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"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 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."

"Infeksi HIV disebabkan oleh interaksi antara virus HIV dengan sel T CD4+ di dalam tubuh manusia. Infeksi HIV yang paling banyak terjadi disebabkan oleh virus HIV-1. Walaupun belum bisa disembuhkan, terdapat terapi efektif untuk pasien terinfeksi HIV, yaitu terapi HAART. Interaksi antara virus HIV-1 dengan sel T CD4+ yang dipengaruhi oleh terapi HAART dapat dimodelkan untuk mengetahui dinamika virus HIV-1 dan sel T CD4+. Skripsi ini membahas model stokastik dinamika HIV-1 dengan terapi HAART. Model stokastik tersebut dimodifikasi dari model deterministik dinamika HIV-1 dengan terapi HAART melalui pemberian gangguan pada parameter. Metode Euler-Maruyama diimplementasikan pada model stokastik dinamika HIV-1 dengan terapi HAART untuk mengaproksimasi solusi model. Analisis solusi aproksimasi dilakukan dengan melihat pengaruh terapi HAART pada model, intensitas noise dan bilangan reproduksi dasar 𝑅0. Dengan digunakan terapi HAART, konsentrasi sel terinfeksi dan partikel virus pada model deterministik mengalami penurunan sampai mencapai titik kesetimbangan bebas penyakit. Pada model stokastik, konsentrasi sel tidak terinfeksi bervariasi di sekitar nilai 𝜆/𝛿, sedangkan konsentrasi sel terinfeksi dan partikel virus menuju nilai 0 secara eksponensial. Intensitas noise mempengaruhi konsentrasi sel yang tidak terinfeksi, sel terinfeksi dan partikel virus dengan efek yang beragam.

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HIV infection is caused by the interaction between the HIV virus and CD4+ T cells in the human body. HIV infection occurs most commonly caused by HIV-1. The HAART therapy is an effective therapy for HIV-infected patients. The model of interaction between the HIV-1 virus and CD4+ T cells that are affected by HAART therapy, is able to determine the dynamics of HIV-1 and CD4+ T cells. This skripsi discusses the stochastic model of the HIV-1 dynamics with HAART therapy. The stochastic model is obtained from the deterministic model of HIV-1 dynamics with HAART therapy by adding perturbation parameter. The Euler- Maruyama method is used to approximate the model solution. The approximate solutions are used to observe the effect of HAART therapy, intensity noise and the basic reproductive number 𝑅0. The HAART therapy shows that the concentration of infected cells and virus particles in deterministic model decreased until it reaches the disease-free equilibrium. In stochastic model, it shows that the concentration of uninfected cells tends to a value determined by 𝜆/𝛿, meanwhile the concentration of infected cells and virus particles tend to 0 exponentially. The intensity noise affects the concentration of uninfected cells, infected cells and virus particles with various effects."

"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."

"[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.]"

"Perhitungan anuitas kontingensi merupakan salah satu komponen penting yang digunakan dalam perhitungan premi di dunia asuransi. Dalam menghitung anuitas, tingkat bunga seringkali diasumsikan konstan. Sedangkan, pada kenyataannya, tingkat bunga cenderung berubah-ubah dalam waktu yang tidak menentu dalam kontrak asuransi jiwa yang umumnya memiliki periode cukup panjang. Oleh karena itu, diperlukan model tingkat bunga stokastik yang dapat menjelaskan randomness atau perilaku keacakan dari perubahan tingkat bunga. Hal ini bertujuan agar perhitungan anuitas kontingensi dapat digambarkan dengan lebih realistis yaitu sesuai dengan perilaku tingkat bunga dalam kehidupan nyata yang fluktuatif. Pada penelitian ini, akan dibangun kelas model tingkat bunga stokastik baru dengan memodelkan force of interest berdasarkan proses compound Poisson secara langsung. Proses compound Poisson yang digunakan dapat menjelaskan random jumps yang terjadi pada tingkat bunga stokastik. Pada penelitian ini ditelaah pembentukan force of interest berdasarkan proses compound Poisson, menelaah bentuk perumusan nilai sekarang, menganalisis fungsi akumulasi force of interest tingkat bunga stokastik, dan menelaah bentuk perumusan Actuarial Present Value (APV) dari anuitas kontingensi yang bersifat diskrit maupun kontinu. Seletah itu, dilakukan ilustrasi perhitungan anuitas kontingensi berdasarkan model tingkat bunga stokastik yang telah dibentuk.

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The calculation of contingency annuities is one of the important components used in calculating premiums in the insurance world. In calculating annuities, the interest rate is often assumed to be constant. Meanwhile, in reality, interest rates tend to fluctuate in an uncertain time in life insurance contracts which generally have a fairly long period. Therefore, we need a stochastic interest rate model that can explain the randomness or random behavior of interest rate changes. It is intended that the calculation of the contingency annuity can be described more realistically, namely in accordance with the fluctuating behavior of interest rates in real life. In this research, a new stochastic interest rate model class be built by modeling the force of interest based on the direct compound Poisson process. The compound Poisson process used can explain the random jumps that occur at the stochastic interest rate. This research examines the formation of force of interest based on the compound Poisson process, examines the form of the present value formulation, analyzes the function of the accumulation of force of interest stochastic interest rates, and examines the form of the formulation of Actuarial Present Value (APV) of discrete or continuous contingency annuities. After that, an illustration of the contingency annuity calculation is carried out based on the stochastic interest rate model that has been formed."

