Introduction to Newton's method
Newton's method, also known as the Newton-Raphson method, is an iterative numerical technique used to find approximate solutions to equations of the form f(x)=0f(x) = 0f(x)=0. The method uses the first derivative of the function to iteratively approach the root. It is based on the idea of linear approximation, using the tangent line at a guessed point to find a better approximation of the root.
How It Works
Initial Guess: Start with an initial guess x0x_0x0 for the root of the equation f(x)=0f(x) = 0f(x)=0.
Iteration: Use the following formula to generate a sequence of approximations:
xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}xn+1=xn−f′(xn)f(xn)
Here, f(xn)f(x_n)f(xn) is the value of the function at xnx_nxn, and f′(xn)f'(x_n)f′(xn) is the value of the derivative of the function at xnx_nxn.
Convergence: Repeat the iteration until the difference between successive approximations is sufficiently small, indicating that the solution has converged to a root.
Geometric Interpretation
The method uses the fact that the tangent line to the curve of f(x)f(x)f(x) at any point xnx_nxn can be used to approximate the function near that point. The x-intercept of this tangent line provides a new approximation xn+1x_{n+1}xn+1 of the root.
Advantages and Limitations
Advantages:
Fast Convergence: When the initial guess is close to the actual root and the function behaves well, Newton's method converges very quickly (quadratically).
Limitations:
Requirement of Derivative: The method requires the computation of the derivative f′(x)f'(x)f′(x), which may not be easy for all functions.
Divergence: If the initial guess is far from the root or if the function's derivative is zero or nearly zero, the method can fail to converge or diverge.
Example in Finance
Newton's method is frequently used in finance for solving equations related to options pricing, particularly when calculating the implied volatility or in the calibration of complex financial models. Below are a few detailed examples of its application in options trading:
1. Calculating Implied Volatility
Implied volatility is a crucial metric in options trading, reflecting the market's expectation of the future volatility of the underlying asset's price. It cannot be directly observed and must be inferred from market prices of options.
Black-Scholes Model
The Black-Scholes model provides a theoretical price for European call and put options. However, the implied volatility is not directly visible in the market; instead, the market price of an option is observed. The relationship between the market price of an option (denoted as CmarketC_{market}Cmarket) and the Black-Scholes price (denoted as CBS(σ)C_{BS}(\sigma)CBS(σ), where σ\sigmaσ is the volatility) is given by:
Cmarket=CBS(σ)C_{market} = C_{BS}(\sigma)Cmarket=CBS(σ)
To find the implied volatility σ\sigmaσ, we solve:
f(σ)=CBS(σ)−Cmarket=0f(\sigma) = C_{BS}(\sigma) - C_{market} = 0f(σ)=CBS(σ)−Cmarket=0
Newton's Method Application
To solve this equation using Newton's method, we need the derivative of CBSC_{BS}CBS with respect to σ\sigmaσ, which is the vega (vvv):
v=∂CBS∂σv = \frac{\partial C_{BS}}{\partial \sigma}v=∂σ∂CBS
The iterative formula for Newton's method becomes:
σn+1=σn−f(σn)v(σn)=σn−CBS(σn)−Cmarketv(σn)\sigma_{n+1} = \sigma_n - \frac{f(\sigma_n)}{v(\sigma_n)} = \sigma_n - \frac{C_{BS}(\sigma_n) - C_{market}}{v(\sigma_n)}σn+1=σn−v(σn)f(σn)=σn−v(σn)CBS(σn)−Cmarket
This iteration continues until the difference between successive estimates of σ\sigmaσ is sufficiently small.
2. Calibration of Stochastic Volatility Models
Stochastic volatility models, such as the Heston model, involve solving systems of non-linear equations to calibrate model parameters based on market data. These models are used to capture the dynamics of volatility, which are not constant over time.
Heston Model Calibration
The Heston model characterizes the underlying asset's price and its volatility as stochastic processes. The market provides option prices, and the model parameters (such as long-term variance, speed of mean reversion, volatility of volatility, etc.) must be calibrated to fit these prices.
Given a set of market prices for different strikes and maturities, the calibration process involves minimizing the difference between the model prices and market prices. Newton's method can be used here to iteratively adjust the model parameters to find the best fit. The system of equations to solve would look like:
fi(θ)=Cmarket,i−Cmodel,i(θ)=0,i=1,…,Nf_i(\theta) = C_{market, i} - C_{model, i}(\theta) = 0, \quad i = 1, \dots, Nfi(θ)=Cmarket,i−Cmodel,i(θ)=0,i=1,…,N
where θ\thetaθ represents the vector of model parameters and Cmodel,i(θ)C_{model, i}(\theta)Cmodel,i(θ) denotes the model price for the iii-th option. The Jacobian matrix, consisting of partial derivatives of the function with respect to the parameters, is used in Newton's method to find the next parameter set:
θn+1=θn−J−1(θn)f(θn)\theta_{n+1} = \theta_n - J^{-1}(\theta_n) f(\theta_n)θn+1=θn−J−1(θn)f(θn)
where JJJ is the Jacobian matrix.
3. American Options and Early Exercise Premium
American options can be exercised at any time before expiration, making their valuation more complex than European options. Newton's method can be employed in the valuation process, especially when using numerical methods like finite differences or trees that involve iterative steps.
Early Exercise Boundary
In the binomial or trinomial tree models for American options, the early exercise premium must be calculated. Newton's method can be used to determine the critical asset price at which it is optimal to exercise the option early. The option's value depends on this critical price, and iteratively adjusting it using Newton's method helps find the optimal early exercise boundary.
Example Scenario
Market Price of the Call Option (CmarketC_{market}Cmarket): $10
Current Stock Price (SSS): $100
Strike Price (KKK): $100
Risk-Free Interest Rate (rrr): 5% (0.05)
Time to Maturity (TTT): 1 year
Newton's Method Iterations
The function to solve is f(σ)=CBS(σ)−Cmarketf(\sigma) = C_{BS}(\sigma) - C_{market}f(σ)=CBS(σ)−Cmarket, where CBS(σ)C_{BS}(\sigma)CBS(σ) is the Black-Scholes price of the call option with volatility σ\sigmaσ. The vega, vvv, is the derivative of the option price with respect to volatility.
Initial Guess: σ0=0.2\sigma_0 = 0.2σ0=0.2 (20%)
The table below shows the iteration process
Explanation
σn\sigma_nσn: The current estimate of the implied volatility.
CBS(σn)C_{BS}(\sigma_n)CBS(σn): The Black-Scholes price calculated with the current volatility estimate.
v(σn)v(\sigma_n)v(σn): The vega, or the sensitivity of the option price to changes in volatility.
f(σn)f(\sigma_n)f(σn): The difference between the Black-Scholes price and the market price.
Update Step: The value subtracted from the current estimate to get the next estimate, calculated as f(σn)v(σn)\frac{f(\sigma_n)}{v(\sigma_n)}v(σn)f(σn).
σn+1\sigma_{n+1}σn+1: The new estimate of the implied volatility.
Conclusion
The iteration process continues until f(σn)f(\sigma_n)f(σn) (the difference between the model price and the market price) is close to zero. In this example, after three iterations, the implied volatility converges to approximately 18.56%. This illustrates how Newton's method can be effectively used to find implied volatility in options trading.