolio over a given time horizon and confidence level, assuming normal market conditions
- VaR can be calculated using historical simulation (using past returns), Monte Carlo simulation (generating random scenarios), or parametric methods (assuming a distribution)
- Limitations of VaR include the assumption of normal distributions, the focus on a single quantile, and the inability to capture tail risks beyond the confidence level
Expected shortfall and tail risk
- is the average loss beyond the VaR threshold, providing a more comprehensive measure of tail risk
- ES is more sensitive to the shape of the loss distribution in the tail and is a coherent risk measure satisfying properties like subadditivity
- Derivatives with non-linear payoffs and high convexity (gamma) can have significant tail risks that may be underestimated by VaR alone
Stress testing derivative portfolios
- involves subjecting a derivatives portfolio to hypothetical extreme market scenarios to assess potential losses
- Scenarios can be historical (replicating past stress events) or hypothetical (simulating plausible shocks), and should cover a range of risk factors relevant to the derivatives held
- Stress tests complement VaR by capturing losses beyond the confidence level and identifying vulnerabilities in the portfolio
- Reverse stress testing can also be used to identify scenarios that would cause a certain level of losses
Regulation of derivative markets
Dodd-Frank Act and derivatives
- The Dodd-Frank Wall Street Reform and Consumer Protection Act was passed in 2010 in response to the financial crisis, with several provisions impacting derivatives markets
- The act mandated central clearing for standardized OTC derivatives, imposed margin requirements for non-cleared swaps, and required trade reporting to increase transparency
- It also established the Volcker Rule prohibiting banks from proprietary trading and restricted investments in hedge funds and private equity funds
- The act created the Consumer Financial Protection Bureau (CFPB) and Financial Stability Oversight Council (FSOC) to monitor systemic risks
European Market Infrastructure Regulation (EMIR)
- EMIR is a European Union regulation introduced in 2012 to regulate OTC derivatives, central counterparties (CCPs), and trade repositories
- Key requirements include mandatory clearing for eligible OTC derivatives, margin and capital requirements for non-cleared derivatives, and reporting of all derivative contracts to trade repositories
- EMIR also sets standards for CCP risk management, governance, and operational resilience to ensure their safety and soundness
- The regulation aims to reduce systemic risk, increase transparency, and protect against market abuse in the derivatives markets
Key Terms to Review (43)
American option: An American option is a type of financial derivative that gives the holder the right to buy or sell an underlying asset at a specified strike price before or on its expiration date. This flexibility allows the holder to exercise the option at any time, making it a valuable tool for hedging and speculation in financial markets. American options can be used on various underlying assets, including stocks, commodities, and indices.
Arbitrage: Arbitrage is the practice of taking advantage of price differences in different markets to make a profit. It involves simultaneously buying and selling an asset in different markets to exploit the price discrepancy, ensuring a risk-free profit under certain conditions. This concept is crucial in maintaining market efficiency, as it helps to equalize prices across different platforms and reduce discrepancies in the valuation of financial derivatives and options.
Asian option: An Asian option is a type of financial derivative where the payoff depends on the average price of the underlying asset over a specified period, rather than just its price at maturity. This averaging feature can reduce volatility and offer a different risk profile compared to traditional options, making them an attractive choice for hedging and speculative strategies in various markets.
Barrier Option: A barrier option is a type of financial derivative whose existence and value depend on the underlying asset's price reaching a certain barrier level. These options can be either 'knock-in' options, which come into existence when the barrier is breached, or 'knock-out' options, which become void if the barrier is crossed. Understanding barrier options is essential as they offer unique risk management opportunities and pricing mechanisms compared to standard options.
Bear spread: A bear spread is an options trading strategy designed to profit from a decline in the price of an underlying asset. This strategy involves simultaneously buying and selling options with different strike prices or expiration dates, which allows the trader to limit potential losses while capitalizing on bearish market sentiment. The bear spread can be executed using either call or put options, making it a flexible approach for investors anticipating a downward move in asset prices.
Binomial model: The binomial model is a mathematical framework used to price options and financial derivatives by simulating the possible price paths of an underlying asset over discrete time intervals. This model creates a binomial tree where each node represents a potential price at a given time, allowing for the assessment of option values based on varying scenarios of price movements. It connects closely with option pricing, as it provides a systematic way to evaluate the potential outcomes and helps in determining fair prices for options in a structured manner.
