• 제목/요약/키워드: American Put Options

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FINITE ELEMENT METHODS FOR THE PRICE AND THE FREE BOUNDARY OF AMERICAN CALL AND PUT OPTIONS

  • Kang, Sun-Bu;Kim, Taek-Keun;Kwon, Yong-Hoon
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제12권4호
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    • pp.271-287
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    • 2008
  • This paper deals with American call and put options. Determining the fair price and the free boundary of an American option is a very difficult problem since they depends on each other. This paper presents numerical algorithms of finite element method based on the three-level scheme to compute both the price and the free boundary. One algorithm is designed for American call options and the other one for American put options. These algorithms are formulated on the system of the Jamshidian equation for the option price and the free boundary. Here, the Jamshidian equation is of a kind of the nonhomogeneous Black-Scholes equations. We prove the existence and uniqueness of the numerical solution by the Lax-Milgram lemma and carried out extensive numerical experiments to compare with various methods.

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수치적 반복 수렴 방법을 이용한 CEV 모형에서의 아메리칸 풋 옵션 가격 결정 (An Iterative Method for American Put Option Pricing under a CEV Model)

  • 이승규;장봉규;김인준
    • 대한산업공학회지
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    • 제38권4호
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    • pp.244-248
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    • 2012
  • We present a simple numerical method for pricing American put options under a constant elasticity of variance (CEV) model. Our analysis is done in a general framework where only the risk-neutral transition density of the underlying asset price is given. We obtain an integral equation of early exercise premium. By exploiting a modification of the integral equation, we propose a novel and simple numerical iterative valuation method for American put options.

RELATIONSHIPS BETWEEN AMERICAN PUTS AND CALLS ON FUTURES CONTRACTS

  • BYUN, SUK JOON;KIM, IN JOON
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제4권2호
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    • pp.11-20
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    • 2000
  • This paper presents a formula that relates the optimal exercise boundaries of American call and put options on futures contract. It is shown that the geometric mean of the optimal exercise boundaries for call and put written on the same futures contract with the same exercise price is equal to the exercise price which is time invariant. The paper also investigates the properties of American calls and puts on futures contract.

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FINITE-DIFFERENCE BISECTION ALGORITHMS FOR FREE BOUNDARIES OF AMERICAN OPTIONS

  • Kang, Sunbu;Kim, Taekkeun;Kwon, Yonghoon
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제19권1호
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    • pp.1-21
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    • 2015
  • This paper presents two algorithms based on the Jamshidian equation which is from the Black-Scholes partial differential equation. The first algorithm is for American call options and the second one is for American put options. They compute numerically free boundary and then option price, iteratively, because the free boundary and the option price are coupled implicitly. By the upwind finite-difference scheme, we discretize the Jamshidian equation with respect to asset variable s and set up a linear system whose solution is an approximation to the option value. Using the property that the coefficient matrix of this linear system is an M-matrix, we prove several theorems in order to formulate a bisection method, which generates a sequence of intervals converging to the fixed interval containing the free boundary value with error bound h. These algorithms have the accuracy of O(k + h), where k and h are step sizes of variables t and s, respectively. We prove that they are unconditionally stable. We applied our algorithms for a series of numerical experiments and compared them with other algorithms. Our algorithms are efficient and applicable to options with such constraints as r > d, $r{\leq}d$, long-time or short-time maturity T.

A SURVEY ON AMERICAN OPTIONS: OLD APPROACHES AND NEW TRENDS

  • Ahn, Se-Ryoong;Bae, Hyeong-Ohk;Koo, Hyeng-Keun;Lee, Ki-Jung
    • 대한수학회보
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    • 제48권4호
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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.

A SPECIFICATION TEST OF AT-THE-MONEY OPTION IMPLIED VOLATILITY: AN EMPIRICAL INVESTIGATION

  • Kim, Hong-Shik
    • 재무관리논총
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    • 제3권1호
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    • pp.213-231
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    • 1996
  • In this study we conduct a specification test of at-the-money option volatility. Results show that the implied volatility estimate recovered from the Black-Scholes European option pricing model is nearly indistinguishable from the implied volatility estimate obtained from the Barone-Adesi and Whaley's American option pricing model. This study also investigates whether the use of Black-Scholes implied volatility estimates in American put pricing model significantly affect the prediction the prediction of American put option prices. Results show that, at long as the possibility of early exercise is carefully controlled in calculation of implied volatilities prediction of American put prices is not significantly distorted. This suggests that at-the-money option implied volatility estimates are robust across option pricing model.

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CLOSED-FORM SOLUTIONS OF AMERICAN PERPETUAL PUT OPTION UNDER A STRUCTURALLY CHANGING ASSET

  • Shin, Dong-Hoon
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제15권2호
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    • pp.151-160
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    • 2011
  • Typically, it is hard to find a closed form solution of option pricing formula under an asset governed by a change point process. In this paper we derive a closed-form solution of the valuation function for an American perpetual put option under an asset having a change point. Structural changes are formulated through a change-point process with a Markov chain. The modified smooth-fit technique is used to obtain the closed-form valuation function. We also guarantee the optimality of the solution via the proof of a corresponding verification theorem. Numerical examples are included to illustrate the results.

