Derivative Valuation Models

Junior - Mehul, what are some extensions of Black Scholes Model that are used by the Industry? 🧐🧐🧐 The Black-Scholes model is a foundational model for option pricing, but several extensions and modifications have been developed and are used in the industry to address its limitations and adapt to more complex financial markets. 😄😄😄 Here are some key extensions: 📚📚📚 1. Stochastic Volatility Models: 🎯🎯 - Heston Model: This model assumes that volatility is stochastic, meaning it varies over time according to its own stochastic process, often modeled as a mean-reverting square-root process. - Hull-White Model: Extends the Black-Scholes model by allowing the volatility to be stochastic and following a lognormal distribution. 2. Local Volatility Models:🎯🎯 - Dupire Model: Assumes that volatility is a deterministic function of both the current asset price and time, allowing for a more accurate fit to the implied volatility surface observed in the market. 3. Jump-Diffusion Models:🎯🎯 - Merton’s Jump-Diffusion Model: Incorporates sudden jumps in asset prices in addition to the continuous paths assumed by the Black-Scholes model. It adds a Poisson jump process to the standard geometric Brownian motion. - Kou Model: Similar to Merton’s model but assumes double exponential jumps, providing a better fit for empirical data on asset returns. 4. Stochastic Interest Rate Models:🎯🎯 - Black-Scholes with Stochastic Interest Rates: Models the interest rate as a stochastic process, which is particularly useful for pricing interest rate derivatives. - Hull-White Interest Rate Model: A specific type of stochastic interest rate model often used in conjunction with the Black-Scholes framework. 5. Variance Gamma Model🎯🎯 - Extends the Black-Scholes model by allowing for changes in the variance of asset returns over time, accounting for observed skewness and kurtosis in return distributions. 6. SABR Model (Stochastic Alpha, Beta, Rho):🎯🎯 - A stochastic volatility model widely used to capture the dynamics of the volatility smile and term structure in options markets. It extends the Black-Scholes model by incorporating stochastic volatility and correlations between the asset price and its volatility. 7. SVI (Stochastic Volatility Inspired) Model:🎯🎯 - Designed to fit the implied volatility surface using a parametrized functional form, allowing for a better capture of market data characteristics. 8. Local-Stochastic Volatility Models:🎯🎯 - LSV Model: Combines the local volatility and stochastic volatility frameworks to provide a more comprehensive model that captures the nuances of market-implied volatilities. These extensions address various limitations of the Black-Scholes model, such as the assumption of constant volatility and the exclusion of jumps or stochastic interest rates, making them more suitable for the complex dynamics observed in real financial markets. #quantitativefinance #financialengineering #quantmodeling

SABR: A Deep-Dive into Stochastic Volatility in Interest Rate Derivatives Volatility is not constant. It smiles, skews, and shifts with market conditions. The SABR model (Stochastic Alpha Beta Rho) was developed to capture this behavior in a tractable, fast-calibrating framework—especially for swaptions and other interest rate derivatives. Here’s a deep look at why SABR is the benchmark in surface modeling and how it works in practice. 1 → Why SABR? Traditional models like Black assume constant volatility and fail to capture observed smiles. Traders needed a model that: → Fits volatility smiles and skews → Calibrates quickly to market data → Extrapolates beyond available strikes → Maintains no-arbitrage structure SABR introduces stochastic volatility over a forward rate, preserving analytical simplicity while increasing realism. 2 → Parameter Structure SABR evolves two correlated processes: one for the forward, one for its volatility. Its parameters shape the surface: → Alpha (α): Sets overall volatility level. → Beta (β): Controls how volatility reacts to the forward rate. • β = 0 → normal; β = 1 → log-normal. → Rho (ρ): Correlation between rate and volatility. • ρ < 0 skews left (as in swaptions); ρ > 0 skews right. → Nu (ν): Volatility of volatility. Adds smile curvature. These four parameters allow SABR to match diverse market smiles across strikes and maturities. 3 → Volatility Surface Dynamics The attached surface shows implied vol vs moneyness and maturity. → Short-tenor smiles are steep, flattening over time. → The skew is driven by ρ — notice asymmetry across strikes. → ν controls curvature — higher ν deepens the smile. This flexibility makes SABR ideal for pricing long-dated or out-of-the-money options. 4 → Calibration in Production SABR is used daily in front-office pricing systems and risk engines. Calibration typically involves: → Input: Swaption vol matrix across expiries and strikes → Goal: Minimize vol or price error → Methods: • Local (slice-by-slice): fast, stable • Global (entire surface): smoother, harder to optimize → Challenges: • Arbitrage-free interpolation • Handling sparse or noisy data • Ensuring smoothness in tenor and strike Because of its closed-form volatility approximation, SABR can be recalibrated throughout the day with minimal computational cost. 5 → Why It’s Still the Standard SABR balances accuracy, speed, and interpretability. → Tractable for calibration → Interpretable for traders and quants → Compatible with risk and pricing systems → Robust in negative-rate regimes (adjusted β) It’s not just elegant—it’s operationally indispensable. #quantitativefinance #SABR #volatilitymodeling #swaptions #ratesderivatives #stochasticvolatility #financialengineering #modelcalibration #volsurfaces #riskanalytics #derivativespricing

