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Option Pricing Methodology

Mathematical foundations and theoretical background of the option pricing models implemented in this dashboard.

Introduction to Option Pricing

Option pricing theory provides a mathematical framework for determining the fair value of derivatives contracts. The fundamental concept is based on no-arbitrage principles, which state that in efficient markets, it should be impossible to construct risk-free portfolios that yield returns exceeding the risk-free rate.

Key Option Value Determinants

  • Underlying asset price (S): Current market price of the asset
  • Strike price (K): Predetermined price at which the option can be exercised
  • Time to expiration (T): Remaining lifespan of the option contract
  • Risk-free interest rate (r): Return on a risk-free investment over the option's lifetime
  • Volatility (σ): Measure of the underlying asset's price fluctuation
  • Dividends (q): Cash distributions from the underlying asset during the option's life

Option Payoffs at Expiration

Call option (right to buy):

\[\text{Payoff}_{\text{call}} = \max(S_T - K, 0)\]

Put option (right to sell):

\[\text{Payoff}_{\text{put}} = \max(K - S_T, 0)\]

Where \(S_T\) is the price of the underlying asset at expiration time T.

Risk-Neutral Valuation

The cornerstone of option pricing is the risk-neutral valuation principle. It states that option prices can be calculated as the expected present value of future payoffs under a special probability measure (the risk-neutral measure \(\mathbb{Q}\)), where all assets earn the risk-free rate.

Risk-Neutral Pricing Formula:

\[V_0 = e^{-rT} \mathbb{E}_{\mathbb{Q}}[ \text{Payoff}(S_T) | \mathcal{F}_0 ]\]

Where \(V_0\) is the option price at time 0,\(r\) is the risk-free rate,\(T\) is the time to expiration, and\(\mathbb{E}_{\mathbb{Q}}[\cdot]\) represents the expectation under the risk-neutral measure given information available at time 0 (\(\mathcal{F}_0\)).

Black-Scholes Model

Developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, the Black-Scholes model is the foundation of modern option pricing theory, providing an elegant closed-form solution for European option prices.

Key Assumptions

  • The underlying asset price follows a geometric Brownian motion with constant drift and volatility
  • Trading occurs continuously with no transaction costs or taxes
  • The risk-free interest rate is constant and known
  • The market permits short-selling with full use of proceeds
  • No arbitrage opportunities exist
  • For standard formulation: no dividends during the option's life

Model Dynamics

Under the risk-neutral measure \(\mathbb{Q}\), the Black-Scholes model assumes the underlying asset price follows a geometric Brownian motion:

\[dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}\]

Where \(S_t\) is the asset price at time t,\(r\) is the risk-free rate,\(\sigma\) is the volatility, and\(W_t^{\mathbb{Q}}\) is a standard Brownian motion under the risk-neutral measure.

Black-Scholes Partial Differential Equation

The option price must satisfy the following partial differential equation:

\[\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0\]

Closed-Form Solution

For European options with no dividends, the Black-Scholes formula provides:

Call option price:

\[C(S, t) = S\, N(d_1) - Ke^{-r(T-t)}\, N(d_2)\]

Put option price:

\[P(S, t) = Ke^{-r(T-t)}\, N(-d_2) - S\, N(-d_1)\]

Where:

\[d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}\]
\[d_2 = d_1 - \sigma\sqrt{T-t}\]

And \(N(x)\) is the cumulative distribution function of the standard normal distribution.

The Black-Scholes model, despite its simplifying assumptions, provides a good first approximation for option prices and serves as the foundation for more sophisticated models.

Heston Stochastic Volatility Model

The Heston model, introduced by Steven Heston in 1993, extends the Black-Scholes framework by allowing volatility to follow its own stochastic process. This addresses a key limitation of Black-Scholes by capturing empirically observed phenomena like volatility clustering, mean reversion, and the volatility smile.

Model Dynamics

Under the risk-neutral measure, the Heston model is described by two coupled stochastic differential equations:

Asset price dynamics:

\[dS_t = (r-q)S_t\,dt + \sqrt{v_t}\,S_t\,dW^S_t\]

Variance dynamics:

\[dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW^v_t\]

With correlation between the Brownian motions:

\[dW^S_t\,dW^v_t = \rho\,dt\]

Model Parameters

  • \(v_0\): Initial variance (square of initial volatility)
  • \(\kappa\) (kappa): Rate of mean reversion of variance - how quickly volatility returns to its long-term mean
  • \(\theta\) (theta): Long-term mean of variance - the level that volatility tends toward over time
  • \(\xi\) (xi/volvol): Volatility of variance - determines the randomness in the volatility process
  • \(\rho\) (rho): Correlation between asset returns and variance changes - typically negative, capturing the leverage effect

The Feller condition \(2\kappa\theta > \xi^2\)ensures that the variance process remains strictly positive. While mathematically important, calibrated parameters may sometimes violate this condition in practice.

