Derivatives Pricing – Terms, Definitions & Topical Outline

In this overview, we look at a comprehensive range of terms and definitions for understanding the pricing of derivatives in finance.

We cover various processes, concepts, and models.

Key Takeaways – Derivatives Pricing

• Derivative pricing involves models like Brownian motion and risk-neutral valuation to predict price movements and valuations, considering market risk and arbitrage opportunities.
• The valuation of swaps, options, and futures integrates factors such as current market data, interest rates, and credit risks, using models like Black-Scholes and Monte Carlo.
• For options, understanding the Greeks and intrinsic/time values is key, whereas for interest rate derivatives, short-rate models like Vasicek and Hull-White are used.

Derivatives Pricing Terms & Definitions (Outline)

The Brownian Model of Financial Markets

The Brownian model of financial markets refers to the use of Brownian motion to model price movements in financial markets.

Suggests that changes in prices are continuous and random.

Rational Pricing Assumptions

Rational pricing assumptions state that the current price of a security reflects the expected value of its future payoffs discounted at a risk-appropriate rate.

Risk-Neutral Valuation

Risk-neutral valuation is a method used in finance to price risky assets by discounting expected payoffs at the risk-free rate, assuming investors are indifferent to risk.

Arbitrage-Free Pricing

Arbitrage-free pricing is a financial concept asserting that assets should be priced such that no arbitrage opportunities exist.

This means, when true, there’s no possibility of risk-free profits.

Valuation Adjustments

Valuation adjustments are modifications made to the fair value of a financial instrument to account for factors such as credit risk and funding costs.

Credit Valuation Adjustment

Credit valuation adjustment (CVA) is the difference between the risk-free portfolio value and the true portfolio value that considers the possibility of a counterparty’s default.

XVA

XVA is a collective term for a series of adjustments in derivative valuations, including:

• CVA (Credit Valuation Adjustment)
• DVA (Debit Valuation Adjustment)
• FVA (Funding Valuation Adjustment)

Yield Curve Modelling

Yield curve modeling involves the mathematical representation of the term structure of interest rates.

This outlines the relationship between interest rates and various maturities.

Multi-Curve Framework

The multi-curve framework is an approach in finance where different yield curves are used for deriving discount factors and forward rates – recognizing the divergence between funding and lending rates.

Bootstrapping

Bootstrapping is a method for constructing a zero-coupon yield curve by sequentially deriving spot rates from the prices of a set of coupon-bearing products.

Construction from Market Data

Construction from market data refers to the process of building financial models or curves using actual market prices and rates as inputs.

Fixed-Income Attribution

Fixed-income attribution is the process of analyzing the performance of a fixed-income investment by breaking down the returns into different contributing factors.

Nelson-Siegel

The Nelson-Siegel model is an empirical formula for fitting the term structure of interest rates with parameters representing level, slope, and curvature of the yield curve.

Principal Component Analysis

Principal component analysis (PCA) is a statistical technique used to simplify the complexity in high-dimensional data by reducing the dimensionality of the data set.

Forward Price Formula

The forward price formula is an equation used to determine the theoretical future price of a commodity or security based on:

• current prices
• interest rates
• dividends, and
• time to maturity

Futures Contract Pricing

Futures contract pricing involves determining the fair value of a futures contract based on the current spot price, the risk-free rate, and the cost of carry.

Swap Valuation

Swap valuation is the process of determining the market value of a swap, considering the present value of its expected future cash flows.

Currency Swap Valuation and Pricing

Currency swap valuation and pricing refer to the process of determining the value of a currency swap.

This involves exchanging principal and interest payments in one currency for those in another.

Interest Rate Swap Valuation and Pricing

Interest rate swap valuation and pricing involve calculating the current value of the swap’s future cash flows discounted at the appropriate market rates.

Variance Swap Pricing and Valuation

Variance swap pricing and valuation involve determining the fair value of a variance swap, which pays out based on the difference between realized and implied volatility.

Asset Swap Spread

Computing the asset swap spread involves calculating the spread over SOFR – or a different interbank lending rate – that a trader will earn for swapping fixed cash flows of a bond for floating cash flows.

