option-pricing

Analytical and numerical option pricing calculator supporting different payoff styles.

Pricing methods implemented:

• Black-Scholes: analytical calculation of the value of a European option using the Black-Scholes model, extended by Merton to allow for the inclusion of a continuous dividend yield.
• Binomial tree: using the binomial options pricing model described by Cox, Ross, and Rubinstein, i.e. the Cox-Ross-Rubinstein (CRR) model (also known as the binomial model)
• Monte Carlo simulation: risk neutral valuation of random price paths as first applied by Boyle, with early exercise handled using the least-squares Monte Carlo approach (with a fixed degree of 2) proposed by Longstaff and Schwartz.

Table of contents

• Installation
• Usage
• Examples
  • Black-Scholes
  • Binomial Tree
  • Monte Carlo simulation
• References

Installation

To install the package run:

npm install option-pricing

or load it directly in the browser using a CDN like unpkg:

<script crossorigin src="https://unpkg.com/option-pricing/client-dist/bundle.js"></script>

Usage

Import the Option class:

import { Option } from "option-pricing";

Create a new Option:

const option = new Option({
  style: "european",     // Style of the option defined by the exercise rights ("european" or "american").
  type: "call",          // Type of the option ("call" or "put").
  initialSpotPrice: 100, // Initial price of the underlying asset (S₀ > 0).
  strikePrice: 105,      // Strike/exercise price of the option (K > 0).
  timeToMaturity: 0.5,   // Time until maturity/expiration in years (τ = T - t > 0).
  volatility: 0.3,       // Underlying volatility (σ > 0).
  riskFreeRate: 0.2,     // [optional=0] Annualized risk-free interest rate continuously compounded (r).
  dividendYield: 0.1,    // [optional=0] Annual dividend yield continuously compounded (q).
});

Then call the price method on the Option, selecting the desired pricing method/model (and passing in the required parameters if a numerical method is selected).

You can use the setter methods provided to update the option parameters.

Pricing method selection and parameters

The available pricing methods are:

• Black-Scholes: "bs" or "black-scholes"
• Binomial tree: "bt" or "binomial-tree"
  • timeSteps: Number of time steps in the tree (> 0).
• Monte Carlo simulation: "mcs" or "monte-carlo-simulation"
  • simulations: Number of simulated price paths (> 0).
  • timeSteps: [optional] Number of time steps to simulate for each path. The default value is 1, a higher value should be chosen for options with early exercise.
  • prngName: [optional] Name of the pseudorandom number generator (PRNG) to use. If not specified then Math.random() will be used. Choices are: "sfc32", "mulberry32", "xoshiro128ss", and "jsf32". See this Stack Overflow answer for details.
  • prngSeed: [optional] Initial seed state of the PRNG. If not specified then a random seed will be created.
  • prngAdvancePast: [optional] Initial amount of numbers generated by the PRNG to discard. The default value is 0, a value of about 15 is recommended by the Stack Overflow answer.

// Black-Scholes
option.price("bs");
option.price("black-scholes");
// Binomial tree
option.price("bt", { timeSteps: 25 });
option.price("binomial-tree", { timeSteps: 100 });
// Monte Carlo simulation
option.price("mcs", { simulations: 10000 }); // uses Math.random() as the PRNG
option.price("monte-carlo-simulation", {
  simulations: 1000,
  timeSteps: 50, 
  prngName: "sfc32",
  prngSeed: "123",
  prngAdvancePast: 15, 
});

Examples

Black-Scholes (analytically priced options)

• European call option

Hull SSM (2014), Problem 15.13, page 166

S₀ = 52, K = 50, τ = 0.25, σ = 0.3, r = 0.12, q = 0

const option = new Option({
  style: "european",
  type: "call",
  initialSpotPrice: 52,
  strikePrice: 50,
  timeToMaturity: 0.25,
  volatility: 0.3,
  riskFreeRate: 0.12,
});
const price = option.price("bs");
console.log(price);
// 5.057...