Key Advantages

  • Captures volatility smiles and skews observed in options markets
  • Models the negative correlation between asset returns and volatility (leverage effect)
  • Allows for mean-reversion in volatility
  • Offers semi-analytical solutions for European options
  • Can be calibrated to match market prices across different strikes and maturities

Solution Methods

Unlike Black-Scholes, the Heston model doesn't have a simple closed-form solution. This application implements two main approaches:

  1. Characteristic Function Method: A semi-analytical approach using the known characteristic function of the log-price under Heston, with numerical integration to obtain option prices.
  2. Monte Carlo Simulation: Numerical simulation of price and variance paths to estimate option prices, particularly useful for path-dependent options.

Monte Carlo Methods

Monte Carlo simulation is a powerful numerical method for option pricing, especially valuable for complex models and path-dependent options where closed-form solutions don't exist or are difficult to compute.

General Procedure

  1. Generate a large number (N) of random price paths for the underlying asset using the model dynamics
  2. Calculate the option payoff at maturity for each simulated path
  3. Average these payoffs to approximate the expected payoff
  4. Discount this average back to present value using the risk-free rate

Mathematically, the Monte Carlo estimate of the option price is:\[V_0 \approx e^{-rT} \frac{1}{N} \sum_{i=1}^{N} \text{Payoff}(S_T^{(i)})\]

Discretization Schemes

For Black-Scholes Model

The Euler scheme for the log-price \(X_t = \ln(S_t)\) is:

\[X_{t+\Delta t} = X_t + (r - q - \frac{\sigma^2}{2})\Delta t + \sigma\sqrt{\Delta t}\,Z\]

Where \(Z \sim N(0,1)\) is a standard normal random variable and \(\Delta t = T/M\) is the time step size for M steps.

For Heston Model

Simulating the Heston model requires special care to handle the coupled SDEs and ensure the variance remains positive. Our implementation uses a Full Truncation Euler scheme:

  1. Generate correlated random numbers:
    \[ Z_1 \sim N(0,1) \quad \text{(independent)}\\ Z_2 = \rho Z_1 + \sqrt{1-\rho^2}\,Z_3 \quad \text{where } Z_3 \sim N(0,1) \text{ (independent)} \]
  2. Truncate variance to ensure positivity:
    \[v_t^+ = \max(v_t, 0)\]
  3. Update variance:
    \[v_{t+\Delta t} = v_t + \kappa(\theta - v_t^+)\Delta t + \xi\sqrt{v_t^+}\sqrt{\Delta t}\,Z_2\]
  4. Update log-price:
    \[X_{t+\Delta t} = X_t + (r - q - \frac{v_t^+}{2})\Delta t + \sqrt{v_t^+}\sqrt{\Delta t}\,Z_1\]

Variance Reduction Techniques

To improve efficiency and accuracy, our implementation includes:

  • Antithetic Variates: For each random path, we generate its "mirror" path by negating the random numbers. This reduces variance without increasing computational cost.
    \[V_0 \approx e^{-rT} \frac{1}{2N} \sum_{i=1}^{N} \left[ \text{Payoff}(S_T^{(i,+)}) + \text{Payoff}(S_T^{(i,-)}) \right]\]
  • Control Variates: Uses analytically solvable variants (like Black-Scholes) as control variates to reduce variance. This is particularly effective for pricing Heston options when parameters like volvol (ξ) or correlation (ρ) are small.
  • Quasi-Random Sequences: Sobol or Halton sequences may replace pseudo-random numbers for faster convergence, especially in higher dimensions.

While Monte Carlo is computationally intensive compared to analytical methods, it offers great flexibility for complex options and models. Our implementation uses vectorized operations and parallelization to improve performance.

Characteristic Function Method

The characteristic function method provides a semi-analytical approach to pricing options under the Heston model. It leverages the fact that while the probability density function of the asset price is not known in closed form, its Fourier transform (the characteristic function) is available analytically.

The Heston Characteristic Function

The characteristic function of the log-price \(X_T = \ln(S_T)\)under the Heston model has the form:

\[\phi(u) = \mathbb{E}[e^{iuX_T}] = e^{C(u,T) + D(u,T)v_0 + iu\ln(S_0)}\]

Where the complex-valued functions C and D are given by:

\[ d = \sqrt{(\kappa - \rho\xi iu)^2 + \xi^2(iu + u^2)}\\ g = \frac{\kappa - \rho\xi iu - d}{\kappa - \rho\xi iu + d}\\ D(u,T) = \frac{\kappa - \rho\xi iu - d}{\xi^2} \cdot \frac{1 - e^{-dT}}{1 - ge^{-dT}}\\ C(u,T) = (r-q)iuT + \frac{\kappa\theta}{\xi^2}\left[(\kappa - \rho\xi iu - d)T - 2\ln\left(\frac{1 - ge^{-dT}}{1 - g}\right)\right] \]

Numerical Considerations

The implementation of these formulas requires careful handling of:

  • Complex square roots (selecting the appropriate branch)
  • Complex logarithms (handling branch cuts)
  • Potential division by zero or near-zero values
  • Numerical overflow in exponential terms

Option Pricing Formula

Using the characteristic function, European option prices can be computed through integral representations. For call options, Heston derived:

\[C(S_0, K, T) = S_0 P_1 - Ke^{-rT}P_2\]

Where the probabilities P1 and P2 are:

\[ P_j = \frac{1}{2} + \frac{1}{\pi} \int_0^\infty \text{Re}\left[\frac{e^{-iu\ln(K)}\phi_j(u)}{iu}\right] du, \quad j = 1, 2 \]

Here, \(\phi_1(u)\) and\(\phi_2(u)\) are modified characteristic functions related to the original \(\phi(u)\). Specifically,\(\phi_2(u) = \phi(u)\) and\(\phi_1(u) = \phi(u-i)/\phi(-i)\).