Black-Scholes Model: The Black-Scholes Model is a mathematical model for pricing financial derivatives, particularly options. It provides a theoretical estimate of the 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. This model is grounded in concepts like Brownian motion and diffusion processes, which are essential for understanding how asset prices evolve over time under uncertainty.
Bull spread: A bull spread is an options trading strategy that involves buying and selling options of the same class with the same expiration date but different strike prices, aiming to profit from a rise in the underlying asset's price. This strategy is typically executed by purchasing a call option at a lower strike price and simultaneously selling another call option at a higher strike price. By doing so, the trader limits potential losses while also capping potential gains.
Butterfly spread: A butterfly spread is an options trading strategy that involves using multiple options contracts to create a position with limited risk and limited profit potential. This strategy typically consists of buying and selling options at three different strike prices, all with the same expiration date, to profit from low volatility in the underlying asset. The goal is to make a profit when the asset's price remains near the middle strike price at expiration.
Call Option: A call option is a financial contract that gives the holder the right, but not the obligation, to buy an underlying asset at a specified price within a set time period. This financial derivative allows investors to speculate on the potential increase in value of the asset, providing opportunities for profit while limiting potential losses. Call options are widely used in investment strategies to manage risk and leverage positions in various markets.
Condor Spread: A condor spread is an options trading strategy that involves buying and selling options with the same expiration date but different strike prices, forming a profit range that resembles a 'condor' shape. This strategy can be constructed using either all calls or all puts and is designed to capitalize on low volatility in the underlying asset, allowing traders to benefit from the price remaining within a certain range.
Covered Call: A covered call is an options trading strategy where an investor holds a long position in an asset, such as stocks, and simultaneously sells call options on that same asset. This approach allows the investor to generate additional income through option premiums while still holding the underlying asset, offering potential profit if the asset's price remains stable or appreciates moderately.
Credit Default Swap (CDS): A credit default swap is a financial derivative that allows an investor to 'swap' or transfer the credit risk of a borrower to another party. Essentially, it acts like insurance against the default of a borrower, where one party pays a periodic fee to another party in exchange for compensation if a specified credit event occurs, such as default or bankruptcy. This mechanism plays a significant role in managing credit risk and is a key component in the broader context of financial derivatives and option pricing.
Credit-linked note (CLN): A credit-linked note (CLN) is a type of debt security that is linked to the credit risk of one or more underlying reference entities. Essentially, CLNs are structured to provide investors with exposure to credit risk while offering a higher yield compared to standard bonds. They combine features of both bonds and credit derivatives, making them a unique investment option that allows issuers to transfer credit risk and investors to gain potentially lucrative returns based on the credit performance of the reference entities.
Delta: Delta is a measure used in options pricing that indicates how much the price of an option is expected to change when the price of the underlying asset changes by one unit. It is a key component in assessing the risk and potential reward associated with options, allowing traders to gauge how sensitive an option's price is to changes in the market. Understanding delta is crucial for effective hedging strategies and for determining the optimal trading strategies in financial derivatives.
Delta Hedging: Delta hedging is a risk management strategy used to reduce the risk associated with price movements in an asset by taking a position in its derivatives, primarily options. This strategy involves adjusting the holdings in the underlying asset to offset changes in the delta of the option, which measures how much the option's price is expected to move in relation to a change in the price of the underlying asset. By employing delta hedging, investors can aim to maintain a neutral position, minimizing potential losses from price fluctuations.
European Option: A European option is a type of financial derivative that grants the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price on a specific expiration date. Unlike American options, which can be exercised at any time before expiration, European options can only be exercised on their expiration date. This characteristic impacts their pricing and strategy compared to other option types.
Expected Shortfall (ES): Expected shortfall (ES) is a risk measure that quantifies the expected loss in value of an investment or portfolio during extreme market conditions, specifically in the tail of the loss distribution. It extends beyond Value at Risk (VaR) by not only considering the potential losses that exceed a certain threshold but also providing an average of those losses, offering a more comprehensive view of risk exposure. ES is particularly valuable for financial risk management and in assessing the performance of risk-sensitive investments.
