• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.4 no.2
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• pp.77-84
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• 2000

Black-Scholes equation arising from option pricing in the presence of cost in trading the underlying asset is derived. The transaction cost is chosen precisely and generalized to reflect the trade in the real world. Furthermore the concept of the bandwidth is introduced to obtain the better rehedging. The model with bandwidth derived in this paper can be used to calculate the more accurate option price numerically even if it is nonlinear and more complicated than the models shown before.

• The Korean Journal of Financial Management
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• v.24 no.4
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• pp.1-43
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• 2007

The traditional Black-Scholes model for option pricing is based on the assumption that the log-return of the underlying asset follows a Brownian motion. But this assumption has been criticized for being unrealistic. Thus, for the last 20 years, many attempts have been made to adopt different stochastic processes to derive new option pricing models. The option pricing models based on L$\acute{e}$vy processes are being actively studied originating from the Gerber-Shiu model driven by H. U. Gerber and E. S. W. Shiu in 1994. In 2004, G. H. L. Cheang derived an option pricing model under multiple L$\acute{e}$vy processes, enabling us to adopt drift and jumps to the Gerber-Shiu model, while Gerber and Shiu derived their model under one L$\acute{e}$vy process. We derive the Gerber-Shiu model which includes drift and jumps under L$\acute{e}$vy processes. By adopting a Gamma distribution, we expand the Heston model which was driven in 1993 to include jumps. Then, using KOSPI200 index option data, we analyze the price-fitting performance of our model compared to that of the Black-Scholes model. It shows that our model shows a better price-fitting performance.

• Journal of the Korean Society for Railway
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• v.10 no.2 s.39
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• pp.137-145
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• 2007

In traditional financial theory, the discount cash flow model(DCF or NPV) operates as the basic framework for most analyses. In doing valuation analysis, the conventional view is that the net present value(NPV) of a project is the measure of the present value of expected net cash flows. Thus, investing in a positive(negative) NPV project will increase(decrease) firm value. Recently, this framework has come under some fire for failing to consider the options of the managerial flexibilities. Real option valuation(ROV) considers the managerial flexibility to make ongoing decisions regarding the implementation of investment projects and the deployment of real assets. The appeal of the framework is natural given the high degree of uncertainty that firms face in their technology investment decisions. This paper suggests an algorithm for estimating volatility of logarithmic cash flow returns of real assets based on the Black-Sholes option pricing model, the binomial option pricing model, and the Monte Carlo simulation. This paper uses those models to obtain point estimates of real option value with the G7- HSR350X(high-speed train).

• The Pure and Applied Mathematics
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• v.27 no.1
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• pp.43-50
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• 2020

We investigate the domain of influence of the local volatility function on the solutions of the general Black-Scholes model. First, we generate the sample paths of underlying asset using the Monte Carlo simulation. Next, we define the inner and outer domains to find the effective volatility region. To confirm the effect of the inner domain, we use the root mean square error for the European call option prices, and then change the values of volatility in the proposed domain. The computational experiments confirm that there is an effective region which dominates the option pricing.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.209-227
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• 2009

We examine a unified approach of calculating the closed form solutions of option price under stochastic volatility models using stochastic calculus and the Fourier inversion formula. In particular, we review and derive the option pricing formulas under Heston and correlated Stein-Stein models using a systematic and comprehensive approach which were derived individually earlier. We compare the empirical performances of the two stochastic volatility models and the Black-Scholes model in pricing KOSPI 200 index options.

• Bulletin of the Korean Mathematical Society
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• v.48 no.4
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• pp.791-812
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• 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
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• v.18 no.1
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• pp.61-74
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• 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.

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

• Journal of Korean Society of Transportation
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• v.35 no.4
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• pp.360-373
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• 2017

The BTO-a projects is the types, which has a demand risk among the type of PPP projects in Korea. When demand risk is realized, private investor encounters financial difficulties due to lower revenue than its expectation and the government may also have a problem in stable infrastructure operation. In this regards, the government has applied various risk sharing policies in response to demand risk. However, the amount of government's risk sharing is the government's contingent liabilities as a result of demand uncertainty, and it fails to be quantified by the conventional NPV method of expressing in the text of the concession agreement. The purpose of this study is to estimate the value of investment risk sharing by the government considering the demand risk in the profit sharing system (BTO-a) introduced in 2015 as one of the demand risk sharing policy. The investment risk sharing will take the form of options in finance. Private investors have the right to claim subsidies from the government when their revenue declines, while the government has the obligation to pay subsidies under certain conditions. In this study, we have established a methodology for estimating the value of investment risk sharing by using the Black - Scholes option pricing model and examined the appropriateness of the results through case studies. As a result of the analysis, the value of investment risk sharing is estimated to be 12 billion won, which is about 4% of the investment cost of the private investment. In other words, it can be seen that the government will invest 12 billion won in financial support by sharing the investment risk. The option value when assuming the traffic volume risk as a random variable from the case studies is derived as an average of 12.2 billion won and a standard deviation of 3.67 billion won. As a result of the cumulative distribution, the option value of the 90% probability interval will be determined within the range of 6.9 to 18.8 billion won. The method proposed in this study is expected to help government and private investors understand the better risk analysis and economic value of better for investment risk sharing under the uncertainty of future demand.

• Journal of Korea Technology Innovation Society
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• v.20 no.3
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• pp.732-753
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• 2017

Recently, when evaluating the technology values in the fields of biotechnology, pharmaceuticals and medicine, we have needed more to estimate those values in consideration of the period and cost for the commercialization to be put into in future. The existing discounted cash flow (DCF) method has limitations in that it can not consider consecutive investment or does not reflect the probabilistic property of commercialized input cost of technology-applied products. However, since the value of technology and investment should be considered as opportunity value and the information of decision-making for resource allocation should be taken into account, it is regarded desirable to apply the concept of real options, and in order to reflect the characteristics of business model for the target technology into the concept of volatility in terms of stock price which we usually apply to in evaluation of a firm's value, we need to consider 'the continuity of stock price (relatively minor change)' and 'positive condition'. Thus, as discussed in a lot of literature, it is necessary to investigate the relationship among volatility, underlying asset values, and cost of commercialization in the Black-Scholes model for estimating the technology value based on real options. This study is expected to provide more elaborated real options model, by mathematically deriving whether the ratio of the present value of the underlying asset to the present value of the commercialization cost, which reflects the uncertainty in the option pricing model (OPM), is divided into the "no action taken" (NAT) area under certain threshold conditions or not, and also presenting the estimation logic for option values according to the observation variables (or input values).