Equity Derivatives Pricing

*** Four Models in Quantitative Finance *** Four models in quantitative finance aren’t just mathematical abstractions—they shape markets, risk strategies, and derivative pricing with precision and elegance. 1. Black-Scholes Model A benchmark in option pricing theory, the Black-Scholes model revolutionized finance by offering a closed-form solution. Key Concepts: • Purpose: To price European-style options without dividends. • Assumptions: Lognormally distributed returns, constant volatility, frictionless markets. Why It Matters • Provides intuitive insights into how time, volatility, and interest rates affect option value. • Basis for volatility surfaces and risk metrics like delta, gamma, and vega. 2. Binomial Tree Model A discrete-time model that builds flexibility into option pricing. Key Concepts: • Structure: Price evolves through “up” and “down” moves in a recombining tree. • Setup Parameters: Time steps, up/down factor, risk-neutral probability. • Pricing Logic: Work backward from terminal payoffs using probabilistic expectations. Advantages: • Flexibility: Works with American options (early exercise). • Intuition: Visual tool to model asset price evolution. • Adaptability: Can incorporate changing volatility or dividends. 3. Monte Carlo Simulation This is a powerful numerical technique for pricing and risk analysis, especially in complex or path-dependent cases. Key Concepts: • Foundation: Simulate thousands of paths for underlying assets using stochastic processes. • Applications: Exotic options, Value-at-Risk (VaR), portfolio stress tests. • Key Elements: Random number generation, payoff averaging, and variance reduction methods. Why It’s Powerful: • Can handle multi-dimensional problems where no analytical solution exists. • Allows incorporation of real-world features, like jumps or stochastic volatility. 4. Finite Difference Method A grid-based numerical technique for solving partial differential equations, like those in the Black-Scholes framework. Key Concepts: • Approach: Replace derivatives with discrete differences (e.g., Δt, ΔS). • Types: • Explicit Method (forward time, centered space) • Implicit Method (backward time, stable for larger steps) • Crank-Nicolson (balanced hybrid of the two) Applications: • Pricing options with barriers, path dependence, or early exercise features. • Handles boundary conditions efficiently. --- B. Noted

Before I went to Grad School, I never heard about a term that is central to the derivatives pricing theory. In the stochastic processes class, our professor explained how you can easily change the probability measure of a random variable using something known as “The Radon-Nikodym Derivative”. A simplified equation that captures the spirit of this change of measure is: z(w) * P (w) = Q(w). If we want to change from the probability measure P to the probability measure Q, we need to reassign probabilities using z, which is the Radon-Nikodym Derivative. Traditionally, when we are working with financial markets applications, P refers to the actuarial or “real world” probability measure, whereas Q refers to the “risk neutral” probability measure. You have to be careful when reaching conclusions on the real world, using probabilities implied from derivatives prices, since you are working under different probability measures. We can use this Radon-Nikodym Derivative, not only on random variables, but also on a whole stochastic process. The Radon-Nikodym Derivative Process is a key element in the Girsanov’s Theorem, which constitutes the basis of the risk neutral pricing of derivative products. There, you transform some random processes into martingales and get simpler expressions to calculate derivative’s prices as conditional expectations. The Radon-Nikodym Derivative concept can not only be applied to these more theoretical procedures, but also to practical matters. Consider for example a range accrual note that pays you a coupon if the spot price is within a range, let’s say USD 150-175, at some fixing dates. Since the range is far away from the current spot price, you will have problems running a traditional Montecarlo simulation to price the product. The reason is that you will have very few paths on that region, hence your price will have a very high variance. If you shift the stochastic process, using a Radon-Nikodym Derivative, you can get more samples in your desire zone. This procedure is called “Importance Sampling” and can be observed in the chart below. Those of you who’d like to get a python code to perform this type of calculations can direct message me or leave your email in the comments and I will send it to you. Sometimes there are concepts that we weren’t aware of but that are central to our job’s daily activities. Have you experienced something similar in the fields of finance, economics or math? #derivatives #montecarlo #markets

Understanding the SABR Model in #Quant #Finance The SABR (Stochastic Alpha Beta Rho) Model is a popular stochastic volatility model used in financial markets to capture the dynamics of implied volatility surfaces, especially for pricing options. Developed by Hagan et al., it’s primarily used for modeling European options and has gained wide adoption due to its ability to fit smile-shaped volatility structures effectively. Key Components of the SABR Model:📚📚📚 1. Stochastic Volatility: Unlike Black-Scholes, the volatility is no longer constant but follows its own stochastic process. 2. Alpha (α): Represents the volatility of the forward price. 3. Beta (β): Controls the elasticity of the model, determining how the volatility reacts to changes in the underlying asset price. Typically, values range between 0 (lognormal) and 1 (normal). 4. Rho (ρ): Correlation between the underlying asset price and its volatility. 5. Nu (ν): The volatility of volatility, which captures how volatile the volatility itself is. Why Use the SABR Model? 📚📚📚 The SABR model is highly flexible, making it useful for a wide range of asset classes, including interest rates, commodities, and FX markets. It's particularly effective at addressing market realities like volatility smiles and skews, which traditional models struggle to capture. Applications: 📚📚📚 Option Pricing: The SABR model helps price European options more accurately by fitting market data more effectively, especially for products where implied volatility exhibits a skew or smile. Interest Rate Derivatives: Widely used in pricing caps, floors, and swaptions due to its ability to reflect the market's volatility dynamics. Limitations: 📚📚 Computational Intensity: While more accurate, SABR is computationally heavier than simpler models like Black-Scholes. Calibration Challenges: Fitting the model to market data can be tricky, especially finding the optimal parameters for alpha, beta, rho, and nu. The SABR model remains a go-to for quant professionals looking to model stochastic volatility and deal with real-world market conditions. As markets evolve, so does the need for more robust models, and SABR remains a cornerstone for modern quant finance solutions. #Quant #Finance #QuantResearch #Portfolio #Derivatives #RiskManagement