Expiration Date: The expiration date is the specific date at which an option contract becomes void and can no longer be exercised. It is a critical feature in the realm of financial derivatives, particularly options, as it defines the time limit within which the holder must decide whether to exercise their right to buy or sell the underlying asset at the predetermined strike price. The expiration date directly affects the time value of an option and plays a significant role in pricing and trading strategies.
Forward-start option: A forward-start option is a type of financial derivative that grants the holder the right to purchase or sell an underlying asset at a specified price, but the exercise of this option begins at a future date rather than at the current time. This unique feature allows investors to gain exposure to price movements over a designated time frame while postponing their decision to exercise the option until a predetermined point in the future, which can provide strategic advantages in managing risk and optimizing investment returns.
Futures: Futures are standardized financial contracts that obligate the buyer to purchase, and the seller to sell, an asset at a predetermined price at a specific future date. These contracts are commonly used for hedging or speculating on the price movement of various assets, such as commodities, currencies, or financial instruments. Futures play a critical role in risk management and can help stabilize prices in markets by allowing participants to lock in prices for future transactions.
Gamma: Gamma is a measure of the rate of change of an option's delta with respect to changes in the underlying asset's price. It reflects the sensitivity of an option's delta to movements in the price of the underlying asset, helping traders assess how much the option's price might change as the underlying asset fluctuates. Understanding gamma is crucial for managing risk and making informed trading decisions in financial derivatives and options pricing.
Gamma Hedging: Gamma hedging is a risk management strategy used to reduce the risk associated with the curvature of an option's price movement as the underlying asset's price changes. This technique involves adjusting a portfolio of options and their underlying assets in response to changes in the delta, which measures how much the price of an option changes when the price of the underlying asset changes. By implementing gamma hedging, traders aim to maintain a neutral position that protects against significant fluctuations in the market.
Hedging: Hedging is a risk management strategy used to offset potential losses or gains in investments by taking an opposite position in a related asset. This practice helps to minimize the impact of price fluctuations on an investment, allowing individuals and institutions to stabilize their financial outcomes. In finance, hedging often involves the use of derivatives like options and futures, while in insurance and risk modeling, it can refer to strategies that balance risk across different entities or portfolios.
Intrinsic value: Intrinsic value refers to the actual, inherent worth of an asset or financial instrument, regardless of its market price. In the context of financial derivatives and option pricing, intrinsic value represents the difference between the underlying asset's current price and the strike price of the option when the option is in-the-money. This concept is crucial for understanding how options are valued and how they behave in various market conditions.
Lookback option: A lookback option is a type of exotic option that allows the holder to 'look back' over time to determine the payoff based on the maximum or minimum price of the underlying asset during the life of the option. This feature makes lookback options unique, as they can provide greater potential for profit compared to standard options by locking in the most favorable price achieved in the past. The flexibility in choosing prices at expiration makes them particularly appealing in financial derivatives and option pricing.
No-arbitrage principle: The no-arbitrage principle states that in efficient markets, identical assets must have the same price, or else arbitrage opportunities will exist, allowing traders to profit without any risk. This principle underlies much of modern financial theory and plays a crucial role in pricing financial derivatives and options. When markets are functioning properly, any discrepancies in pricing should quickly be corrected by traders seeking to exploit these opportunities.
Options: Options are financial derivatives that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price before or at a certain expiration date. They play a crucial role in risk management and trading strategies, allowing investors to hedge positions or speculate on price movements without the need to own the actual asset.
Protective Put: A protective put is an options trading strategy where an investor buys a put option for an asset they already own to protect against potential losses. This strategy allows the investor to limit downside risk while still maintaining the potential for upside gains from the asset's appreciation. It serves as a form of insurance, providing peace of mind and reducing the overall risk exposure associated with holding the underlying asset.
Put Option: A put option is a financial derivative that gives the holder the right, but not the obligation, to sell a specified amount of an underlying asset at a predetermined price, known as the strike price, before or on a specified expiration date. This option allows investors to hedge against declines in the price of the underlying asset, thereby providing a form of insurance. It plays a crucial role in options pricing and can be used strategically in various financial strategies to manage risk.
Quanto option: A quanto option is a type of financial derivative that allows investors to gain exposure to the price movement of an underlying asset in one currency while receiving payouts in another currency, typically at a fixed exchange rate. This structure helps manage foreign exchange risk, as the investor does not have to deal with fluctuating currency values, making it appealing for hedging or speculation.
