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"[Model Black-Scholes merupakan model pertama penentuan harga opsi. Terdapat asumsi-asumsi yang harus dipenuhi pada model Black-Scholes, salah satunya volatilitas yang konstan. Karena asumsi tersebut, maka nilai implied volatility berdasarkan model Black-Scholes akan sama untuk setiap harga opsi. Implied volatilty dipengaruhi oleh harga strike dan waktu jatuh tempo. Namun, pada skripsi ini, implied volatility dibatasi pada pengaruh harga strike saja dan hubungan antara implied volatility dengan harga strike diinterpretasikan dalam kurva smile. Bentuk kurva smile berbeda-beda tergantung pada data observasi nilai opsi di pasar dan bentuknya seperti senyum (smile), skew, atau smirk. Dengan mempelajari kurva smile, seorang investor dapat mempertimbangkan risiko berinvestasi opsi. Pada skripsi ini dibahas bagaimana cara menentukan implied volatility Heston yang diinterpretasikan dalam kurva smile. Untuk dapat menentukan implied volatility Heston, diperlukan harga opsi Heston yang disubstitusi ke model harga opsi Black-Scholes. Untuk memperoleh harga opsi Heston, dilakukan penurunan harga opsi saham Heston berdasarkan model pergerakan harga saham Heston. Kemudian, dengan menghitung beberapa nilai implied volatility Heston yang diperoleh dengan menggunakan harga strike yang berbeda, dapat dibentuk kurva smile Heston. Hasil analisis kurva smile dari implied volatility Heston menggunakan data Anglo American Shares dengan selang harga strike dan tingkat bunga bebas risiko yang berbeda serta waktu jatuh tempo yang tetap adalah sebuah kurva smile yang berbentuk smirk.

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Black-Scholes model is the first option pricing model. There are some assumptions that need to be satisfied in Black-Scholes model, one of them is the constant volatility. Because of that assumption, implied volatility from Black-Scholes model will be same for all option price. Implied volatility depends on strike price and time to maturity. However, in this skripsi, implied volatility is bounded by strike price only and the relation between implied volatility and strike price is interpreted in smile curve. The shapes of smile curve is vary through observed option price effect and its shape looks like smile, skew, or smirk. With studying smile curve, an investor can consider the risk of investing an option. This skripsi will study how to determine Heston implied volatility which is interpreted in smile curve. Heston option price which is substituted to Black-Schole model is needed to determine Heston implied volatility. For that purpose, deriving Heston option pricing model based on Heston stock price model is needed to be done. Then, by calculating some of implied volatilities that have different strike price, smile curve can be made. The analysis result of Anglo American Shares data with different in Strike Price interval and risk-free rates but same in maturity time (1 year) is a smirk shaped smile curve.;Black-Scholes model is the first option pricing model. There are some assumptions that need to be satisfied in Black-Scholes model, one of them is the constant volatility. Because of that assumption, implied volatility from Black-Scholes model will be same for all option price. Implied volatility depends on strike price and time to maturity. However, in this skripsi, implied volatility is bounded by strike price only and the relation between implied volatility and strike price is interpreted in smile curve. The shapes of smile curve is vary through observed option price effect and its shape looks like smile, skew, or smirk. With studying smile curve, an investor can consider the risk of investing an option. This skripsi will study how to determine Heston implied volatility which is interpreted in smile curve. Heston option price which is substituted to Black-Schole model is needed to determine Heston implied volatility. For that purpose, deriving Heston option pricing model based on Heston stock price model is needed to be done. Then, by calculating some of implied volatilities that have different strike price, smile curve can be made. The analysis result of Anglo American Shares data with different in Strike Price interval and risk-free rates but same in maturity time (1 year) is a smirk shaped smile curve.;Black-Scholes model is the first option pricing model. There are some assumptions that need to be satisfied in Black-Scholes model, one of them is the constant volatility. Because of that assumption, implied volatility from Black-Scholes model will be same for all option price. Implied volatility depends on strike price and time to maturity. However, in this skripsi, implied volatility is bounded by strike price only and the relation between implied volatility and strike price is interpreted in smile curve. The shapes of smile curve is vary through observed option price effect and its shape looks like smile, skew, or smirk. With studying smile curve, an investor can consider the risk of investing an option. This skripsi will study how to determine Heston implied volatility which is interpreted in smile curve. Heston option price which is substituted to Black-Schole model is needed to determine Heston implied volatility. For that purpose, deriving Heston option pricing model based on Heston stock price model is needed to be done. Then, by calculating some of implied volatilities that have different strike price, smile curve can be made. The analysis result of Anglo American Shares data with different in Strike Price interval and risk-free rates but same in maturity time (1 year) is a smirk shaped smile curve., Black-Scholes model is the first option pricing model. There are some assumptions that need to be satisfied in Black-Scholes model, one of them is the constant volatility. Because of that assumption, implied volatility from Black-Scholes model will be same for all option price. Implied volatility depends on strike price and time to maturity. However, in this skripsi, implied volatility is bounded by strike price only and the relation between implied volatility and strike price is interpreted in smile curve. The shapes of smile curve is vary through observed option price effect and its shape looks like smile, skew, or smirk. With studying smile curve, an investor can consider the risk of investing an option. This skripsi will study how to determine Heston implied volatility which is interpreted in smile curve. Heston option price which is substituted to Black-Schole model is needed to determine Heston implied volatility. For that purpose, deriving Heston option pricing model based on Heston stock price model is needed to be done. Then, by calculating some of implied volatilities that have different strike price, smile curve can be made. The analysis result of Anglo American Shares data with different in Strike Price interval and risk-free rates but same in maturity time (1 year) is a smirk shaped smile curve.]"

