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,...

As a mathematician looking at exotic derivatives, I find the TPF, or Target Redemption Forward, to be a fascinating study in asymmetric payoffs and path dependency. On the surface, the pitch to a corporate hedger is incredibly appealing: lock in a sequence of forward rates that are significantly better than the current market, with the caveat that the structure automatically terminates once a predetermined "target profit" is accumulated. But when we break down the underlying mathematics, the risk profile is brutally one-sided. A TPF is essentially a strip of short puts and long calls (often with leveraged downside) chained together by a path-dependent knock-out barrier. For the quants and risk managers pricing and hedging these books, the challenges are immense: 🔹 The Accumulated State Variable: The knock-out isn’t just based on the spot price at a given time; it’s dependent on the integral of all previous fixing outcomes. 🔹 The Asymmetry of Duration: If the market moves in the client’s favor, the target profit is hit quickly, the trade knocks out, and the expected duration collapses. If the market moves against the client, the trade stays alive, leaving the dealer short massive amounts of levered, deep-in-the-money risk at the worst possible time. 🔹 Complex Sensitivities: Pricing these requires robust Monte Carlo frameworks or multidimensional PDE solvers. You aren't just managing standard Delta and Gamma; you are intensely exposed to the forward volatility skew, Vanna, and the dynamic correlation between fixing dates. When the market gaps through the strike, the sudden explosion in Delta and negative Gamma is exactly what causes these products to blow up so spectacularly during unexpected macro shocks. I’m curious to hear from the FX traders, structurers, and risk analysts on my feed: Given the heavy reliance on local and stochastic volatility models to price that forward skew, how are you currently managing the sudden Vega and Gamma expansions when a TPF book is threatened by a major spot breakdown? Are you dynamically hedging the individual fixings, or relying purely on macro overlays? #QuantitativeFinance #CapitalMarkets #Derivatives #FXTrading #RiskManagement #Quants #FinancialEngineering #MathFinance #ExoticOptions #TargetRedemptionForward

Chapter 3 of Shreve's Stochastic Calculus for Finance I - State Prices Chapter 2 handed me the risk-neutral measure as a pricing device. Chapter 3 asks the follow-up: what connects the real world to that pricing world? The answer is a single object, the Radon-Nikodým derivative, which is leading to the most intuitive idea in the book: state prices. The core idea. The actual measure P and the risk-neutral measure P̃ assign different probabilities to the same outcomes. The object converting one into the other is: Z(ω) = P̃(ω) / P(ω) In the binomial model: Z(ω₁…ω_N) = (p̃/p)^#H · (q̃/q)^#T -- Theorem 3.1.1(properties of the change of measure) E[Z] = 1, Ẽ[Y] = E[Z·Y], E[Y] = Ẽ[Y/Z], Z is the dictionary between the two worlds. The Radon-Nikodým process. Zₙ = Eₙ[Z] (a martingale under P, Lemma 3.2.5) -- Theorem 3.2.7(the conditional "Bayes" formula: Ẽₙ[Y] = (1/Zₙ)·Eₙ[Zₘ·Y]) The workhorse for computing risk-neutral conditional expectations under actual probabilities. The payoff, state prices (3.2.6 / 3.2.7). The state-price density: ζₙ = Zₙ / (1+r)ⁿ Then a derivative is priced under the real-world measure: Vₙ = (1/ζₙ)·Eₙ[ ζ_N · V_N ] The state price of one outcome, the value today of $1 delivered only in that state, is P̃(ω)/(1+r)ᴺ. Section 3.3 is the line that matters most for risk. Shreve draws a sharp distinction. For pricing a derivative, you need only the risk-neutral measure. But for risk and asset management, you care about the actual probability, the real chance of a catastrophe. The twist: those portfolios contain derivatives, whose scenario values must still be computed risk-neutral. So real risk work uses both, actual probabilities to weigh how likely a scenario is, risk-neutral pricing to value the book inside it. He then introduces the other philosophy, the Capital Asset Pricing Model, built not on replication but on utility functions, which is nondecreasing, concave U capturing the risk/return trade-off: U(αx + (1−α)y) ≥ α·U(x) + (1−α)·U(y) with the HARA family and the index −U''(x)/U'(x). The intuition lands in one example: a gamble paying 1 or 99 on a coin flip has expected payoff 50, yet a risk-averse agent prefers a certain 50. By Jensen's inequality, E[U(X)] ≤ U(E[X]): here E[ln X] = 2.30 < ln(E X) = 3.91. Concavity is risk aversion, made mathematical. The risk lesson. State prices make risk aversion visible: a dollar in a bad state, when the market's down and you need it most, is worth more than a dollar in a good state. That's why ζ is high in bad states. The state-price density is the market's pricing kernel, real-world probabilities reweighted by how much each state hurts. The deeper takeaway: pricing lives in the risk-neutral world, but risk management never leaves the real one. #QuantitativeFinance #StochasticCalculus #StatePrices #RiskManagement #FinancialEngineering #FRM #FinancialMaths

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