Risk-neutral valuation: Risk-neutral valuation is a financial concept used to price derivatives by assuming that all investors are indifferent to risk. In this framework, the expected returns on risky assets are adjusted using a risk-neutral measure, allowing for the valuation of financial derivatives based on their expected payoffs discounted at the risk-free rate. This approach simplifies the pricing of options and other derivatives, as it removes the need to account for individual risk preferences.
Straddle: A straddle is an options trading strategy that involves buying both a call option and a put option with the same strike price and expiration date. This strategy allows an investor to profit from significant price movements in either direction, whether the underlying asset increases or decreases in value. Straddles are particularly useful in situations where the investor expects high volatility but is uncertain about the direction of the price movement.
Strangle: A strangle is an options trading strategy that involves buying both a call option and a put option with the same expiration date but different strike prices. This strategy allows traders to profit from significant price movements in either direction, making it a popular choice when volatility is expected. By holding both options, the trader can benefit from large price changes, regardless of whether the asset's price goes up or down.
Stress Testing: Stress testing is a risk management tool used to evaluate the resilience of financial systems and institutions under extreme conditions. It involves simulating adverse scenarios to assess potential impacts on capital, liquidity, and overall financial stability. This technique is crucial for understanding vulnerabilities and ensuring that organizations can withstand severe economic shocks.
Strike Price: The strike price, also known as the exercise price, is the predetermined price at which an option holder can buy (in the case of a call option) or sell (in the case of a put option) the underlying asset when exercising the option. It is a critical element in options trading as it directly influences the option's intrinsic value and is essential for determining whether an option is in-the-money, at-the-money, or out-of-the-money. Understanding strike price helps in making informed trading decisions regarding financial derivatives.
Swaps: Swaps are financial derivatives in which two parties exchange cash flows or financial instruments over a specified period based on predetermined conditions. These agreements are used to manage risk, speculate on changes in interest rates or currencies, and improve liquidity. By allowing parties to swap their cash flow streams, swaps facilitate better alignment with their financial strategies and needs.
Theta: Theta is a measure of the rate at which the price of an option decreases as it approaches its expiration date, representing the time decay of the option's value. It quantifies the impact of time on the pricing of options, and is critical for traders and investors to understand because it affects their strategies regarding holding or exercising options. A high theta indicates that an option's price will decrease significantly over time, while a low theta suggests that the option retains its value longer as expiration approaches.
Time Value: Time value refers to the concept that the value of money is not static and can change over time, primarily due to the potential earning capacity of that money. This idea is crucial in finance, as it highlights how a dollar today is worth more than a dollar in the future because of its ability to earn interest or generate returns. Time value is a foundational principle in understanding how financial derivatives and option pricing operate, since the longer the time until an option expires, the more potential there is for value creation through price movements.
Total return swap (TRS): A total return swap (TRS) is a financial contract in which one party transfers the total economic performance of an asset, including income and capital gains, to another party in exchange for regular cash flows, often based on a fixed or floating interest rate. This instrument allows investors to gain exposure to an asset without actually owning it, making it useful for hedging and speculation. TRSs are commonly used in the context of financial derivatives and option pricing as they can affect the valuation and risk profiles of portfolios.
Underlying asset: An underlying asset is a financial instrument or security that serves as the basis for a financial derivative. This asset can take many forms, including stocks, bonds, commodities, currencies, or indices, and its value directly influences the price of the derivative products like options and futures. Understanding the characteristics of the underlying asset is essential for pricing derivatives and managing associated risks.
Value-at-risk (VaR): Value-at-risk (VaR) is a statistical measure used to assess the potential loss in value of an asset or portfolio over a defined period for a given confidence interval. It provides a way to quantify financial risk, helping in understanding the worst-case scenario under normal market conditions. VaR connects closely with risk management techniques, where it can be applied in simulation methods to estimate potential losses, particularly in pricing financial derivatives and assessing individual or collective risks within insurance models.
Vega: Vega is a measure of an option's sensitivity to changes in the volatility of the underlying asset. It indicates how much the price of an option is expected to change when the volatility of the underlying asset increases or decreases by one percentage point. Understanding vega is crucial for traders and investors as it helps in assessing the risk associated with options positions, especially in uncertain market conditions.