Most people look at the Black-Scholes equation and see a way to find the "fair value" of an option. But when you strip away the stochastic calculus and look at the mechanics, you realize it’s actually a P&L decomposition. It doesn't tell you what the option should be worth; it tells you how to manufacture that value dynamically. I drew this sketch to visualize what is actually happening under the hood of the PDE. 1. The Engine (Taylor Expansion): The top section shows the reality of risk. Your P&L is driven by Time (Theta), Direction (Delta), and Convexity (Gamma). 2. The Cost of Business (The PDE): The equation everyone memorizes is really just a "No Free Lunch" constraint. It simplifies to: Theta + Gamma + Interest = 0 In plain English: The money you lose every day by holding the option (Time Decay) must be exactly offset by the money you make trading the volatility (Gamma), minus your financing costs. The Insight: If you are a market maker, you aren't betting on the price. You are managing a relationship between Time and Movement. 🔹 If the market doesn't move, Theta eats you alive. 🔹 If the market moves more than implied, Gamma pays the bills. The model isn't predicting the future. It's quantifying the "break-even" volatility you need to survive the time decay. When you look at a model, do you see a "Crystal Ball" (prediction) or a "Thermometer" (measurement)? #QuantitativeFinance #BlackScholes #Derivatives #RiskManagement #Mathematics #CapitalMarkets #OptionsTrading #FinancialEngineering

Term Structure Models The first QF models were developed for equity prices. GBM & Black-Scholes modelled equities & their options. A natural extension of these models would have been models for bond prices. But this is not what happened. Bonds are issued with many maturities with varying liquidity making them difficult to build models for. But bond prices can be calculated if you know the IRs applicable for its CFs. So instead of building models for bond prices, researchers focused on models for IRs at different maturities. The models became known as term structure models. The first TSMs were the short rate models (SRMs). Vasicek, CIR and HW are examples. SRMs use parameters such as drift, volatility and mean reversion to model a short rate, equivalent to the red overnight (1D) rate in the TS diagram below, to project an IR curve that is based on assumptions about IRs in the market generally. The objective with all curve building is to closely match observable market rates. The SRMs though rarely did this as the SRM curve-building process did not take observable data as inputs. Instead, they tried to match the observable TS after the curve was built. Once SRM-curves were built, they were used for two purposes: 1) To price illiquid bonds: The fitted, continuous curves allowed rates at any maturity to be obtained and fed into a bond pricing routine. The top RH example below shows red rates r1 to r5 being obtained from the fitted SRM curve and passed into the bond pricing eqn to generate the price for the illiquid bond. 2) To price derivatives: The bottom RH diagram shows SRM-derived rates being fed into a MC simulation for a Bermudan swaption. Thousands of rates are simulated. They determine both the early-exercise and maturity date CFs of the Bermudan. The valuation of the Bermudan is the avg of the discounted CFs. Money markets and derivatives trading in the 1980s and 90s led to methodology changes in curve-building. The first was that instruments such as deposits, futures, FRAs and swaps replaced bond yields as the calibrating instruments for TSMs. The second was that SRMs were replaced by a new family of TSMs called fwd rate models (FRMs). While SRMs were easy to work with mathematically, the resulting curves did not fit the observable rates very well. And the fact that they did not take the observable curve as a model input also meant that they did not take advantage of the information content of the curve. A curve has embedded fwd rates which have their own vols. The relationships btw fwd rates are captured in correlations. The move to building FRMs whose calibration routines were able to reproduce both observable fwd rates and the vols of those fwd rates happened gradually. The HJM framework was able to reproduce the initial fwd rates but not their vols (as implied from traded caplets & swaptions). The LMM model that succeeded it was able to reproduce both the fwd rates and their vols.

As part of exploring financial engineering and risk management, along with fixed income markets, I went through the original article that introduced the Black-Derman-Toy(BDT) short-rate model, and it was a useful reminder of how much modern fixed income modeling still rests on the first principles. The BDT framework remains one of the most elegant bridges between observable yield curves and arbitrage free pricing of interest rate derivatives. The paper develops a one-factor, recombining binomial lattice for the short rate is explicitly calibrated to the initial term structure and an exogeneously specified volatility surface. At its core, the model assumes that the log of the short rate follows a binomial process, allowing interest rates to remain strictly positive while preserving analytical tractability. Calibration proceeds sequentially at each time step, node-specific short rates are chosen so that the model reproduces both the observed zero-coupon bond prices and the term structure of yield volatilities. This structure makes the BDT model particularly well suited for pricing treasury bond options, caps, floors and callable fixed income instruments, where consistency with the current yield curve is non negotiable. From a RISK perspective, the BDT model highlights a subtle but critical point that model risk dominates parameter risk in one-factor short rate frameworks. By construction, all yield curve movements are driven by a single source of uncertainity, implicitly assuming perfect correlation across maturities. This can materially understate exposure to curve twists and butterfly shifts in stress scenarios. Moreover, volatility is an input rather than an output, making valuations highly sensitive to how the volatility term structure is specified. In modern times, BDT is best viewed as the baseline valution model which is useful in intuition, benchmarking and clean arbitrage free pricing, but that one should be complemented with multi-factor models and robust stress testing when managing convexity, optionality and tail risk in fixed income portfolios. #FixedIncome #InterestRateModels #FinancialEngineering #RiskManagement #QuantFinance #Derivatives #YieldCurve #ModelRisk #BDT #TreasuryMarkets