Credit Default Swap Pricing and Valuation

Credit default swap pricing and valuation entail calculating the price of a CDS contract.

This provides insurance against the default of a particular credit entity.

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 on or before a specified date.

Put–Call Parity (Arbitrage Relationships for Options)

Put-call parity is a financial principle that defines a relationship between the price of European put and call options with the same strike price and expiration date.

Intrinsic Value, Time Value

Intrinsic value is the difference between the underlying asset’s price and the strike price of an option.

Time value is the additional premium paid for the possibility of price movement.

Moneyness

Moneyness is the term used to describe the position of an underlying asset’s current price relative to the strike price of an option.

Pricing Models

Pricing models are mathematical models used to calculate the theoretical value of financial instruments based on various factors.

Black–Scholes Model

The Black-Scholes model is a mathematical model for pricing European-style options and estimates the variation over time of financial instruments.

Black Model

The Black model (aka the Black-76 model) is a variant of the Black-Scholes option pricing model that applies to futures contracts.

Binomial Options Model

The binomial options model provides a method for the valuation of options using a discrete-time framework, based on iterating over possible price movements.

Implied Binomial Tree

An implied binomial tree is a binomial model that is calibrated to market prices, reflecting the implied volatility of the underlying asset.

Edgeworth Binomial Tree

The Edgeworth binomial tree is an extension of the binomial options pricing model that incorporates skewness and kurtosis into the distribution of asset returns.

Monte Carlo Option Model

The Monte Carlo option model uses simulation to calculate the value of options with multiple sources of uncertainty or with complex features.

Implied Volatility, Volatility Smile

Implied volatility represents the market’s view of the likelihood of changes in a given security’s price.

The volatility smile is a graph of implied volatility plotted against strike prices.

Local Volatility

Local volatility is a model where volatility is a function of both the current asset level and time, used to price derivatives.

Stochastic Volatility

Stochastic volatility models assume that volatility is a random process, characterized by its own specific behavior and properties.

Constant Elasticity of Variance Model

The constant elasticity of variance model is a stochastic volatility model where the volatility of the underlying asset varies with the asset price.

It’s considered an alternative to Black-Scholes.

Heston Model

The Heston model is a stochastic volatility model where the volatility of the asset is itself a random process driven by a second stochastic process (i.e., the premise that volatility is not constant and can change unpredictably over time).

The reasoning behind this is:

• randomness of prices
• volatility clustering
• mean reversion (vol reverts to a long-term average over time)
• correlation between asset returns and volatility
• flexibility in distribution (Heston model can produce distributions in fat tails and skewness)

Stochastic Volatility Jump

Stochastic volatility jump models combine both stochastic volatility and jumps in the return process.

This helps to capture the asset’s price behavior more accurately.

SABR Volatility Model

The SABR volatility model describes a system where both the volatility and the asset price follow stochastic processes and are interdependent.

Markov Switching Multifractal

The Markov switching multifractal is a model for volatility that incorporates multifractal processes into the pricing of financial assets.

The logic behind the Markov switching multifractal model is to account for the observed phenomenon that financial asset volatility exhibits irregular bursts and variations over different time scales, which are captured using multifractal processes governed by Markov chains.

The Greeks

The Greeks are quantitative measures that describe how the price of an option or other derivative is expected to change based on various factors.

Finite Difference Methods for Option Pricing

Finite difference methods for option pricing are numerical methods for solving the differential equations that describe option pricing models.

Vanna–Volga Pricing

Vanna-Volga pricing is an approach used to approximate the price of options, considering the volatility smile and the sensitivities of the option’s delta and vega.

Trinomial Tree

A trinomial tree is a model for the valuation of options, where each node in the tree represents possible paths that the price of the underlying can take.

Implied Trinomial Tree

The implied trinomial tree is a variant of the trinomial tree model for option pricing that is calibrated to match market prices and volatilities.

Garman-Kohlhagen Model

The Garman-Kohlhagen model is an adaptation of the Black-Scholes model that is used to price European options on currencies.

