• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1497-1530
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• 2016

External barrier options are two-asset options with stochastic variables where the payoff depends on one underlying asset and the barrier depends on another state variable. The barrier state variable determines whether the option is knocked in or out when the value of the variable is above or below some prescribed barrier level. This paper derives the explicit analytic solution of the chained option with an external single or double barrier by utilizing the probabilistic methods - the reflection principle and the change of measure. Before we do this, we examine the closed-form solution of the external barrier option with a single or double-curved barrier using the methods of image and double Mellin transforms. The exact solution of the external barrier option price enables us to obtain the pricing formula of the chained option with the external barrier more easily.

• Communications of the Korean Mathematical Society
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• v.31 no.1
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• pp.199-207
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• 2016

A chained option is a barrier option activated in the event that the underlying asset price crosses barrier or barriers prior to maturity in a specified order. In this paper, we study the pricing of chained options with the quanto property called the "Quanto chained option". A quanto chained option is a chained option starting at time when the foreign exchange rate has the multiple crossing of specified barriers. We provide closed-form formulas for valuing the quanto chained options based on probabilistic approach.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1501-1509
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• 2011

In this study, assume that the stock price obeys the stochastic differential equation driven by mixed fractional Brownian motion, and the short rate follows the Vasicek model. Then, the Black-Scholes partial differential equation is held by using fractional Ito formula. Finally, the pricing formulae of the barrier option are obtained by partial differential equation theory. The results of Black-Scholes model are generalized.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.439-461
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• 2020

A step-up option is a newly developed financial instrument that simultaneously provides higher security and profitability. This paper introduces two step-up options: step-up type1 and step-up type2 options, and derives the option pricing formulas using the Laplace transform. We assume that the underlying equity price follows a regime-switching model that reflects the long-term maturity of these options. The option prices are calculated for the two types of funds, a pure stock fund composed of risky assets only and a mixed fund composed of stocks and bonds, to reflect possible variety in the fund underlying asset mix. The impact of changes in the model parameters on the option prices is analyzed. This paper provides information crucial to product developments.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.345-360
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• 2018

In finance, barrier options are options contracts with a payoff that depends on whether the price of the underlying asset hits a predetermined barrier level during the option's lifetime. Based on exotic options and random fluctuations of interest rates in the marketplace, we consider discount barrier options with a stochastic interest rate driven by the Hull-White process. This paper derives the closed-form solutions of the discount barrier option and the discount double barrier option using Mellin transform methods and the PDE (partial differential equation) method of images.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.15 no.1
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• pp.1-17
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• 2011

A new method for option pricing based on the trinomial tree method is introduced. The new method calculates the local average of option prices around a node at each time, instead of computing prices at each node of the trinomial tree. Local averaging has a smoothing effect to reduce oscillations of the tree method and to speed up the convergence. The option price and the hedging parameters are then obtained by the compact scheme and the Richardson extrapolation. Computational results for the valuation of European and American vanilla and barrier options show superiority of the proposed scheme to several existing tree methods.

• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.285-294
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• 2008

A new Monte Carlo method is presented to compute the prices of barrier options on stocks. The key idea of the new method is to use an exit probability and uniformly distributed random numbers in order to efficiently estimate the first hitting time of barriers. It is numerically shown that the first hitting time error of the new Monte Carlo method decreases much faster than that of standard Monte Carlo methods.

• Industrial Engineering and Management Systems
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• v.10 no.2
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• pp.170-177
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• 2011

We present an improved binomial method for pricing financial deriva-tives by using cell averages. After non-overlapping cells are introduced around each node in the binomial tree, the proposed method calculates cell averages of payoffs at expiry and then performs the backward valuation process. The price of the derivative and its hedging parameters such as Greeks on the valuation date are then computed using the compact scheme and Richardson extrapolation. The simulation results for European and American barrier options show that the pro-posed method gives much more accurate price and Greeks than other recent lattice methods with less computational effort.

• Proceedings of the Korean Statistical Society Conference
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• 2003.10a
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• pp.73-78
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• 2003

This presentation derives a distribution function of the terminal value and running maximum of two-dimensional Brownian motion {X(t) = (X$_1$(t), X$_2$(T))', t > 0}. One random variable of the joint distribution is the terminal time value of the Brownian motion {X$_1$(t), t > 0}. The other random variable is the partial-time running maximum of the Brownian motion {X$_2$(t), t > 0}. With this distribution function, this presentation also derives an explicit pricing formula for a barrier option whose monitoring period of the option starts at an arbitrary date and ends at another arbitrary date before maturity.

• Journal of the Korean Statistical Society
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• v.33 no.2
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• pp.245-254
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• 2004

This paper derives a distribution function of the terminal value and running maximum of two-dimensional Brownian motion {X($\tau$) = (X$_1$($\tau$), X$_2$ ($\tau$))', $\tau$ 〉0}. One random variable of the joint distribution is the terminal time value, X$_1$ (T). The other random variable is the maximum of the Brownian motion {X$_2$($\tau$), $\tau$〉} between time s and time t. With this distribution function, this paper also derives an explicit pricing formula for an outside barrier option whose monitoring period starts at an arbitrary date and ends at another arbitrary date before maturity.