옵션 가치 및 민감도 평가 방법: 속도와 정확도 개선에 대한 고찰 (Option Pricing and Sensitivity Evaluation Methodology: Improvement of Speed and Accuracy)

  • 최영수;오세진;이원창
    • Communications for Statistical Applications and Methods
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    • 제15권4호
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    • pp.563-585
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    • 2008
  • 본 연구는 다양하고 복잡해지는 파생상품 추세에 상응하는 적절한 가치평가에 대한 연구의 필요성을 인지하고 가격 및 민감도 평가에 있어서 속도와 정확도를 향상시키는데 그 의의를 두고자 한다. 몬테카를로 시뮬레이션에서 의사난수 대신 저불일치수열인준난수를 이용하면 시행횟수의 감소와 정확도 개선이 가능한데, 미국형 옵션이나 경로의존형 상품 등 다차원의 난수가 필요할 경우 기존의 준난수를 사용하면 상관관계가 증가하는 문제로 적용에 한계가 있다. 이런 단점을 보완하기 위해 문제를 발생시키는 차원의 난수를 제외시켜 상관계수를 특정값 이하로 제어하는 새로운 방법을 고안하여 다차원 상품에 적용이 가능토록 하였고 미국형 풋옵션에 적용하여 새로운 방법의 유용성을 검증하였다. 또한, 몬테카를로 시뮬레이션에서 민감도 계산방법으로 우도비율법과 경로의존형 근사방법을 사용하면 속도 및 정확도가 개선됨을 보인다. 이러한 결과는 최근 시장의 추세인 기초자산이나 위험요소가 여러 개인 경우 그리고 경로의존형 및 조기상환형 상품 등에 적용 가능토록하여 몬테카를로 시뮬레이션 방법에 있어 가장 큰 단점으로 지적되는 수행시간을 단축시키고 민감도 계산의 오차를 줄여줌을 보여준다. 또한, 2개 이하의 기초자산으로 이루어진 파생상품의 가치 및 민감도 평가에 가장 효율적인 수치해석적 방법론으로 알려져 있는 유한차분법의 적용시 격자생성구간의 설정이 매우 중요하다는 사실을 비대칭 나비형스프레드에 적용하여 실증적으로 보인다.

한국전쟁의 교훈과 대비 -병력수(兵力數) 및 부대수(部隊數)를 중심으로- (The lesson From Korean War)

