Maximum Likelihood Estimation Parameter Model Heston Berdasarkan Optimasi Limited-Memory BFGS dengan Kendala

Anam, Hanif Fauzul (2025) Maximum Likelihood Estimation Parameter Model Heston Berdasarkan Optimasi Limited-Memory BFGS dengan Kendala. Other thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Harga dan volatilitas aset dasar dari instrumen derivatif dimodelkan dalam model Heston sebagai proses stokastik bivariat yang berkorelasi. Model ini dikembangkan untuk mengatasi keterbatasan model-model penetapan harga opsi sebelumnya serta memungkinkan representasi dinamika pasar yang lebih realistis sehingga menjadi salah satu model yang paling banyak digunakan dalam penetapan harga opsi. Estimasi parameter model Heston dilakukan dalam penelitian ini melalui metode maximum likelihood estimation (MLE). Metode ini memunculkan permasalahan optimasi kompleks dengan kendala pada nilai parameter. Algoritma limited-memory BFGS (L-BFGS) yang telah disesuaikan untuk permasalahan optimasi berkendala digunakan untuk menyelesaikan masalah ini. Estimasi parameter dilakukan berdasarkan data historis nilai indeks Standard & Poor's 500 (S&P 500) dan Cboe Volatility Index (VIX). Nilai parameter yang terestimasi kemudian diterapkan pada rumus penetapan harga opsi beli indeks S&P 500. Harga teoretis yang dihasilkan dibandingkan dengan harga pasarnya guna mengevaluasi kinerja model di bawah asumsi premi risiko nol menggunakan metriks squared error (SE) dan root-mean-square error (RMSE). Penelitian ini menyajikan demonstrasi komprehensif mengenai penerapan metode estimasi parameter model Heston berbasis maximum likelihood dengan optimasi L-BFGS berkendala beserta aplikasinya dalam penetapan harga opsi beli Eropa. Hasil penelitian mencakup implementasi metode estimasi, perolehan estimasi parameter model Heston, analisis karakteristik konvergensi algoritma optimasi, serta evaluasi galat harga berdasarkan variasi indeks kesepakatan dan waktu jatuh tempo.
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The price and volatility of the underlying asset in derivative instruments are modeled in the Heston framework as a correlated bivariate stochastic process. This model was developed to address the limitations of earlier option pricing models and to enable a more realistic representation of market dynamics, making it one of the most widely used models in option pricing. In this study, the parameters of the Heston model are estimated using the maximum likelihood estimation (MLE) method. This approach gives rise to a complex constrained optimization problem due to the parameter restrictions required for financial interpretability. The problem is solved using a constrained variant of the limited-memory BFGS (L-BFGS) algorithm. Parameter estimation is performed based on historical data from the Standard & Poor’s 500 (S&P 500) index and the Cboe Volatility Index (VIX). The estimated parameters are then applied to the Heston characteristic function-based pricing formula to compute the theoretical prices of European call options on the S&P 500 Index. These theoretical prices are compared with market prices to evaluate the model's performance under the zero risk premium assumption using the squared error (SE) and root-mean-square error (RMSE) metrics. This research provides a comprehensive demonstration of the application of maximum likelihood-based Heston parameter estimation using constrained L-BFGS optimization and its use in European call option pricing. The findings of this research include the implementation of the estimation method, the resulting parameter estimates of the Heston model, an analysis of the convergence characteristics of the optimization algorithm, and an evaluation of pricing errors across different strike indices and maturities.

Item Type:	Thesis (Other)
Uncontrolled Keywords:	Model Heston, Volatilitas stokastik, Penetapan harga opsi, Fungsi karakteristik, MLE, L-BFGS, S&P 500, VIX, Heston model, Stochastic volatility, Option pricing, Characteristic function.
Subjects:	H Social Sciences > HG Finance > HG4012 Mathematical models
Divisions:	Faculty of Science and Data Analytics (SCIENTICS) > Mathematics > 44201-(S1) Undergraduate Thesis
Depositing User:	Hanif Fauzul Anam
Date Deposited:	30 Jul 2025 09:42
Last Modified:	30 Jul 2025 09:42
URI:	http://repository.its.ac.id/id/eprint/121234

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