"Model Heston merupakan salah satu model yang sangat populer untuk menghitung harga opsi. Namun, keakuratan model tersebut sangat bergantung pada parameter model yang digunakan. Oleh karena itu, pemilihan model parameter sama pentingnya dengan model itu sendiri. Salah satu cara untuk menemukan parameter model Heston terbaik adalah dengan cara meminimumkan fungsi eror antara hara opsi model dengan harga opsi yang berlaku di pasar. Cara seperti ini disebut kalibrasi. Implementasi kalibrasi model Heston dengan algoritma differential evolution (DE) dapat dilakukan dengan enam langkah. Langkah pertama, yaitu menentukan data harga opsi yang digunakan. Langkah-langkah selanjutnya yaitu menentukan metode perhitungan model Heston, fungsi eror, variasi dan parameter kontrol DE, serta kondisi terminasinya. Langkah terakhir, DE diimplementasikan untuk mendapatkan parameter model. Hasil simulasi lima puluh kali kalibrasi pada data harga opsi artifisial menunjukan DE telah cukup baik dalam mengkalibrasi empat dari lima jenis data harga opsi yang digunakan. Lebih jauh lagi, kalibrasi menggunakan lima puluh data harga opsi saham Apple Inc juga memberikan hasil yang cukup baik.

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The Heston Model is one of the most popular model for option pricing. Yet, its accuracy is highly depend on choosing model parameters. Thus, choosing model parameters is important as the model itself. One way to choose the best model parameters is minimizing eror function between the model price and the market price. Such a way is called calibration. Calibrating Heston model with differential evolution (DE) algorithm can be implemented in six steps. First, decide the option price data used for calibration. Then, choose a method for evaluating option price by Heston Model, error function for calibration, variation and control parameter for DE, Also terminating condition of the algorithm. The last, Implement DE to get pameters of the model. The result of fifty times calibration with DE was good enough in four of five artifisial data used. Moreover, calibration using fifty option price of The Apple Inc data also show a good result."

