Option Pricing 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

Adding this book to my library - Just completed Options, Futures, and Other Derivatives (Tenth Edition) by John C. Hull and Sankarshan Basu — a cornerstone text for anyone serious about derivatives, risk management, and quantitative finance. - As someone who has cleared FRM Level 1 and is building advanced risk models (including Monte Carlo-based VaR, CVaR, and stress testing for fixed income portfolios), this book offered both breadth and depth in topics directly relevant to my work. - Some of the technical areas I found especially valuable include: — Stochastic calculus foundations – Ito’s Lemma, Wiener processes, and how they feed into derivative pricing. — Partial Differential Equations (PDEs) – the derivation and application of the Black-Scholes-Merton PDE and its extensions for options pricing. — Numerical methods – finite difference techniques, binomial and trinomial trees for pricing complex derivatives. — Volatility modelling – implied volatility surfaces, volatility smiles/skews, and stochastic volatility models. — Interest rate models – Vasicek, CIR, and HJM frameworks, with applications in bond and swap pricing. — Credit derivatives – valuation of CDS, counterparty credit risk, and credit valuation adjustments (CVA). — Risk metrics – integrating derivatives into portfolio VaR, Expected Shortfall, and stress scenarios. — Regulatory context – Basel III/IV implications for derivatives exposure and clearing. - The clarity with which Hull links mathematical models to market practice makes this not just an academic read but a practical reference I will continue to use in risk modeling, portfolio hedging, and derivatives analytics. - This completes another step in strengthening my toolkit for analyzing and managing market risk — and reinforces the importance of continuous learning in finance. What’s the one advanced finance or quantitative risk management book you would recommend I read next? #FRM #RiskManagement #Derivatives #QuantFinance #Options #Futures #Swaps #VaR #FinancialMarkets #ContinuousLearning John Hull

Pricing American Options with Finite Difference Methods The BSM equation describes how an option value, V, changes over time (dV/dt). It is a PDE that describes the inter-relationships of multiple variables. The most important relationship is the one btw the underlying price (say a stock price, S) and the value V of the option written on it. The green line in diagram A illustrates the relationship for a call option. As S rises so does V. It is a smooth, but non-linear relationship. At the strike price K, V begins to rise at a faster rate, rising to the point that it becomes a 45-degree angle on the S-V axis. At this point S is deep in-the-money. The green line has a slope of 1 and V and S are now behaving like they belong to the same traded instrument. The red line is a tangent to the green line. It can be used to approximate how much V changes for a small change in S, dS. The red tangent line is touching the green curve at the strike price K implying the stock price S in the example is at-the-money. This slope of the red line is the delta of the option. The difference btw the green and red line is the curvature of the option. Curvature determines a) how quickly delta changes and b) the size of the curvature error when delta is used to approximate changes in the option value for a given change in S. The curvature of an option is also known as its gamma. American options can be exercised on any date btw now and their maturity. A call option holder will exercise if they think that the early exercise value, Max(S-K, 0), is greater than the BSM-calculated value. When valuing an American option, a method is therefore needed which compares the EEV to the BSM value on each day prior to the maturity date. An FDM grid achieves this. It maps the multi-variate relationships in the BSM continuous world to an equivalent set of relationships in the BSM discrete world. In this discrete world a plausible range for S is divided into fixed price intervals, ΔS. The time-to-maturity T is divided into daily time intervals, ΔT. The nodes of the FDM grid (C below) contain the BSM option prices for each combination of ΔS and ΔT. FDMs take advantage of the fact that the terminal value V for the call option is known given the forward price of S. That is, V = max (S-K, 0). The FDM then works backwards from the terminal date to determine the BSM option values V(ΔS, ΔT) at each node on the grid. It does this using the discrete world version of the BSM PDE. In this world of time-steps and fixed stock price increments, delta and gamma are calculated using BSM difference quotients (see B) and plugged into the discrete BSM PDE. As the discrete BSM PDE works backwards from the terminal value, the option value at each node is calculated as the max of the EEV & the BSM value. The difference eqn keeps working backwards an upwards until the node at the top LHS of the grid is reached. The value of this node is the price of the American call option.

