Equity Derivatives Pricing

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

Option Pricing with Quantum Mechanical Methods I first encountered a formal treatment of pricing financial derivatives using the framework of quantum mechanics in Baaquie’s book Quantum Finance when it was published. Over the years, the term “quantum finance” has appeared more frequently in literature. I paid limited attention to this line of work until the paper discussed below, which caught my interest by addressing a well-known problem using the language of quantum mechanics. The paper proposes an option pricing model that converts the Fokker–Planck equation into the Schrödinger equation, yielding both the return distribution and a closed-form solution for European options. The model shows that S&P 500 returns follow a Laplace distribution with power-law tails and that quantum methods outperform GBM-based models in explaining return dynamics and put option prices. Findings: -The paper proposes an option pricing model inspired by quantum mechanics to address the long-standing puzzle of overpriced put options. -The authors reformulate the stock return dynamics by transforming the Fokker–Planck equation into a Schrödinger equation. -This framework yields an explicit probability density function for stock returns and a closed-form solution for European option prices. -Empirical results suggest that S&P 500 index returns follow a Laplace distribution with power-law tail behavior rather than a Gaussian distribution. -The quantum-mechanics-based model outperforms traditional GBM-based models in fitting both index returns and observed put option prices. -The findings indicate that high put option prices observed in the market are close to fair value when modeled within this quantum framework. Reference: Minhyuk Jeong, Biao Yang, Xingjia Zhang, Taeyoung Park & Kwangwon Ahn, A quantum model for the overpriced put puzzle, Financial Innovation (2025) 11:130 Join a community of 7,000+ quants—subscribe to the newsletter! https://lnkd.in/gVFDBTCK #options #volatility #quantitativefinance ABSTRACT Put options are known to be priced unusually high in the market, which we refer to as the overpriced put puzzle . This study proposes a quantum model (QM) that can explain such high put option prices as fair prices. Starting from a stochastic differential equation of stock returns, we convert the Fokker–Planck equation into the Schrödinger equation. To model the market force that always draws excess returns back to equilibrium, we specify a diffusion process corresponding to a QM with a delta potential. The results demonstrate that stock returns follow a Laplace distribution and exhibit power law in the tail. We then construct a closed-form solution for European put option pricing, determining that our model better explains the returns of the S&P 500 index and its corresponding put option prices than do geometric Brownian motion-based models. This study has significant implications for investors and risk managers,...

For decades, options were viewed as derivative instruments prices derived from the underlying, offering no predictive information beyond the market's consensus volatility. That view is obsolete. Modern research demonstrates that options prices contain forward-looking information about tail risk, jump probabilities, and even future equity returns. The challenge is extracting that signal from noisy market prices. The most robust predictive framework is the volatility risk premium strategy: sell index options (typically out-of-the-money puts) to harvest the premium that reflects the market's overestimation of tail risk. But the edge has been arbed away through crowding. The new frontier is term structure of skew signals. When near-term skew is steep relative to long-term skew, it predicts near-term downside realization. When the term structure inverts (long-term skew steeper than short-term), it predicts a delayed volatility event. More sophisticated models use machine learning on options chains directly feeding the entire surface of implied volatilities across strikes and maturities into gradient boosting models to predict future realized volatility or equity returns. The features are not just levels but shapes: convexity, curvature, and the relative pricing of out-of-the-money puts versus calls. The firms that master options predictive models are not trading options as derivatives; they are trading options as primary information sources about market expectations. #OptionsTrading #VolatilityRiskPremium #ImpliedSkew #PredictiveModels #MachineLearning #QuantResearch #TailRisk