"Investor perlu memiliki strategi dalam menentukan harga opsi wajar untuk sebuah opsi. Salah satu strategi yang dapat digunakan adalah mempelajari model harga opsi Heston. Dalam model harga opsi diperlukan nilai-nilai parameter yang harus ditentukan terlebih dahulu melalui kalibrasi. Kalibrasi dapat dipandang sebagai masalah optimasi nonlinear, yakni dengan meminimumkan nilai fungsi objektif yang terkait. Algoritma Particle Swarm Optimization merupakan salah satu metode iteratif yang dapat digunakan dalam menentukan solusi masalah optimasi nonlinear. Selanjutnya hasil kalibrasi digunakan untuk menentukan harga wajar sebuah opsi. Data yang digunakan dalam penelitian ini adalah data 50 harga opsi pasar saham Apple Inc (AAPL). Berdasarkan hasil implementasi yang dilakukan, algoritma Particle Swarm Optimization menunjukkan kinerja yang cukup baik dalam aproksimasi nilai parameter model harga Opsi Heston.

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Investors should have a strategy to determine a fair price for an option. One of the strategy that can be applied is by studing the Heston option pricing model. In the option pricing model, there are some required parameter values that should be determined by using the calibration. The calibration can be considered as a nonlinear optimization problem by minimizing the value of a related objective function. Particle swarm optimization algorithm is one of iterative methods that can be used in the calibration of model?s parameters. Furthermore, the results of calibration can be used to determine the price of an option. The data used in this research is consist of 50 stock market option prices of Apple Inc. Based on the results the implementation, particle swarm optimization algorithm shows a good performance."

"Dalam dunia investasi dikenal sekuritas derivatif, yaitu alat keuangan yang nilainya bergantung pada alat keuangan lain yang tercantum di dalamnya. Salah satu contoh sekuritas derivatif adalah opsi call Eropa. Untuk menghitung nilai opsi call Eropa digunakan formula Black-Scholes. Semua parameter dalam formula Black-Scholes, yaitu harga aset dasar saat ini, suku bunga bebas resiko, harga patokan, dan waktu jatuh tempo dapat diketahui secara langsung, kecuali volatilitas. Dalam skripsi ini akan dibahas bagaimana mengestimasi volatilitas dengan menggunakan implied volatility, yaitu nilai volatilitas yang di dapat dengan menggunakan bantuan nilai opsi yang didapat dari pasar. Estimasi implied volatility dengan metode numerik, yaitu metode Bisection dan metode Newton-Raphson. "

"Penentuan harga opsi (option pricing) memegang peranan penting pada perdagangan saham agar dapat membuat keputusan yang dapat memperoleh keuntungan yang optimal baik untuk pembeli maupun penjual opsi. Salah satu model pasar yang dapat digunakan pada option pricing ini adalah model Black-Scholes dengan volatilitas stokastik dari harga saham yang berdasarkan proses Ornstein-Uhlenbeck. Model ini digunakan agar dapat menggambarkan sifat dari volatilitas yang ada pada pasar saham sesungguhnya. Untuk mengaproksimasi harga opsi call Eropa berdasarkan model tersebut, digunakan metode Euler-Maruyama. Diteliti juga laju konvergensi dari aproksimasi tersebut. Kemudian, dilakukan analisis terhadap hasil simulasi harga opsi menggunakan beberapa fungsi volatilitas harga saham yang berdasarkan proses Ornstein-Uhlenbeck. Hasil simulasi menunjukkan bahwa pemilihan fungsi volatilitas pada model pasar perlu dipertimbangkan lebih lanjut karena berkaitan dengan konsep mean-reversion yang diharapkan dari volatilitas pasar saham di dunia nyata.

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Option pricing holds a crucial role in trading to make decision that would lead to the best benefit for both the option buyer and seller. The market model that could be used for option pricing is Black-Scholes model with stochastic stock prices volatility driven by Ornstein-Uhlenbeck process. This model is used in order to reflect the properties of the volatility in the real market. In this short thesis, Euler-Maruyama method is used to approximate the price of the European call option based on that model. The rate of convergence of the approximation is also determined. The simulation of the option price approximation is performed with some Ornstein-Uhlenbeck-driven volatility functions for the stock price model. The result of the simulation shows that the choice of the volatility function for the stock price model needs to be scrutinized since it is related to the mean-reversion concept that is expected from the stock prices volatility in real market."

