Sungchul Lee

• Doctor of Philosophy
• Professor at Yonsei University

In this paper, we propose an unconditionally stable numerical technique for a multi-dimensional Black-Scholes equation to price an option with high accuracy. The proposed scheme uses the operator splitting method to reduce the multi-dimensional partial differential equation to a set of one-dimensional sub-problems. The computational domain is discr...

We propose an unconditionally stable numerical algorithm, which uses the Feynman-Kac formula of the Black-Scholes equation to obtain accurate option prices and hedge parameters. We discretize the asset and time using uniform grid points. We approximate the option values by piecewise quadratic polynomials for each time step and integrate them analyt...

This paper investigates a novel optimization problem motivated by sparse, sustainable and stable portfolio selection. The existing benchmark portfolio via the Dantzig type optimization is used to construct a sparse, sustainable and stable portfolio. Based on the formulations, this paper proposes two portfolio selection methods, west and north portf...

In order to improve the forecasting accuracy of the volatilities of the markets, we propose the hybrid models based on artificial neural networks with multi-hidden layers in this paper. Specifically, the hybrid models are built using the estimated volatilities obtained from GARCH family models and Google domestic trends (GDTs) as input variables. W...

We consider the problem of constructing a perturbed portfolio by utilizing a benchmark portfolio. We propose two computationally efficient portfolio optimization models, the mean-absolute deviation risk and the Dantzig-type, which can be solved using linear programing. These portfolio models push the existing benchmark toward the efficient frontier...

This paper presents a deep learning method for faster magnetic resonance imaging (MRI) by reducing k-space data with sub-Nyquist sampling strategies and provides a rationale for why the proposed approach works well. Uniform subsampling is used in the time-consuming phase-encoding direction to capture high-resolution image information, while permitt...

This paper presents a deep learning method for faster magnetic resonance imaging (MRI) by reducing k-space data with sub-Nyquist sampling strategies and provides a rationale for why the proposed approach works well. Uniform subsampling is used in the time-consuming phase-encoding direction to capture high-resolution image information, while permitt...

In this paper, we propose a stochastic method to project the public pension fund in the public pension system (PPS). For this we introduce the stochastic differential equations for the three parts: the premium revenue, the benefit expenditure, and the fund process. From these we show that the solution of the aggregated fund process is the sum of lo...

A Barrier option is an option whose payoff depends on the underlying asset prices during the life of the option. Most Barrier option pricing usually assumes the continuous monitoring of the barrier. However, Barrier options traded in markets are discretely monitored and in this discretely monitoring case there are no closed form solutions available...

Sullivan (2000) used the Gauss–Legendre quadrature and the Chebyshev approximation to price the American put option. Starting from his work we construct a systematic way of pricing the Bermudan option with long-term maturity using numerical integrations. At each exercise time we approximate the option values multiplied by the discounted transition...

The limiting distribution for the linear placement statistics under the null hypotheses has been provided by Orban and Wolfe [9] and Kim [5] when one of the sample sizes goes to infinity, and by Kim, Lee and Wang [6] when the sample sizes of each group go to infinity simultaneously. In this paper we establish the generalized Kruskal-Wallis one-way...

We consider the valuation of options with stressed-beta in a reduced form model. Under this two-state beta model, we provide the analytic pricing formulae for the European options and American options as the integral forms. Specifically, we provide the integral representation of the early exercise premium of an American put option. We use the quadr...

In this paper we derive the pricing formula for the exchange option value in a two-state Poisson CAPM. A two-state Poisson CAPM models the stochastic market environment. We also provide examples and graphs to illustrate our result.

Using the Poisson point process we model how the SiPM works and derive the non-linear response formula of the SiPM. Using this non-linear response formula we are able to capture the mean and variance saturation phenomena near the infinity and the linear behavior of the mean and variance near 0.We also introduce a different model of the SiPM. Under...

Orban and Wolfe (1982) and Kim (1999) provided the limiting distribution for linear placement statistics under null hypotheses only when one of the sample sizes goes to infinity. In this paper we prove the asymptotic normality and the weak convergence of the linear placement statistics of Orban and Wolfe (1982) and Kim (1999) when the sample sizes...

In this paper, estimating upper bounds of large deviation probabilities for the increments of d-dimensional Gaussian processes in Hölder type norm, we obtain functional limit results for the increments of d-dimensional Gaussian processes in the Hölder type norm.

Let $L$ be the Euclidean functional with $p$-th power-weighted edges. Examples include the sum of the $p$-th power-weighted lengths of the edges in minimal spanning trees, traveling salesman tours, and minimal matchings. Motivated by the works of Steele, Redmond and Yukich (1994, 1996) have shown that for $n$ i.i.d. sample points $\{X_1,...,X_n\}$...

For n independent, identically distributed uniform points in [0, 1] d , d ≥ 2, let L n be the total distance from the origin to all the minimal points under the coordinatewise partial order (this is also the total length of the rooted edges of a minimal directed spanning tree on the given random points). For d ≥ 3, we establish the asymptotics of t...

For n independent, identically distributed uniform points in [0, 1]^d, d ≥ 2, let L_n be the total distance from the origin to all the minimal points under the coordinatewise partial order (this is also the total length of the rooted edges of a minimal directed spanning tree on the given random points). For d ≥ 3, we establish t...

Let \({\cal P}\) be the Poisson point process with intensity 1 in R d and let \({\cal P}_n\) be \({\cal P} \cap [-n/2,n/2]^d\). We obtain a strong invariance principle for the total length of the nearest-neighbor graph on \({\cal P}_n\).