Monte Carlo & Derivative Pricing What's going on behind the Monte Carlo simulations when pricing a derivative? The price, mathematically, is just the discounted average payoff. Thus the payoff function is important. And pricing is a game of averages. Take the European call option example. The option pays off at expiry if the value of shares ST is greater than the strike price K. Thus the payoff function is max(ST-K, 0). Then the Monte Carlo is essentially simulating multiple possible values of terminal share prices ST, from a given starting price S0, volatility sigma, and some randomness assumed under the risk neutral measure. Kind of like the Dr. Strange of quant world. With the given simulated share prices at time T, we can apply the payoff function - to calculate the option payoff for each simulation. For simulated ST bigger than K, the payoff is positive (above the red line in illustration). If ST is below K, then no payoff. Finally, we can get the average payoff across all the simulations at time T. And account for time value of money, by discounting the average payoff to time 0. Think of this as some sort of insurance against a rising share price which pays off above a certain value, representing claims. This insurance comes at some premium price, and the premium price is just the average value of potential claims amount, that's all. The challenge is knowing how the claims will manifest, or how share prices will evolve and what values will have the option payoff. The Monte Carlo pricing can then be used to further obtain simulated option Greeks, like the option delta. Option delta is the change in option value for a unit change in shares - simulate the option price for S0+1, and S0-1, then calculate the change in option price for both cases, and take the average. Similarly other Greeks can be simulated. Last but not least, for European call this is only an illustration. The Black Scholes formula exist. Monte Carlo is more useful for exotic options. In summary, the Monte Carlo: 1. Simulates share prices under the risk neutral distribution 2. The option price depends on the payoff function 3. The price is just an average 4. Option Greeks can be simulated. Happy Sunday! PS: In case you want to try. The example parameters: S0=10, K=10, vol=0.3, r=5%, T=1.

An Introduction to Option Pricing Theory for European, American and Exotic Derivatives. 🚨 Some topics I covered in this document: 𝟭. 𝗜𝗻𝘁𝗿𝗼𝗱𝘂𝗰𝘁𝗶𝗼𝗻 𝘁𝗼 𝗢𝗽𝘁𝗶𝗼𝗻𝘀: - An option is a financial derivative giving the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price on or before a specific date. - Properties of options. - Differences between American and European Options. 𝟮. 𝗦𝘁𝗼𝗰𝗵𝗮𝘀𝘁𝗶𝗰 𝗣𝗿𝗼𝗰𝗲𝘀𝘀𝗲𝘀 𝗳𝗼𝗿 𝗔𝘀𝘀𝗲𝘁 𝗣𝗿𝗶𝗰𝗲𝘀: - Discussion of Geometric Brownian Motion and how this leads to a general model for stock/asset prices. - The Risk-Neutral Measure and Martingales. 𝟯. 𝗘𝘂𝗿𝗼𝗽𝗲𝗮𝗻 𝗢𝗽𝘁𝗶𝗼𝗻𝘀 𝗣𝗿𝗶𝗰𝗶𝗻𝗴: - Payoff formulae for European options. - Derivation sketch of the Black-Scholes model. - Black-Scholes Formula. 𝟰. 𝗔𝗺𝗲𝗿𝗶𝗰𝗮𝗻 𝗢𝗽𝘁𝗶𝗼𝗻𝘀 𝗣𝗿𝗶𝗰𝗶𝗻𝗴: - Key features about American options. - Binomial Tree pricing model for American options. 𝟱. 𝗘𝘅𝗼𝘁𝗶𝗰 𝗢𝗽𝘁𝗶𝗼𝗻𝘀: - Barrier Options, Asian Options and Digital Options are discussed. 𝟲. 𝗠𝗲𝗮𝘀𝘂𝗿𝗶𝗻𝗴 𝗢𝗽𝘁𝗶𝗼𝗻 𝗦𝗲𝗻𝘀𝗶𝘁𝗶𝘃𝗶𝘁𝗶𝗲𝘀 𝘄𝗶𝘁𝗵 𝘁𝗵𝗲 𝗚𝗿𝗲𝗲𝗸𝘀: - Brief discussion and definition of the Greeks (first-order derivatives). 𝟳. 𝗔𝗱𝘃𝗮𝗻𝗰𝗲𝗱 𝗣𝗿𝗶𝗰𝗶𝗻𝗴 𝗠𝗼𝗱𝗲𝗹𝘀: - The Black-Scholes model is limited due to its assumption of constant volatility since real markets exhibit volatility smiles and skews. Some models that account for this are discussed. - Jump-Diffusion Models - The Merton Jump-Diffusion Model. - The Heston Model Extension. If you're interested in #finance, #quantfinance, #riskmanagement or #actuarialscience, feel free to follow me: Armandt Erasmus as I intend to post more on these topics. 🙂