"Skripsi ini menganalisis interest rate, stock returns, dan implied volatility terhadap Credit Default Swap (CDS) di Indonesia. Penelitian ini bertujuan untuk mengetahui pengaruh interest rate, stock returns, dan implied volatility sebagai determinan terhadap Credit Default Swap (CDS) di Indonesia. Sebagai proxy untuk interest rate menggunakan yield SUN dengan tenor 10 tahun, stock returns menggunakan return IHSG dan implied volatility menggunakan vstoxx index.Penelitian ini adalah penelitian kuantitatif dengan desain eksplanatif dan menggunakan program statistik SPSS versi 17. Pengumpulan data sebagian besar dilakukan dengan mengambil data dari Bank Indonesia.Hasil penelitian ini membuktikan bahwa terdapat pengaruh signifikan dari interest rate, stock returns, dan implied volatility terhadap CDS spreads di Indonesia.

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This research analyze interest rate, stock returns, and implied volatility to credit default swap (CDS) in Indonesia. The aim of this research, is to get the conclusion about the influence of interest rate, stock returns, and implied volatility as the determinants to CDS in Indonesia. As a proxy, for interest rate use government bond (SUN) yields with maturities ten years, stock returns use IHSG returns, and implied volatility use vstoxx index.

This research was quantitative with design explanative and using software SPSS version 17. Data were collected from Bank Indonesia.

The result of this research proved that there was significant influence of interest rate, stock returns, and implied volatility to credit default swap (CDS) spreads in Indonesia."

"Tugas akhir ini mengkaji harga opsi put Eropa dengan aset dasar zero-coupon bond dan tingkat bunga diasumsikan mengikuti model Vasicek dengan jump. Kajian dilakukan dengan mengkonstruksi kembali persamaan harga opsi put Eropa tersebut. Ukuran jump didefinisikan mengikuti distribusi mixed-exponential. Dengan memanfaatkan infinitesimal generator dan konsep martingale dapat dikonstruksi transformasi Laplace dari distribusi model Vasicek dengan jump. Kemudian, dengan menggunakan hasil transformasi Laplace dari distribusi model Vasicek dengan jump dan konsep equivalent martingale measure dapat dikonstruksi persamaan harga opsi put Eropa dengan aset dasar zero-coupon bond.

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This undergraduate thesis examines the price of European put options with underlying asset zero-coupon bond and the interest rate following the Vasicek model with jump. The study was conducted by reconstructing the European put option pricing equation. The jump size is defined following a mixed-exponential distribution. By utilizing infinitesimal generators and martingale concepts, Laplace transform is constructed for the distribution of the Vasicek model with jump. Then, using the results of the Laplace transform for the distribution of the Vasicek model with jump and the concept of equivalent martingale measure, the European put option pricing equation with underlying asset zero-coupon bond is constructed."

"Skripsi ini menggunakan model market Black-Scholes yang dimodifikasi dengan volatilitas stokastik yang dipengaruhi oleh proses Ornstein-Uhlenbeck untuk menentukan harga European option, baik call option maupun put option. Model dikonstruksi dari kasus umum sampai kasus khusus, yaitu harga aset dan volatilitas adalah proses yang tidak saling berkorelasi. Solusi analitik dari harga European option diturunkan untuk kasus khusus dari model market yang dilengkapi minimal martingale measure dengan menggunakan inverse transformasi bilateral Laplace. Eksistensi dan uniqueness dari inverse transformasi bilateral Laplace dari fungsi probabilitas dianalisis terlebih dahulu sebelum menggunakan transformasi integral tersebut untuk menurunkan solusi analitik. Skripsi ini juga membahas bentuk alternatif dari solusi analitik harga European option dengan menggunakan inverse alternatif Post-Widder.