"ABSTRAKPenentuan harga opsi sering dimodelkan menggunakan persamaan Black-Scholes dimana harga aset pada persamaan Black-Scholes dirumuskan dengan gerak Geometrik-Brownian. Namun gerak Geometrik-Brownian sering tidak konsisten terhadap harga pasar aktual karena tidak ada pengelompokan rezim dalam modelnya constant return rate . Model threshold autoregressive diadaptasi pada gerak Geometrik-Brownian sehingga parameter dari gerak Geometrik-Brownian berganti-ganti setiap terjadi regime-switching. Regime-switching ditandai dengan pergerakan force of interest dari harga aset yang mengikuti gerak Brownian. Asumsi pasar tidak lengkap menyebabkan ada tak hingga satuan ukur risk-neutral. Satuan ukur risk-neutral yang diinginkan, didapatkan menggunakan metode transformasi Esscher dan minimal entropy martingale measure MEMM . Pada akhirnya harga opsi dapat dihitung menggunakan satuan ukur risk-neutral yang telah didapatkan.

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ABSTRACTOption pricing is often modeled using the Black Scholes equation where the asset price of the Black Scholes equation is formulated by Geometric Brownian motion. But Geometric Brownian motion is often inconsistent with market prices because there is no regime grouping in the model constant return rate . The threshold autoregressive model is adapted to Geometric Brownian motion so that the parameters of Geometric Brownian motion will alternate between regimes. Regime switching are detected by the movement of force of interest from the price of the underlying assets. The market assumption is incomplete causing there are infinite existences of risk neutral measure. The desired risk neutral measure, obtained using the Esscher transformation method and a minimum entropy martingale measure MEMM . In the end, the option price can be calculated using the risk neutral measure that already obtained."

"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."

"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."

"Pada kondisi tertentu, harga saham dapat mengalami fluktuasi yang cukup tajam (lompatan). Jika model harga saham tidak memperhatikan kemungkinan terjadinya lompatan, prediksi harga saham kurang dapat mencerminkan kondisi yang sebenarnya. Karena itu dibutuhkan model Jump-Diffusion (JD) yang dapat menangkap lompatan tersebut. Salah satu model JD adalah model Pareto-Beta Jump- Diffusion (PBJD). Model yang diusulkan oleh C.A Ramezani dan Y. Zeng (1998) ini merupakan perluasan model Merton Jump-Diffusion (MJD) (1976). Waktu muncul lompatan ke atas dan ke bawah masing-masing direpresentasikan oleh suatu proses Poisson, sedangkan besar lompatan ke atas berdistribusi Pareto(ηu), dan besar lompatan ke bawah berdistribusi Beta(ηd,1). Model PBJD memiliki koefisien difusi s konstan yang menyatakan volatilitas model. Pada tesis ini volatilitas konstan s diganti menjadi volatilitas stokastik mengikuti model Heston (1993). Model PBJD dengan volatilitas stokastik selanjutnya disebut sebagai model PBJDVS, dan model PBJDVS ini berbentuk sebuah sistem Persamaan Diferensial Stokastik (PDS). Tesis ini membahas bagaimana menentukan solusi analitik model PBJDVS. Berdasarkan solusi analitik tersebut, ditentukan probability density function (pdf) log-return saham satu periode model PBJDVS. Selain itu, berdasarkan solusi analitik, dilakukan simulasi lintasan (sample path) harga saham dan simulasi bentuk pendekatan kurva pdf log-return saham satu periode model PBJDVS. Hasil simulasi lintasan harga saham model PBJDVS dengan parameter tertentu dapat menunjukkan adanya lompatan.

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In certain conditions, stock price can fluctuate highly (jump). If a stock price model ignores the possiblity of jump, the stock price prediction can?t adequately reflect the real condition. That is why Jump-Diffusion (JD) model that can catch the jump is needed. One of the JD models is Pareto-Beta Jump-Diffusion (PBJD) model. The model, proposed by C.A Ramezani and Y. Zeng (1998), is an extension of Merton Jump-Diffusion (MJD) (1976) model. Each of the up and down jump occurence times are represented by a Poisson process, while the up-jump magnitudes are distributed Pareto(ηu), and the down-jump magnitudes are distributed Beta(ηd,1). PBJD model has constant diffusion coeficient s that means volatility of the model. In this thesis the constant volatility s is replaced by a stochastic volatility following Heston model (1993). PBJD model with stochastic volatility will be called PBJDVS model, and this PBJDVS model has the form of Stochastic Differential Equation (SDE) system. This thesis discusses how to determine analytic solution of PBJDVS model. Based on this analytic solution, probability density function (pdf) for one-period stock log-returns of PBJDVS model is determined. In addition, based on analytic solution, sample path of stock price and approximate curve of pdf for one-period stock log-returns of PBJDVS model are simulated. The simulation of sample path of stock price with certain parameters can show jump."