In this paper, we study path properties of a d-dimensional Gaussian process with the usual Euclidean norm, via estimating upper bounds of large deviation probabilities on the suprema of the Gaussian process.

Let PnP_n be the Poisson point process with intensity 1 in [–n,n] d . We prove the law of the iterated logarithm for the total length of the nearest neighbor graph on PnP_n .

Suppose each edge of the complete graph Kn is assigned a random weight chosen independently and uniformly from the unit interval [0,1]. A minimal spanning tree is a spanning tree of Kn with the minimum weight. It is easy to show that such a tree is unique almost surely. This paper concerns the number Nn([alpha]) of vertices of degree [alpha] in the...

Let A,|A|⩽n, be a subset of [0,1]d, and let L(A,[0,1]d,p) be the length of the minimal matching, the minimal spanning tree, or the traveling salesman problem on A with weight function w(e)=|e|p. In the case 1⩽p

Consider the complete graph Kn on n vertices and the n-cube graph Qn on 2n vertices. Suppose independent uniform random edge weights are assigned to each edges in Kn and Qn and let and denote the unique minimal spanning trees on Kn and Qn, respectively. In this paper we obtain the Gaussian tail for the number of edges of and with weight at most t/n...

Consider the random assignment (or bipartite matching) problem with iid uniform edge costs t(i,j). Let A n be the optimal assignment cost. Just recently does Aldous give a rigorous proof that EA n →ζ(2). We establish the upper and lower bounds for VarA n , i.e., there exist two strictly positive but finite constants C 1 and C 2 such that C 1 n -5/2...

In this paper, we establish a martingale inequality and develop the symmetry argument to use this martingale inequality. We apply this to the length of the longest increasing subsequences and the independence number of sparse random graphs.

Let {Xi: i[greater-or-equal, slanted]1} be i.i.d. uniform points on [-1/2,1/2]d, d[greater-or-equal, slanted]2, and for 0<p<[infinity]. Let L({X1,...,Xn},p) be the total weight of the minimal spanning tree on {X1,...,Xn} with weight function w(e)=ep. Then, there exist strictly positive but finite constants [beta](d,p), C3=C3(d,p), and C4=C4(d,p) su...

Let X i : i ≥ 1 be i.i.d. points in ℝ d , d ≥ 2, and let T n be a minimal spanning tree on X 1 ,…, X n . Let L ( X 1 ,…, X n ) be the length of T n and for each strictly positive integer α let N ( X 1 ,…, X n ;α) be the number of vertices of degree α in T n . If the common distribution satisfies certain regularity conditions, then we prove central...

Let {X_i : i ≥ 1} be i.i.d. points in R^d, d ≥ 2, and let T_n be a minimal spanning tree on {X₁,...,X_n}. Let L({X₁,...,X_n}) be the length of T_n and for each strictly positive integer α let N({X₁,...,X_n};α) be the number of vertices of degree α...

Let {Xi: i[greater-or-equal, slanted]1} be i.i.d. points in , d[greater-or-equal, slanted]2, and let LMM({X1,...,Xn},p), LMST({X1,...,Xn},p), LTSP({X1,...,Xn},p), be the length of the minimal matching, the minimal spanning tree, the traveling salesman problem, respectively, on {X1,...,Xn} with weight function w(e)=ep. If the common distribution sat...

Let ${X_i: i \geq 1}$ be i.i.d. with uniform distribution $[- 1/2, 1/2]^d, d \geq 2$, and let $T_n$ be a minimal spanning tree on ${X_1, \dots, X_n}$. For each strictly positive integer $\alpha$, let $N({X_1, \dots, X_n}; \alpha)$ be the number of vertices of degree $\alpha$ in $T_n$. Then, for each $\alpha$ such that $P(N({X_1, \dots, X_{\alpha+1}...

We consider the power laws of certain limiting values in greedy lattice animals which were introduced by Cox, Gandolfi, Griffin, and Kesten (1993) and Gandolfi and Kesten (1994). We study the behavior of the limiting values as we change the parameter p.

Let {X v: v ∈ Z d}, d≥2, be i.i.d. positive random variables with the common distribution F which satisfy, for some a>0, ∫ x d (log+x)d+a dF(x)<∞ Define $$M_n = \max \left\{ {\sum\limits_{\upsilon \in \pi } {X_\upsilon } {\kern 1pt} :\pi {\text{ a selfavoiding path of length }}n{\text{ starting at the origin}}} \right\}$$ $$N_n = \max \left\{ {\sum...

Let ${X_i, 1 \leq i < \infty}$ be i.i.d. with uniform distribution on $[0, 1]^d$ and let $M(X_1, \dots, X_n; \alpha)$ be $\min {\sum_{e \epsilon T'} |e|^{\alpha}; T' \text{a spanning tree on ${X_1, \dots, X_n}$}}$. Then we show that for $\alpha > 0$, $$\frac{M(X_1, \dots, X_n; \alpha) - EM (X_1, \dots, X_n; \alpha)}{n^{(d-2 \alpha)/2d}} \to N(0, \s...

Let $\{X_\nu: \nu \in \mathbb{Z}^d\}$ be i.i.d. positive random variables with $E\{X^d_0(\log^+ X_0)^{d+\varepsilon}\} < \infty$ for some $\varepsilon > 0$ and $d \geq 2$. Define $M_n$ and $N_n$ by \begin{align*} M_n &= \max\big\{\sum_{\nu \in \pi} X_\nu: \pi \text{a self-avoiding path of length}\quad n \\ \text{starting at the origin}\big\},\\ N_n...