*** 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

Understanding Volatility Surfaces in Quantitative Finance In quantitative finance, pricing derivatives accurately hinges on more than just a simple volatility number. Market-implied volatility is not constant across strikes and maturities — it bends, twists, and reshapes. This non-uniformity gives rise to the volatility surface, a foundational concept for modern pricing, risk, and hedging models. 1. What is a Volatility Surface? ➤ A volatility surface maps implied volatility across strike prices (moneyness) and time to maturity ➤ Rather than assuming volatility is fixed (as in Black-Scholes), the market provides different volatilities for each option, leading to complex, 3D surfaces ➤ These surfaces evolve over time and reflect market sentiment, supply-demand imbalances, and expectations of future uncertainty 2. Why is it Crucial in Quantitative Finance? ➤ Risk-Neutral Pricing: Derivative prices must be consistent with observed market quotes. Vol surfaces allow models to reproduce current option prices precisely ➤ Dynamic Hedging: Changes in volatility skew/smile impact hedging portfolios — traders calibrate models daily to the surface to remain delta/gamma/vega neutral ➤ Stress Testing: Shifts or distortions in surfaces help quantify the PnL impact under market stress scenarios 3. Key Modeling Approaches ➤ Local Volatility Models (e.g., Dupire) → Assume volatility is a function of strike and time, producing path-dependent dynamics → Common in equity derivatives where volatility smile is pronounced ➤ Stochastic Volatility Models (e.g., Heston) → Treat volatility itself as a random process, introducing correlation with the asset → Captures volatility clustering and mean reversion — relevant in FX and commodities ➤ SABR Model → Widely used in interest rate derivatives → Accurately models volatility smile for swaptions and bond options ➤ LV-LSV Hybrids → Combine local and stochastic frameworks to better reflect complex dynamics, particularly in exotic option pricing 4. Where Does This Matter in Industry? ➤ Equity desks calibrate surfaces daily to quote volatility for exotic structures (barriers, autocallables) ➤ FX markets use surfaces for dual digitals, touch/no-touch options, and structured forwards ➤ Interest rate desks model swaption vol cubes and collars using SABR-based interpolation ➤ Model risk teams monitor surface arbitrage violations — ensuring prices are free from butterfly/calendar spread inconsistencies Volatility surfaces are not just about smoothing market quotes — they’re blueprints of risk perception, tools for calibration, and the canvas on which almost every pricing model is painted. In practice, they separate theoretical elegance from operational robustness. #QuantitativeFinance #VolatilitySurface #LocalVolatility #StochasticVolatility #SABR #OptionsPricing #MarketRisk #QuantResearch #Derivatives #RiskManagement

While Black-Scholes gives a neat analytical formula for vanilla options, things get complicated fast when you move to exotic options (like barrier, Asian, or lookback options). The issue? Path dependency. In vanilla options, the payoff depends only on the asset price at a single moment (maturity). In exotic options, the payoff is influenced by the entire path the asset price takes over a period. This means our probability models must capture this continuous, complex movement. Enter the power of partial differential equations (PDEs), the mathematical discipline often used to model fluid dynamics, heat flow, and diffusion in physics. 🔹 Instead of a simple trajectory, think of exotic pricing as a multidimensional diffusion problem. 🔹 A path-dependent barrier option isn't just a threshold; it’s an absorbing or reflecting boundary condition on the PDE's solution space. 🔹 Solving these systems often pushes boundaries, requiring sophisticated numerical methods like finite difference or Monte Carlo simulations on high-performance computing clusters. The "engine" under the hood of quant finance isn't just data analysis; it's a deep, theoretical mathematical framework made real by computational engineering. What’s the biggest challenge you’ve faced in implementing numerical PDE methods for path-dependent derivatives? #QuantitativeFinance #CapitalMarkets #PDEs #ExoticOptions #MathInFinance #ComputationalFinance #FinancialEngineering #BlackScholes #DerivativesPricing