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The Term Structure of Commodity Prices: A Mathematical and Computational Perspective

The term structure of commodity prices refers to the relationship between the prices of futures contracts for the same commodity with different maturities. This relationship is often depicted as a curve, with the price of a futures contract increasing as its maturity approaches. This is known as normal backwardation. However, in some cases, the price of a futures contract can decrease as its maturity approaches. This is known as contango.

The term structure of commodity prices is influenced by many factors, including:

  • The cost of storage and financing: Storing and financing a commodity can be costly, so hedgers who need to deliver a commodity at a future date are willing to pay a premium for futures contracts with longer maturities. This helps explain the normal backwardation often observed in commodity futures markets.
  • Expectations about future supply and demand: If market participants expect future supply to exceed future demand, they will be willing to sell futures contracts with longer maturities at a discount. This can lead to contango.
  • Speculative activity: Speculators can also play a role in shaping the term structure of commodity prices. For example, if speculators believe a commodity is underpriced, they may buy futures contracts with longer maturities in anticipation of future price increases. This can lead to normal backwardation.

The Mathematical Foundations of the Term Structure of Commodity Prices

The mathematics behind the term structure of commodity prices is relatively complex. However, a simplified version can be derived using the following assumptions:

  • The commodity is storable.
  • The cost of storage and financing is constant.
  • There is no risk of spoilage.
  • The market is in equilibrium, meaning no excess supply or demand exists for any futures contract.

Under these assumptions, the price of a futures contract with maturity T can be expressed as follows:

F_T = E_t[S_T] - K

where:

  • FT​ is the price of the futures contract with maturity T.
  • Et​[ST​] is the expected spot price of the commodity at maturity T, given the information available at time t.
  • K is the cost of storage and financing the commodity from time t to time T.

This equation states that the price of a futures contract is equal to the expected spot price of the commodity at maturity minus the cost of storage and financing.

C++ Code and Unit Tests for Commodity Futures Pricing

The following C++ code shows a simple implementation of the term structure of the commodity prices model described above:

term_structure_test.cpp

#include "gtest/gtest.h"
#include "term_structure.h"

class TermStructureTest : public testing::Test {
public:
  TermStructureTest() {}

protected:
  TermStructure term_structure_;

  void SetUp() override {
    this->term_structure_ = TermStructure(0.1);
  }

  void TearDown() override {}
};

TEST_F(TermStructureTest, GetFuturesPrice) {
  EXPECT_EQ(100, this->term_structure_.GetFuturesPrice(1));
  EXPECT_EQ(99.9, this->term_structure_.GetFuturesPrice(2));
  EXPECT_EQ(99.8, this->term_structure_.GetFuturesPrice(3));
}

To call the tests, we can use the following command:

g++ term_structure.cpp term_structure_test.cpp -o term_structure_test
./term_structure_test

term_structure.cpp

#include <vector>

class TermStructure {
public:
  TermStructure(double cost_of_storage_and_financing) {
    this->cost_of_storage_and_financing_ = cost_of_storage_and_financing;
    this->futures_prices_ = std::vector<double>(10000, 0.0);
  }

  double GetFuturesPrice(int maturity) const {
    if (maturity > this->futures_prices_.size()) {
      return 0.0;
    }

    return this->futures_prices_[maturity - 1];
  }

private:
  void CalculateFuturesPrices() {
    for (int i = 0; i < this->futures_prices_.size(); i++) {
      this->futures_prices_[i] = E_t[S_i] - this->cost_of_storage_and_financing_;
    }
  }

  double cost_of_storage_and_financing_;
  std::vector<double> futures_prices_;
};

int main() {
  TermStructure term_structure(0.1);

  // Calculate the futures prices for all maturities.
  term_structure.CalculateFuturesPrices();

  // Get the futures price for a specific maturity.
  double futures_price = term_structure.GetFuturesPrice(1);

  // Print the futures price to the console.
  std::cout << "Futures price: " << futures_price << std::endl;

  return 0;
}

To compile and run the program, we can use the following commands:

g++ term_structure.cpp -o term_structure
./term_structure

The term structure of commodity prices is important for understanding how commodity markets work. It can also be used to develop trading strategies. For example, a trader who believes the futures market is in contango may sell futures contracts with longer maturities and buy futures contracts with shorter maturities. This strategy is known as a roll yield strategy.

The term structure of commodity prices is a complex topic, but it is important for anyone who wants to understand commodity markets. Market participants can make more informed decisions about their trading activities by understanding the factors that influence the term structure.

Ali Kayani

https://www.linkedin.com/in/ali-kayani-silvercoder007/

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