11.1 Pricing financial derivatives

Pricing financial derivatives is a complex yet crucial aspect of quantitative finance. It involves valuing instruments whose worth stems from underlying assets like stocks or commodities. Understanding key concepts like arbitrage, time value of money, and risk-neutral valuation forms the foundation.

Advanced models go beyond basic assumptions to capture real-world market behavior. Techniques like stochastic calculus, numerical methods, and sophisticated pricing models enable more accurate valuation of complex derivatives. Proper calibration, validation, and risk management are essential for applying these tools effectively in practice.

Foundations of derivative pricing

• Derivative pricing is a critical aspect of financial engineering and quantitative finance, providing a framework for valuing and managing financial instruments whose value is derived from underlying assets
• Understanding the foundational concepts and theories behind derivative pricing is essential for effectively applying these techniques in real-world scenarios and developing robust pricing models

Key financial concepts

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• Derivatives are financial instruments whose value is derived from the performance of underlying assets (stocks, bonds, commodities, currencies, or interest rates)
• Arbitrage is the practice of taking advantage of price discrepancies in different markets to generate risk-free profits
  • Absence of arbitrage is a fundamental assumption in many pricing models
• Market efficiency suggests that asset prices reflect all available information, making it difficult to consistently outperform the market

Time value of money

• The time value of money is the concept that money available now is worth more than an identical sum in the future due to its potential earning capacity
• Present value (PV) is the current worth of a future sum of money or stream of cash flows given a specified rate of return
  • $PV = \frac{FV}{(1+r)^n}$, where $FV$ is the future value, $r$ is the discount rate, and $n$ is the number of periods
• Future value (FV) is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today
  • $FV = PV(1+r)^n$

Risk-neutral valuation

• Risk-neutral valuation is a pricing approach that assumes investors are indifferent to risk and require no additional return for bearing risk
• Under risk-neutral valuation, the expected return of all securities is equal to the risk-free interest rate
• This approach allows for the valuation of derivatives by discounting the expected payoff at the risk-free rate, simplifying the pricing process
• The risk-neutral probability measure adjusts the actual probabilities of future outcomes to account for the risk premium, enabling risk-neutral valuation

Stochastic calculus for derivatives

• Stochastic calculus is a branch of mathematics that deals with the study of stochastic processes, which are processes that evolve over time with random outcomes
• It provides a framework for modeling and analyzing the behavior of financial assets and derivatives, which often exhibit random fluctuations

Brownian motion

• Brownian motion, also known as a Wiener process, is a continuous-time stochastic process that models random movements and is a key building block for many financial models
• It is characterized by the following properties:
  • Continuous paths
  • Independent increments
  • Normally distributed increments with mean 0 and variance proportional to the time interval
• The standard Brownian motion $W_t$ has a mean of 0 and a variance of $t$ at time $t$

Ito's lemma

• Ito's lemma is a crucial tool in stochastic calculus that allows for the computation of the differential of a function of a stochastic process
• It is used to derive the dynamics of a process that is a function of another stochastic process, such as the price of a derivative that depends on the underlying asset price
• For a function $f(t, x)$ of a stochastic process $X_t$ that follows an Ito process $dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$, Ito's lemma states that:
  • $df(t, X_t) = \left(\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma\frac{\partial f}{\partial x}dW_t$

Stochastic differential equations

• Stochastic differential equations (SDEs) are differential equations that incorporate random processes, such as Brownian motion, to model the dynamics of a system with uncertainties
• In finance, SDEs are used to model the price dynamics of assets and derivatives, capturing the randomness and volatility of the markets
• The general form of an SDE is:
  • $dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$
  • where $\mu(t, X_t)$ is the drift term, representing the deterministic part of the process, and $\sigma(t, X_t)$ is the diffusion term, representing the random part of the process
• SDEs are often solved using numerical methods, such as the Euler-Maruyama scheme or more advanced techniques like the Milstein method

Black-Scholes model

• The Black-Scholes model is a widely used mathematical model for pricing European-style options, developed by Fischer Black, Myron Scholes, and Robert Merton
• It provides a closed-form solution for the price of European call and put options based on the underlying asset price, time to expiration, risk-free interest rate, and volatility

Assumptions and limitations

• The Black-Scholes model relies on several assumptions, including:
  • The underlying asset price follows a geometric Brownian motion with constant drift and volatility
  • No dividends are paid during the option's lifetime
  • There are no transaction costs or taxes
  • The risk-free interest rate is constant and known
  • The option can only be exercised at expiration (European-style)
• These assumptions limit the model's applicability in real-world scenarios, where factors like dividends, transaction costs, and early exercise (for American-style options) may be present