The stochastic volatility, stochastic interest rates and random jumps model: Bakshi, Cao and Chen (1997) set out to empirically test various options pricing models that existed at that time. To do so, they developed a new, very general model, with a semi-closed solution in the style of Heston (1993), in which volatility, interest rates and jumps were random. This model allows the simultaneous testing of the performance of the different models it contains: Black-Scholes (BS), stochastic interest rates (SI), stochastic volatility (SV), stochastic volatility plus stochastic interest rates (SVSI) and stochastic volatility plus jumps (SVJ). Among their results we can highlight: - Judged on internal parameter consistency, all models are misspecified, with the SVJ the least and the BS the most. - Out-of-sample pricing errors are the highest for the BS, the second highest for the SV and the lowest for the SVJ. In Bueno-Guerrero and Clark (2024), Steven Clark and I added Lévy jumps to the generalized Black-Scholes (GBS) model of Bueno-Guerrero (2019) and compare the resulting model with the Merton (1976) jump-diffusion model and with the general model of Bakshi, Cao and Chen (1997) (which we call in the paper the Heston model with jumps). The result of the comparison is that, on an equal parameter count basis, our GBS model with CGMY jumps provides the best in-sample fits, across three different volatility environments, compared to any of the alternative models. The formulas below are from the general model of Bakshi, Cao and Chen (1997). In them: - v_t is the instantaneous variance of returns (conditional of no jumps occurring) - W_1t and W_2t are two standard and correlated Q-Brownian motions - J_t is the percentage jump size (conditional on jump occurring) - q_t is a Poisson jump counter with intensity λ - κ_v, η_v and σ_v are, respectively, the mean-reversion speed, the long-run mean, and the volatility coefficient of the variance process - κ_r, η_r and σ_r are, respectively, the mean-reversion speed, the long-run mean, and the volatility coefficient of the risk-free interest rate process - W_rt is a Q-Brownian motion uncorrelated with any other process. References: - Bakshi, Cao and Chen (1997): "Empirical Performance of Alternative Option Pricing Models", Journal of Finance, 52, 2003–2049. - Bueno-Guerrero (2019): "Black–Scholes and Heston Models with Stochastic Interest Rates and Term Structure of Volatilities", Journal of Derivatives, 27, 32–48. - Bueno-Guerrero and Clark (2024): "Option Pricing under a Generalized Black–Scholes Model with Stochastic Interest Rates, Stochastic Strings, and Lévy Jumps", Mathematics, 12, 82. - Heston (1993): "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options", Review of Financial Studies, 6, 327–343. - Merton (1976): "Option Pricing when Underlying Stock Returns are Discontinuous", Journal of Financial Economics, 3, 125–144. #finance #mathematics #markets

The "High Growth" Trap 📉 If I offered you two call options on two different stocks, both trading at $100 today with identical volatility... Stock A: Expected to grow 20% this year. Stock B: Expected to grow 2% this year. Would you pay a higher premium for the option on Stock A? Most traders instinctively say "Yes." The logic seems sound: "If Stock A has a higher expected return, it has a higher probability of finishing In-The-Money. Therefore, the option must be worth more." It feels like betting on a faster horse. You assume the growth rate (mu) is a key input for the price. THE REALITY The math disagrees. If volatility and spot prices are identical, the price of those two options is exactly the same. This is the counter-intuitive heart of Risk-Neutral Pricing. In the world of derivatives valuation, the expected growth rate of the underlying asset is irrelevant. It vanishes from the equation. THE PROOF Let's break down why "The Drift" disappears. The Setup: You sell a Call option on Stock A. The Hedge: To protect yourself, you buy Delta amount of the stock. The Result: If the stock rips higher (+20%), you lose on the option, but your long stock position makes money. By dynamically adjusting this hedge (Delta Hedging), you eliminate the directional risk. Because you have hedged away the risk, you are no longer entitled to the risky return (the 20% growth). You are only entitled to the Risk-Free Rate. The option isn't priced based on where the stock is going. It is priced based on the cost of the hedge. However, we must remember that models are not markets. Risk-Neutral Valuation relies on the critical assumption of a Complete Market. A market is "Complete" only if every possible risk can be perfectly hedged (replicated) using existing assets. In reality, markets are often Incomplete: Illiquidity (e.g., Credit Risk): In distressed markets, you often cannot find a counterparty to execute the hedge. Discrete Time: Models assume we can rebalance continuously. Real life happens in discrete steps, creating "hedging error." Transaction Costs: In the real world, rebalancing a hedge isn't free. If the market is incomplete, the risk cannot be fully hedged away. When risk remains, risk preferences (and the real-world drift) start to matter again. We don't price derivatives based on what we hope will happen. We price them based on the cost of manufacturing a hedge. But as Risk Managers, we must never forget that the "Complete Market" is a theoretical luxury, not a practical reality. #QuantFinance #RiskManagement #MarketRisk #CreditRisk #Derivatives #Quant #BlackScholes #ModelRisk #FinancialEngineering #CQF

From Black-Scholes to Heston: Why Stochastic Volatility Matters Financial markets taught us one hard truth: volatility is not constant. Yet, for decades, we priced risk as if it were. Stochastic Volatility models changed that conversation. Instead of treating volatility as a fixed input, they model it as a random process evolving over time — just like asset prices themselves. Why does this matter? Because markets exhibit: • Volatility clustering • Leverage effect (falling prices → rising volatility) • Fat tails • Volatility smiles & skews Constant-vol models simply cannot explain these realities. Among stochastic frameworks, the Heston Model (1993) became the industry standard — offering: Mean-reverting variance dynamics Correlation between price and volatility shocks Semi-closed form solutions for option pricing Practical calibration for real-world trading desks In derivatives pricing and risk management, this is not academic elegance — it is survival. When volatility itself becomes stochastic, markets are no longer one-dimensional. Hedging becomes incomplete. Variance risk premium emerges. Risk measurement deepens. The real insight? Risk is not just about price movement. It is about the movement of uncertainty itself. Stochastic volatility models help us price that uncertainty. #QuantFinance #Derivatives #RiskManagement #StochasticVolatility #HestonModel #FinancialEngineering #VolatilitySmile