*** 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

⚙️ Finite Differences for Option Pricing 📈 Differential equations show up everywhere in quant finance — and they’re nothing to fear. In this video, I break down how we use finite differences to solve pricing PDEs when no closed-form solution exists. Here’s what I cover 👇 🎯 Why PDEs Matter in Quant Finance Every derivatives model — Black-Scholes, Heston, local vol, stochastic vol — leads to a pricing PDE. - If the PDE has a closed form (like Black-Scholes), great - But when it doesn’t — and more realistic models don’t — we need numerical methods to recover the price 🧠 The Core Idea: Approximating Derivatives Finite differences start with the definition of a derivative: take the limit as h → 0. But on a computer, h doesn’t need to be infinitesimal — just small enough. This simple approximation lets us turn any derivative into algebra we can compute directly. From this, we can iteratively “paint” the pricing function point by point. 📈 From ODEs to PDEs Once you understand the idea on an ordinary differential equation (ODE), the extension to PDEs is almost identical: - Discretize the inputs (e.g., stock price & time) - Use finite difference formulas to approximate all partial derivatives - Plug those approximations into the pricing PDE Iterate across the grid to recover the full pricing surface. This works for the heat equation, Black-Scholes, Heston — anything. ⚠️ The One Thing That Matters: Stability Your grid must satisfy stability conditions, or errors explode as you iterate. When the grid is good, the numerical solution converges toward the true one; when it’s bad, it diverges. The video walks through both cases visually. 🧩 The Quant Takeaway Finite differences are simple, powerful, and essential. Once you understand: - derivative approximations - grid discretization - stability - iterative updates you can build your own solvers for Black-Scholes, Heston, or any pricing PDE. If you’re serious about mastering the math, probability, and modeling behind real decision-making — check out 👉 https://quantguild.com 🚀 Inside Quant Guild 📚 90+ lessons across math, finance & probability ⚙️ Adaptive practice engine + gamified learning 💼 Interview questions with full solutions 🎮 Trading simulations and quant games 🎓 Built by quants, for aspiring and professional quants ⭐ Start learning — for free 🎥 Watch the full video below 👇 https://lnkd.in/emYfXj2V

The Smile and Vol Curve Differences: Heston vs. Local Vol for Cliquet Pricing Local Volatility (LV) and the Heston model offer improvements over Black-Scholes for pricing path-dependent options like cliquets. However, they capture the volatility smile and volatility curve differently, resulting in potentially significant price discrepancies. The Volatility Smile: Key Differences: Smile Consistency: LV: Smile is most accurate for calibrated strike prices, potentially deviating for others. Heston: Smile is more dynamic and consistent across strikes due to stochastic volatility. Smile Shape: LV: Limited control over smile shape beyond the calibration data. Heston: Model parameters offer greater control over smile shape and skewness. The Volatility Curve: Key Differences: Curve Explicitness: LV: No explicit volatility curve, only localized implied volatilities. Heston: Generates a full volatility curve based on model parameters. Volatility Dynamics: LV: Volatility remains static after calibration. Heston: Volatility fluctuates randomly over time. Heston Model Advantages for Cliquet Pricing: Path Dependence: Cliquets rely on the underlying asset price following a specific path. The Heston model, with its stochastic volatility, can better reflect potential volatility changes along that path, leading to more accurate pricing for path-dependent cliquets. Volatility Jumps: The Heston model allows for volatility jumps, which can significantly impact cliquet payoffs. If the market anticipates potential volatility spikes, the Heston model can capture this risk and price the cliquet accordingly. Smile Consistency: As discussed earlier, the Heston model generates a more consistent smile across different strike prices. This can be crucial for cliquets with strike prices outside the range used for LV calibration. The Trade-Off: The decision between LV and Heston boils down to a trade-off between: Accuracy: Heston offers potentially more accurate pricing for complex cliquets. Efficiency: LV is computationally faster and easier to calibrate. Check out our Master Class on "Derivative Pricing with Stochastic Volatility Model" which was taken by Somdip Datta (ex VP Quant, Goldman Sachs NY and PhD Princeton, IIT Kgp Alum). Masterclass Link - https://lnkd.in/gxagsZhX