• Bulletin of the Korean Mathematical Society
• /
• v.55 no.4
• /
• pp.1241-1261
• /
• 2018

We derive discrete time model of the geometric fractional Brownian motion. It provides numerical pricing scheme of financial derivatives when the market is driven by geometric fractional Brownian motion. With the convergence analysis, we guarantee the convergence of Monte Carlo simulations. The strong convergence rate of our scheme has order H which is Hurst parameter. To obtain our model we need to convert Wick product term of stochastic differential equation into Wick free discrete equation through Malliavin calculus but ours does not include Malliavin derivative term. Finally, we include several numerical experiments for the option pricing.

• Korean Journal of Mathematics
• /
• v.14 no.2
• /
• pp.185-202
• /
• 2006

In this paper, using the Black-Scholes analysis, we will derive the partial differential equation of convertible bonds with both non-stochastic and stochastic interest rate. We also find numerical solutions of convertible bonds equation with known interest rate using the finite element method.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.5
• /
• pp.1129-1141
• /
• 2019

We propose a stochastic delay financial model which describes influences driven by historical events. The underlying is modeled by stochastic delay differential equation (SDDE), and the delay effect is modeled by a stopping time in coefficient functions. While this model makes good economical sense, it is difficult to mathematically deal with this. Therefore, we circumvent this model with similar delay effects but mathematically more tractable, which is by the backward time integration. We derive the option pricing equation and provide the option price and the perfect hedging portfolio.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.4
• /
• pp.791-812
• /
• 2011

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European option can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up until now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These approaches typically provide numerical or approximate analytic methods to find the price and the boundary. Topics included in this survey are early approaches(trees, finite difference schemes, and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, analytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochastic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.26 no.2
• /
• pp.121-137
• /
• 2022

In this paper, we investigate an efficient hybrid method for solving two-asset time fractional Black-Scholes partial differential equations. The proposed method is based on the Crank-Nicolson the radial basis functions methods. We show that, this method is convergent and we obtain good approximations for solution of our problems. The numerical results show high accuracy of the proposed method without needing high computational cost.

• Journal of Korean Society of Industrial and Systems Engineering
• /
• v.22 no.51
• /
• pp.151-161
• /
• 1999

For many decades, Deterministic DCF approach has been widely used to evaluate investment opportunities. Under new manufacturing conditions involving uncertainty and risk, the DCF approach is not appropriate. In DCF, Risk is incorporated in two ways: certainty equivalent method, risk adjusted discount rate. This paper proposes a determination method of the Risk Adjusted Discount Rate for economically decision making advanced manufacturing technologies. Conventional DCF techniques typically use discount rate which do not consider the difference in risk of differential investment options and periods. Due to their relative efficiency, advanced manufacturing technologies have different degree of risk. The risk differential of investments is included using $\beta$ coefficient of capital asset pricing model. The comparison between existing and proposed method investigated. The DCF model using proposed risk adjusted discount rate enable more reasonable evaluation of advanced manufacturing technologies.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.14 no.4
• /
• pp.211-224
• /
• 2010

We define some asset models which are useful for portfolio construction in various terms of time. Our asset models are geometric jump-diffusions defined by the solutions of stochastic differential equations which are decomposed by various terms of time basically. We also can study pricing and hedging strategy of options in our models roughly.

• Journal of Korean Institute of Industrial Engineers
• /
• v.21 no.3
• /
• pp.299-311
• /
• 1995

Conventional discounted cash flow techniques fail to capture the risk associated with investments. This paper proposes an annual cash flow model that considers risk, cost structure and inventory liquidation in the evaluation of investment alternatives. The risk differential of investments is included using the capital asset pricing model while the stochastic version of the cost-volume-profit approach is used to consider inventory liquidation and cost structure. Tradeoffs between fixed and variable costs have been investigated, and portrayed using iso-cash flow curves. The proposed cash flow model has been developed, in particular, to enable an accurate evaluation of advanced manufacturing systems.

• The Korean Journal of Financial Management
• /
• v.8 no.2
• /
• pp.165-208
• /
• 1991

This paper develops an intertemporal general equilibrium model of asset pricing with taxes under the noisy and the incomplete information structure and examines theoretically the stochastic behavior of general equilibrium asset prices in a one-good, production, and exchange economy in continuous time markets. The important features of the model are its integration of real and financial markets and the analysis of the effects of differential tax rates between ordinary income and capital gains. The model developed here can provide answers to a wide variety of questions about stochastic structure of asset prices and the effect of tax on them.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.18 no.1
• /
• pp.61-74
• /
• 2014

This paper presents accurate and efficient numerical methods for calculating the sensitivities of two-asset European options, the Greeks. The Greeks are important financial instruments in management of economic value at risk due to changing market conditions. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a finite difference method and resulting discrete equations are solved by means of an operator splitting method. For Delta, Gamma, and Theta, we investigate the effect of high-order discretizations. For Rho and Vega, we develop an accurate and robust automatic algorithm for finding an optimal value. A cash-or-nothing option is taken to demonstrate the performance of the proposed algorithm for calculating the Greeks. The results show that the new treatment gives automatic and robust calculations for the Greeks.