Pricing European options

• The Black-Scholes formula for the price of a European call option is:
  • $C(S, t) = SN(d_1) - Ke^{-r(T-t)}N(d_2)$
  • where $d_1 = \frac{\ln(\frac{S}{K}) + (r + \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$ and $d_2 = d_1 - \sigma\sqrt{T-t}$
• The price of a European put option can be obtained using put-call parity:
  • $P(S, t) = Ke^{-r(T-t)}N(-d_2) - SN(-d_1)$
• In these formulas, $S$ is the current underlying asset price, $K$ is the option strike price, $r$ is the risk-free interest rate, $T-t$ is the time to expiration, $\sigma$ is the volatility of the underlying asset, and $N(\cdot)$ is the standard normal cumulative distribution function

Greeks and risk management

• Greeks are sensitivity measures that quantify how option prices change with respect to various parameters, such as the underlying asset price, volatility, and time to expiration
• The main Greeks include:
  • Delta ($\Delta$): measures the sensitivity of the option price to changes in the underlying asset price
  • Gamma ($\Gamma$): measures the sensitivity of Delta to changes in the underlying asset price
  • Vega ($\mathcal{V}$): measures the sensitivity of the option price to changes in the underlying asset's volatility
  • Theta ($\Theta$): measures the sensitivity of the option price to changes in the time to expiration
  • Rho ($\rho$): measures the sensitivity of the option price to changes in the risk-free interest rate
• Greeks are essential tools for risk management, as they help option traders and risk managers understand and hedge the risks associated with their option positions

Numerical methods for derivatives

• While closed-form solutions like the Black-Scholes formula exist for some simple cases, many derivative pricing problems require numerical methods to obtain approximate solutions
• Numerical methods are particularly useful when dealing with complex payoffs, path-dependent options, or when the assumptions of analytical models are violated

Binomial option pricing

• The binomial option pricing model is a discrete-time model that approximates the continuous-time price dynamics of the underlying asset using a binomial tree
• At each time step, the asset price can move up or down by a certain factor, with associated probabilities
• The option price is then computed by working backwards through the tree, discounting the expected payoff at the risk-free rate
• As the number of time steps increases, the binomial model converges to the continuous-time Black-Scholes model

Monte Carlo simulations

• Monte Carlo simulations are a flexible and powerful numerical method for pricing derivatives, particularly those with complex payoffs or path-dependent features
• The method involves simulating a large number of possible price paths for the underlying asset using random sampling and then averaging the discounted payoffs to obtain the option price
• Monte Carlo simulations can easily incorporate various stochastic processes, such as jump-diffusion or stochastic volatility models
• Variance reduction techniques, such as antithetic variates or control variates, can be used to improve the efficiency and accuracy of the simulations

Finite difference methods

• Finite difference methods are used to solve the partial differential equations (PDEs) that arise in derivative pricing, such as the Black-Scholes PDE
• These methods discretize the PDE on a grid in time and space, approximating the derivatives using finite differences
• Common finite difference schemes include the explicit, implicit, and Crank-Nicolson methods, each with different stability and accuracy properties
• Finite difference methods are particularly useful for pricing American-style options, which involve free boundary problems due to the early exercise feature

Advanced pricing models

• While the Black-Scholes model provides a foundation for derivative pricing, its assumptions may not always hold in practice, leading to the development of more advanced pricing models that capture additional features of the market

Jump-diffusion models

• Jump-diffusion models extend the Black-Scholes model by incorporating sudden jumps in the asset price, which can represent events like earnings surprises or macroeconomic shocks
• The price dynamics are modeled as a combination of a continuous diffusion process (Brownian motion) and a jump process (Poisson process)
• The Merton jump-diffusion model is a well-known example, where the jump sizes are assumed to follow a normal distribution
• Other jump-diffusion models, such as the Kou model or the double-exponential jump-diffusion model, consider different jump size distributions to better fit market data

Stochastic volatility models

• Stochastic volatility models relax the assumption of constant volatility in the Black-Scholes model, allowing the volatility of the underlying asset to vary over time according to a random process
• These models can capture the volatility smile and skew observed in option markets, where implied volatilities vary with strike price and time to expiration
• Popular stochastic volatility models include:
  • Heston model: assumes the volatility follows a Cox-Ingersoll-Ross (CIR) process
  • SABR model: assumes a stochastic volatility process correlated with the asset price process
  • GARCH models: model the volatility as an autoregressive process based on past volatility and returns
• Stochastic volatility models often require numerical methods, such as Monte Carlo simulations or finite difference methods, for pricing and calibration

Interest rate models

• Interest rate models are used to price derivatives whose payoffs depend on the evolution of interest rates, such as bonds, swaps, and interest rate options
• These models describe the dynamics of the term structure of interest rates, which represents the relationship between interest rates and their maturities
• Short-rate models, such as the Vasicek model or the Cox-Ingersoll-Ross (CIR) model, specify the dynamics of the instantaneous short-term interest rate
• Heath-Jarrow-Morton (HJM) framework models the evolution of the entire forward rate curve, providing a more flexible and consistent approach to interest rate modeling
• LIBOR market models (LMMs) are used to price interest rate derivatives based on the dynamics of forward LIBOR rates, which are the benchmark rates used in many financial contracts