Numerical Integration

The integrals for P1 and P2 are computed numerically. Our implementation:

  • Uses adaptive quadrature methods for accurate integration
  • Truncates the infinite integration at an appropriate upper limit (typically 100-200) where the integrand becomes negligibly small
  • Adjusts the number of integration points based on option characteristics (more points for short-dated options or strikes far from current price)
  • Implements extensive caching mechanisms to avoid recalculating the characteristic function multiple times with the same parameters

This semi-analytical approach is generally more efficient than Monte Carlo simulation for standard European options and provides greater accuracy, particularly when proper numerical techniques are applied.

Model Calibration

Calibration is the process of finding model parameters that best fit observed market prices. For the Heston model, this involves determining the optimal set {κ, θ, ξ, ρ, v0} that minimizes the difference between model-generated prices and market prices.

The Calibration Problem

Objective Function:

\[\min_{\kappa, \theta, \xi, \rho, v_0} \sum_{i=1}^n w_i\left(P^{\text{model}}_i(\kappa, \theta, \xi, \rho, v_0) - P^{\text{market}}_i\right)^2\]

Where wi are weights assigned to each option, typically giving higher importance to more liquid options or those closer to at-the-money.

Calibration Strategy

  1. Data Preparation: Filter and clean market data, removing illiquid options or those with invalid prices.
  2. Parameter Constraints: Define reasonable bounds for each parameter:
    • κ (kappa): typically 0.1 to 10.0
    • θ (theta): related to long-term implied volatility, often 0.01 to 0.25
    • ξ (xi): volatility of volatility, typically 0.1 to 2.0
    • ρ (rho): correlation, typically -0.95 to 0.0
    • v0: initial variance, often close to squared ATM implied volatility
  3. Multiple Initial Guesses: Start optimization from several different parameter sets to avoid getting trapped in local minima.
  4. Optimization Algorithm: Use gradient-based methods (e.g., L-BFGS-B) for efficiency, with fallback to derivative-free methods (e.g., Nelder-Mead) for robustness.
  5. Weighting Scheme: Apply higher weights to at-the-money options which are more liquid and informative about the volatility structure.

Implementation Optimizations

  • Caching: Store and reuse intermediate calculations to avoid redundant computations during the optimizer's iterations.
  • Early Stopping: Reject poor parameter sets quickly before full price computation.
  • Stratified Sampling: For large option chains, use representative subsets during optimization to improve speed.
  • Parallel Processing: Calculate model prices for multiple options simultaneously.

Calibration Quality Assessment

We evaluate calibration quality using several metrics:

Root Mean Squared Error (RMSE)

\[\text{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^n (P^{\text{model}}_i - P^{\text{market}}_i)^2}\]

Mean Absolute Error (MAE)

\[\text{MAE} = \frac{1}{n}\sum_{i=1}^n |P^{\text{model}}_i - P^{\text{market}}_i|\]

Mean Relative Error (MRE)

\[\text{MRE} = \frac{1}{n}\sum_{i=1}^n \frac{|P^{\text{model}}_i - P^{\text{market}}_i|}{P^{\text{market}}_i} \times 100\%\]

Maximum Absolute Error

\[\text{MaxError} = \max_i |P^{\text{model}}_i - P^{\text{market}}_i|\]

A successful calibration typically achieves an RMSE below 1-2% of the underlying asset price, with particularly good fits for options near the money.

Further Resources

For those interested in deepening their understanding of option pricing theory and implementation:

Books

  • Hull, J. C. (2021). Options, Futures, and Other Derivatives (11th ed.). Pearson - The industry standard textbook for derivatives pricing
  • Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley - Excellent coverage of stochastic volatility models
  • Wilmott, P. (2006). Paul Wilmott on Quantitative Finance. Wiley - Comprehensive with focus on computational methods
  • Lewis, A. L. (2000). Option Valuation Under Stochastic Volatility. Finance Press - Detailed coverage of Fourier methods

Research Papers

  • Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities.Journal of Political Economy, 81(3)
  • Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility.Review of Financial Studies, 6(2)
  • Gatheral, J., & Jacquier, A. (2014). Arbitrage-free SVI volatility surfaces.Quantitative Finance, 14(1)
  • Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model.Journal of Computational Finance, 11(3)

Online Resources

Try the Pricing Dashboard