• European put option

Hull SSM (2014), Problem 17.7, page 187

S₀ = 0.52, K = 0.5, τ = 0.6667, σ = 0.12, r = 0.04, q = 008

const option = new Option({
  style: "european",
  type: "put",
  initialSpotPrice: 0.52,
  strikePrice: 0.5,
  timeToMaturity: 0.6667,
  volatility: 0.12,
  riskFreeRate: 0.04,
  dividendYield: 0.08,
});
const price = option.price("black-scholes");
console.log(price);
// 0.0162...

• American call option on a non-dividend paying stock

"Since it is never optimal to exercise early an American call option on a non-dividend-paying stock (see Section 11.5), equation (15.20) [the Black-Scholes-Merton pricing formula] is the value of an American call option on a non-dividend-paying stock." (Hull, 2014, p. 359)

That is, the price of an American call option on a non-dividend paying stock is the same as the price of a European call option.

const option = new Option({
  style: "american",
  type: "call",
  initialSpotPrice: 52,
  strikePrice: 50,
  timeToMaturity: 0.25,
  volatility: 0.3,
  riskFreeRate: 0.12,
});
const price = option.price("bs");
console.log(price);
// 5.057...

• American put option

"no exact analytic formula for the value of an American put option on a non-dividend-paying stock has been produced" (Hull, 2014, p. 359)

So, the result when attempting to analytically price an American put option will be undefined.

const option = new Option({
  style: "american",
  type: "put",
  initialSpotPrice: 52,
  strikePrice: 50,
  timeToMaturity: 0.25,
  volatility: 0.3,
  riskFreeRate: 0.12,
});
const price = option.price("bs");
console.log(price);
// undefined

Binomial tree

• American put option

Hull SSM (2014): Problem 13.17, page 142

S₀ = 1500, K = 1480, τ = 1, σ = 0.18, r = 0.04, q = 0.025, time steps = 2

const option = new Option({
  style: "american",
  type: "put",
  initialSpotPrice: 1500,
  strikePrice: 1480,
  timeToMaturity: 1,
  volatility: 0.18,
  riskFreeRate: 0.04,
  dividendYield: 0.025,
});
const price = option.price("bt", { timeSteps: 2 });
console.log(price);
// 78.413...

Monte Carlo simulation

• Compare to the exact European call option price of 5.057

S₀ = 52, K = 50, τ = 0.25, σ = 0.3, r = 0.12, q = 0

const option = new Option({
  style: "european",
  type: "call",
  initialSpotPrice: 52,
  strikePrice: 50,
  timeToMaturity: 0.25,
  volatility: 0.3,
  riskFreeRate: 0.12,
});
const params = {
  simulations: 10000,
  prngSeed: "123",
  prngAdvancePast: 15,
};
const sfc32 = option.price("mcs", {
  ...params,
  prngName: "sfc32",
});
console.log(sfc32);
// 5.073740074053896
const mulberry32 = option.price("mcs", {
  ...params,
  prngName: "mulberry32",
});
console.log(mulberry32);
// 5.007275545013275
const xoshiro128ss = option.price("mcs", {
  ...params,
  prngName: "xoshiro128ss",
});
console.log(xoshiro128ss);
// 5.021720720229929
const jsf32 = option.price("mcs", {
  ...params,
  prngName: "jsf32",
});
console.log(jsf32);
// 5.1499321549907

References

• Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654.
• Boyle, P.P. (1977) Options: A Monte Carlo Approach. Journal of Financial Economics, 4, 323-338.
• Cox, J.C., Ross, S.A., & Rubinstein, M. (1979). Options pricing: A simplified approach. Journal of Financial Economics, 7, 229-263.
• Hull (2014): Hull, J. (2014) Options, Futures and Other Derivatives. 9th Edition, Prentice Hall, Upper Saddle River.
• Hull SSM (2014): Hull, J. (2014) Student Solutions Manual for Options, Futures, and Other Derivatives. 9th Edition, Prentice Hall, Upper Saddle River.
• Longstaff, F., & Schwartz, E. (2001). Valuing American Options by Simulation: A Simple Least-Squares Approach. Review of Financial Studies. 14. 113-47.
• Merton, R. C. (1973). Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4(1), 141–183.
• Stack Overflow: Seeding the random number generator in Javascript