Lattice Model

The lattice model is a technique used in financial valuation to represent the possible paths of the price of an asset or an option over time.

Margrabe’s Formula

Margrabe’s formula is used to value an option to exchange one risky asset for another risky asset.

Carr–Madan Formula

The Carr-Madan formula is used to price European call and put options using Fourier transforms which allow for fast and accurate computation.

Pricing of American Options

Pricing of American options involves determining the value of options that can be exercised at any time before expiration, which is more complex than pricing European options.

We discussed this more here.

Often requires a binomial pricing model or solving for a free boundary problem.

These next six terms cover the pricing of American options.

Barone-Adesi and Whaley

Barone-Adesi and Whaley is a model for pricing American options that uses an approximation technique to simplify the calculation.

Bjerksund and Stensland

Bjerksund and Stensland provide an analytical approximation formula for the price of American options, improving upon earlier models.

Black’s Approximation

Black’s approximation is a method for estimating the value of American options by relating them to their European counterparts.

Least Square Monte Carlo

The Least Square Monte Carlo method is used for pricing American options by simulating a large number of potential future paths for the underlying asset.

Roll-Geske-Whaley

Roll-Geske-Whaley is a method for pricing American call options on stocks that pay known dividends using an analytical approximation.

Longstaff–Schwartz Model

The Longstaff-Schwartz model is a method for valuing American options by using a least squares approach for early exercise decisions.

Optimal Stopping

Optimal stopping in finance refers to the problem of determining the best time to take a particular action in order to maximize an expected reward or minimize a cost.

Interest Rate Derivatives

Interest rate derivatives are financial instruments whose value is derived from the interest rates or interest rate indices.

Black Model for Interest Rate Derivatives

The Black model for interest rate derivatives is used to price options on bond futures and interest rate caps and floors.

Caps and Floors

Caps and floors are types of interest rate derivatives that are used to hedge against rising or falling interest rates, respectively.

Swaptions

Swaptions are options granting the holder the right but not the obligation to enter into an interest rate swap.

Bond Options

Bond options are financial derivatives that give the holder the right to buy or sell a bond at a specified price on or before the option expiration date.

Short-Rate Models

Short-rate models are used to model the future evolution of interest rates by describing the current interest rate or the short rate’s dynamics.

Rendleman–Bartter Model

The Rendleman-Bartter model is a type of short-rate model that assumes that the short rate follows a stochastic process with constant volatility.

Vasicek Model

The Vasicek model is an interest rate model that describes the evolution of interest rates using a mean-reverting stochastic process.

Ho–Lee Model

The Ho-Lee model is a short-rate model that is a single-factor model of the term structure of interest rates, which assumes no arbitrage opportunities.

Hull–White Model

The Hull-White model is an extension of the Vasicek model that allows the short rate’s volatility and mean reversion to be time-dependent.

Cox–Ingersoll–Ross Model

The Cox-Ingersoll-Ross model describes the evolution of interest rates using a mean-reverting square-root process.

Black–Karasinski Model

The Black-Karasinski model is a short-rate model where the short rate follows a lognormal process, ensuring that rates remain positive.

Black–Derman–Toy Model

The Black-Derman-Toy model is a one-factor model of the term structure of interest rates and is used to price bond options.

Kalotay–Williams–Fabozzi Model

The Kalotay-Williams-Fabozzi model is a framework for the analysis and valuation of bonds and interest rate derivatives.

Chen Model

The Chen model is a short-rate model that considers the mean-reverting process and the stochastic volatility of the short rate.

Forward Rate-Based Models

Forward rate-based models use forward rates, which are future rates implied by current term structures, to model and predict future interest rate movements.

LIBOR Market Model (Brace–Gatarek–Musiela Model, BGM)

The LIBOR market model, also known as the BGM model, is used to model the evolution of the forward LIBOR rates and is widely used in the pricing of interest rate derivatives.

LIBOR no longer exists as a financial instrument, but it could be applied to other interbank rates like SOFR.

Heath–Jarrow–Morton Model (HJM)

The Heath-Jarrow-Morton model is a framework for modeling forward interest rates that directly specifies the drift and volatility of the forward rates.