Spread option pricing

Introduction

A spread option is an option written on the difference of two underlying assets, with the following payoff function

$$ \max(S_2 - S_1 - K, 0) $$

where $S_2$ and $S_1$ are the two underlying risky assets, and $K$ is the strike of the Spread option. Spread option can be seen as one of the simplest generalization of the classical call and put options.

These options can be traded on an exchange, but the majority of the trading volume comes from over-the-counter transactions. They are popular because they are designed to hedge against adverse movements in several indexes. Due to their versatility, spread options are utilized across various markets, including fixed income, currency and foreign exchange, commodity futures, and energy markets.

C++ library

The following spread_pricing library expose some pricing function for price spread option with an European payoff and where the underlying are following a GBM. In particular the ExchangeOption and the GBMSpreadOption classes expose the pricing and greek calculation functionalities.

Build the library

In order to build the library you need install various dependencies

• autodiff library: needed for the Greeks calculation, leveraging automatic differentiation, for the margrabe option MargrabeOption class. But autodiff depends itself on other libraries:
  • Eigen3: lightweight C++ template library for linear algebra (sudo apt install libeigen3-dev)
  • Catch2
  • PyBind (sudo apt -y install python3-pybind11)
• googletest: used a testing framework, if it is not installed locally, it will be imported by Cmake using FetchContent

Once you have installed all the dependencies then to build the library locally, the workflow to follow is the following

git clone https://github.com/paolodelia99/spread-option-pricing
cd spread-option-pricing
source scripts/build.sh -bt <Release/Debug> -c <gcc/clang> -v <compiler_version>

While to run the test the run-tests.sh script in the scripts folder has been provided

source scripts/run-tests.sh -bt <Release/Debug> -c <gcc/clang> -v

Use the library

An example usage of the MargrabeOption class is the following:

#include <iostream>
#include "SpreadMarket.h"
#include "MargrabeOption.h"

int main() {
    auto s1 = new float(100);
    auto s2 = new float(100);
    auto t = new float(100);
    auto spd_data = std::make_shared<SpreadMarketData<float>>(SpreadMarketData(s1, s2, t));

    const float vol1 = 0.2;
    const float vol2 = 0.3;
    const float corr = 0.0;
    const float r = 0.0;

    MargrabeOption mrg_option(spd_data, vol1, vol2, r, corr);

    const float price = mrg_option.getSpreadPrice();
    std::cout << "Spread Option Price: " << price << std::endl;

    // Greeks
    // Deltas
    const auto [d_s1, d_s2] = mrg_option.getDeltas();
    std::cout << "Delta 1(dP/dS_1): " << d_s1 << std::endl;
    std::cout << "Delta 2(dP/dS_2): " << d_s2 << std::endl;

    //Gammas
    const auto [dd_s1, dd_s2] = mrg_option.getGammas();
    std::cout << "Gamma 1(d^2P/dS_1^2): " << dd_s1 << std::endl;
    std::cout << "Gamma 2(d^2P/dS_2^2): " << dd_s2 << std::endl;

    // Cross Gamma
    const float cross_gamma = mrg_option.getCrossGamma();
    std::cout << "Cross Gamma (d^2P/dS_1 dS_2): " << cross_gamma << std::endl;

    return 0;
}

Authors

D'Elia Paolo