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This undergraduate thesis consider the modified Black-Scholes model of financial market with stochastic volatility driven by Ornstein-Uhlenbeck process to price a European option, both call option and put option. The model is constructed from general case to special case, in which asset price and volatility are uncorrelated process. The analytic solution of European option price formula is derived for the special case of the market with respect to the minimal martingale measure using inverse bilateral Laplace transform. Existence and uniqueness of inverse bilateral Laplace transform with respect to probability function will be analyzed before using the integral transform to derive the analytic solution. This undergraduate thesis also provides an alternative form of analytic solusion of the European option price formula using Post-Widder inversion formula."

"ABSTRACTPenentuan harga opsi penting untuk meningkatkan pendapatan untuk keperluan investasi. Pada skripsi ini, metode dimension reduction Monte Carlo diimplementasikan pada model six-factor cross currency untuk menentukan harga opsi Eropa. Model six-factor cross currency merupakan suatu model berdimensi tinggi yang biasa diselesaikan menggunakan metode Monte Carlo. Akan tetapi, metode Monte Carlo membutuhkan jumlah simulasi yang besar.  Dimension Reduction Monte Carlo merupakan suatu pendekatan yang digunakan untuk mereduksi jumlah dimensi pada model berdimensi tinggi secara one-way coupling. Metode tersebut dapat digunakan untuk mereduksi dimensi dari model six-factor cross currency dari 6 menjadi 1. Melalui pendekatan dimension reduction, hanya satu variabel yang perlu diaproksimasi. Pada kasus tersebut, dipilih variabel variansi dari spot foreign exchange rate dan nilainya diaproksimasi menggunakan metode Milstein.

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ABSTRACTOption pricing determination is important in order to increase profit for investment. In this thesis defend, the dimension reduction Monte Carlo method is implemented to determine put and call European option pricing under a six-factor cross currency model.  A six-factor cross currency model is a high-dimensional model which is usually solved using Monte Carlo. However, Monte Carlo requires hugh numbers of simulations. Dimension reduction Monte Carlo is an approach for reducing the dimension of high-dimensional models with one-way coupling. It can be applied to reduce the dimension of six-factor cross currency from 6 to 1. By the dimension reduction approach, only the factor that is conditioned on is needed to be approximated. In this case, the variance of spot foreign exchange rate is chosen as the factor that is conditioned on and its value is approximated using the Milstein method."

"Penelitian ini bertujuan untuk memberikan persepsi baru dalam pengukuran ketahanan pangan dengan menggunakan volatilitas harga pangan. Pendekatan data iklim digunakan untuk menganalisis pola volatilitas harga bersama dengan variabel sosial dan ekonomi. Proyeksi perubahan iklim dilakukan untuk melihat wilayah yang rentan dalam produksi pangan selanjutnya persepsi petani diukur untuk melihat kesiapan mereka untuk menghadapi perubahan iklim. Ditemukan pola adaptasi petani terhadap dampak perubahan iklim pada ketidakpastian harga pangan khususnya tingkat rumah tangga. Selanjutnya data dinamika iklim digunakan sebagai nilai input untuk pembuatan model prediksi volatilitas harga pangan. Dengan menggunakan metode statistik dan teknik jaringan syaraf tiruan, estimasi volatilias harga berhasil dihasilkan berserta dengan nilai prediksinya, luaran yang dihasilkan digunakan untuk mengoptimalkan sistem pangan yang lebih berkelanjutan.

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This study aims to provide a new perception of food security by using food price volatility to measure sustainability. The climate data approach is used to analyze patterns of price volatility along with social and economic variables. Climate change projections are carried out to look at areas that are vulnerable to food production, then farmers' perceptions are measured to see their readiness to face climate change. The pattern of farmers' adaptation to climate change on food price uncertainty, especially at the household level, was found. Furthermore, climate dynamics data are used as input values for prediction models of food price volatility. The Model using statistical methods and neural network techniques, the estimation of price volatility is successfully generated along with the predictive value, and the resulting output is used to optimize a more sustainable food system."