Calibration and model validation

• Calibration is the process of estimating the parameters of a pricing model to match the observed market prices of derivatives
• Model validation involves testing the performance and accuracy of a pricing model using historical data and out-of-sample testing

Parameter estimation techniques

• Maximum likelihood estimation (MLE) is a common approach for calibrating models, where the parameters are chosen to maximize the likelihood of observing the market data given the model
• Least-squares optimization minimizes the sum of squared differences between the model prices and the market prices, finding the parameters that provide the best fit
• Bayesian methods, such as Markov Chain Monte Carlo (MCMC), can be used to estimate parameters by combining prior beliefs about the parameters with the observed market data to obtain a posterior distribution
• Gradient-based optimization techniques, such as the Levenberg-Marquardt algorithm, are often employed to efficiently search for the optimal parameters

Model testing and validation

• In-sample testing evaluates the model's performance on the data used for calibration, providing a measure of how well the model fits the observed prices
• Out-of-sample testing assesses the model's ability to predict prices on data not used in the calibration process, giving an indication of the model's generalization performance
• Backtesting involves applying the model to historical data and comparing the model's predictions with the actual realized prices to gauge the model's accuracy and robustness
• Cross-validation techniques, such as k-fold cross-validation, can be used to assess the model's performance by repeatedly splitting the data into training and testing sets

Handling market data

• Market data, such as option prices, implied volatilities, and interest rates, are essential inputs for calibrating and validating pricing models
• Data preprocessing steps, such as filtering outliers, interpolating missing values, and smoothing noisy data, may be necessary to ensure the quality and consistency of the input data
• Implied volatility surfaces, which represent the implied volatilities of options across different strike prices and maturities, are often used as calibration targets for stochastic volatility models
• Handling market data requires careful consideration of data sources, data synchronization, and the treatment of weekends and holidays to ensure accurate and reliable results

Hedging and risk management

• Hedging is the practice of taking offsetting positions to reduce the risk exposure of a portfolio to market movements
• Risk management involves identifying, measuring, and controlling the various risks associated with derivative positions, such as market risk, counterparty risk, and operational risk

Delta hedging

• Delta hedging is a strategy that aims to neutralize the sensitivity of an option position to small changes in the underlying asset price
• It involves continuously adjusting the position in the underlying asset to offset the delta of the option position
• The delta of an option represents the rate of change of the option price with respect to the underlying asset price
• In practice, delta hedging is often implemented using a combination of the underlying asset and risk-free bonds, with the proportions determined by the option's delta and the desired level of risk reduction

Gamma and vega hedging

• Gamma hedging targets the second-order sensitivity of an option position to changes in the underlying asset price, as measured by the option's gamma
• It involves adjusting the position in the underlying asset to offset the non-linear exposure to price movements, which becomes more significant for larger price changes
• Vega hedging focuses on managing the exposure to changes in the underlying asset's volatility, as measured by the option's vega
• Vega hedging can be achieved by taking positions in other options with offsetting vega exposures or by using volatility derivatives, such as variance swaps or volatility index futures

Value-at-Risk (VaR)

• Value-at-Risk (VaR) is a widely used risk measure that quantifies the potential loss of a portfolio over a given time horizon at a specified confidence level
• For example, a 99% 10-day VaR of $1 million means that there is a 99$1 million over the next 10 days
• VaR can be estimated using various methods, such as historical simulation, Monte Carlo simulation, or parametric approaches (e.g., variance-covariance method)
• Expected Shortfall (ES) or Conditional Value-at-Risk (CVaR) is an alternative risk measure that quantifies the average loss beyond the VaR threshold, providing a more comprehensive view of the tail risk
• VaR and ES are important tools for setting risk limits, allocating capital, and communicating risk to stakeholders, but they have limitations, such as the assumption of normal market conditions and the potential for underestimating extreme events

Exotic and complex derivatives

• Exotic derivatives are non-standard financial instruments that have more complex payoff structures or features compared to plain vanilla options
• They are often tailored to meet specific investor needs or to take advantage of particular market conditions

Path-dependent options

• Path-dependent options are exotic options whose payoffs depend not only on the final price of the underlying asset but also on the path followed by the asset price over the life of the option
• Examples of path-dependent options include:
  • Asian options: the payoff is based on the average price of the underlying asset over a specified period
  • Barrier options: the option's existence or payoff depends on whether the underlying asset price reaches or breaches a predetermined level (barrier) during the option's lifetime
  • Lookback options: the payoff is based on the maximum or minimum price achieved by the underlying asset over the option's lifetime
• Pricing and hedging path-dependent options often require numerical methods, such as Monte Carlo simulations or finite difference methods, to account for the continuous monitoring of the asset price path

Multi-asset options

• Multi-asset options are derivatives whose payoffs depend on