  • 윤일영
    • 안보군사학연구
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    • 통권8호
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    • pp.49-168
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    • 2010
  • Just before the Korean War, the total number of the North Korean troops was 198,380, while that of the ROK(Republic of Korea) army troops 105,752. That is, the total number of the ROK army troops at that time was 53.3% of the total number of the North Korean army. As of December 2008, the total number of the North Korean troops is estimated to be 1,190,000, while that of the ROK troops is 655,000, so the ROK army maintains 55.04% of the total number of the North Korean troops. If the ROK army continues to reduce its troops according to [Military Reform Plan 2020], the total number of its troops will be 517,000 m 2020. If North Korea maintains the current status(l,190,000 troops), the number of the ROK troops will be 43.4% of the North Korean army. In terms of units, just before the Korean War, the number of the ROK army divisions and regiments was 80% and 44.8% of North Korean army. As of December 2008, North Korea maintains 86 divisions and 69 regiments. Compared to the North Korean army, the ROK army maintains 46 Divisions (53.4% of North Korean army) and 15 regiments (21.3% of North Korean army). If the ROK army continue to reduce the military units according to [Military Reform Plan 2020], the number of ROK army divisions will be 28(13 Active Division, 4 Mobilization Divisions and 11 Local Reserve Divisions), while that of the North Korean army will be 86 in 2020. In that case, the number of divisions of the ROK army will be 32.5% of North Korean army. During the Korean war, North Korea suddenly invaded the Republic of Korea and occupied its capital 3 days after the war began. At that time, the ROK army maintained 80% of army divisions, compared to the North Korean army. The lesson to be learned from this is that, if the ROK army is forced to disperse its divisions because of the simultaneous invasion of North Korea and attack of guerrillas in home front areas, the Republic of Korea can be in a serious military danger, even though it maintains 80% of military divisions of North Korea. If the ROK army promotes the plans in [Military Reform Plan 2020], the number of military units of the ROK army will be 32.5% of that of the North Korean army. This ratio is 2.4 times lower than that of the time when the Korean war began, and in this case, 90% of total military power should be placed in the DMZ area. If 90% of military power is placed in the DMZ area, few troops will be left for the defense of home front. In addition, if the ROK army continues to reduce the troops, it can allow North Korea to have asymmetrical superiority in military force and it will eventually exert negative influence on the stability and peace of the Korean peninsular. On the other hand, it should be reminded that, during the Korean War, the Republic of Korea was attacked by North Korea, though it kept 53.3% of troops, compared to North Korea. It should also be reminded that, as of 2008, the ROK army is defending its territory with the troops 55.04% of North Korea. Moreover, the national defense is assisted by 25,120 troops of the US Forces in Korea. In case the total number of the ROK troops falls below 43.4% of the North Korean army, it may cause social unrest about the national security and may lead North Korea's misjudgement. Besides, according to Lanchester strategy, the party with weaker military power (60% compared to the party with stronger military power) has the 4.1% of winning possibility. Therefore, if we consider the fact that the total number of the ROK army troops is 55.04% of that of the North Korean army, the winning possibility of the ROK army is not higher than 4.1%. If the total number of ROK troops is reduced to 43.4% of that of North Korea, the winning possibility will be lower and the military operations will be in critically difficult situation. [Military Reform Plan 2020] rums at the reduction of troops and units of the ground forces under the policy of 'select few'. However, the problem is that the financial support to achieve this goal is not secured. Therefore, the promotion of [Military Reform Plan 2020] may cause the weakening of military defence power in 2020. Some advanced countries such as Japan, UK, Germany, and France have promoted the policy of 'select few'. However, what is to be noted is that the national security situation of those countries is much different from that of Korea. With the collapse of the Soviet Unions and European communist countries, the military threat of those European advanced countries has almost disappeared. In addition, the threats those advanced countries are facing are not wars in national level, but terrorism in international level. To cope with the threats like terrorism, large scaled army trops would not be necessary. So those advanced European countries can promote the policy of 'select few'. In line with this, those European countries put their focuses on the development of military sections that deal with non-military operations and protection from unspecified enemies. That is, those countries are promoting the policy of 'select few', because they found that the policy is suitable for their national security environment. Moreover, since they are pursuing common interest under the European Union(EU) and they can form an allied force under NATO, it is natural that they are pursing the 'select few' policy. At present, NATO maintains the larger number of troops(2,446,000) than Russia(l,027,000) to prepare for the potential threat of Russia. The situation of japan is also much different from that of Korea. As a country composed of islands, its prime military focus is put on the maritime defense. Accordingly, the development of ground force is given secondary focus. The japanese government promotes the policy to develop technology-concentrated small size navy and air-forces, instead of maintaining large-scaled ground force. In addition, because of the 'Peace Constitution' that was enacted just after the end of World War II, japan cannot maintain troops more than 240,000. With the limited number of troops (240,000), japan has no choice but to promote the policy of 'select few'. However, the situation of Korea is much different from the situations of those countries. The Republic of Korea is facing the threat of the North Korean Army that aims at keeping a large-scale military force. In addition, the countries surrounding Korea are also super powers containing strong military forces. Therefore, to cope with the actual threat of present and unspecified threat of future, the importance of maintaining a carefully calculated large-scale military force cannot be denied. Furthermore, when considering the fact that Korea is in a peninsular, the Republic of Korea must take it into consideration the tradition of continental countries' to maintain large-scale military powers. Since the Korean War, the ROK army has developed the technology-force combined military system, maintaining proper number of troops and units and pursuing 'select few' policy at the same time. This has been promoted with the consideration of military situation in the Koran peninsular and the cooperation of ROK-US combined forces. This kind of unique military system that cannot be found in other countries can be said to be an insightful one for the preparation for the actual threat of North Korea and the conflicts between continental countries and maritime countries. In addition, this kind of technology-force combined military system has enabled us to keep peace in Korea. Therefore, it would be desirable to maintain this technology-force combined military system until the reunification of the Korean peninsular. Furthermore, it is to be pointed out that blindly following the 'select few' policy of advanced countries is not a good option, because it is ignoring the military strategic situation of the Korean peninsular. If the Republic of Korea pursues the reduction of troops and units radically without consideration of the threat of North Korea and surrounding countries, it could be a significant strategic mistake. In addition, the ROK army should keep an eye on the fact the European advanced countries and Japan that are not facing direct military threats are spending more defense expenditures than Korea. If the ROK army reduces military power without proper alternatives, it would exert a negative effect on the stable economic development of Korea and peaceful reunification of the Korean peninsular. Therefore, the desirable option would be to focus on the development of quality of forces, maintaining proper size and number of troops and units under the technology-force combined military system. The tableau above shows that the advanced countries like the UK, Germany, Italy, and Austria spend more defense expenditure per person than the Republic of Korea, although they do not face actual military threats, and that they keep achieving better economic progress than the countries that spend less defense expenditure. Therefore, it would be necessary to adopt the merits of the defense systems of those advanced countries. As we have examined, it would be desirable to maintain the current size and number of troops and units, to promote 'select few' policy with increased defense expenditure, and to strengthen the technology-force combined military system. On the basis of firm national security, the Republic of Korea can develop efficient policies for reunification and prosperity, and jump into the status of advanced countries. Therefore, the plans to reduce troops and units in [Military Reform Plan 2020] should be reexamined. If it is difficult for the ROK army to maintain its size of 655,000 troops because of low birth rate, the plans to establish the prompt mobilization force or to adopt drafting system should be considered for the maintenance of proper number of troops and units. From now on, the Republic of Korean government should develop plans to keep peace as well as to prepare unexpected changes in the Korean peninsular. For the achievement of these missions, some options can be considered. The first one is to maintain the same size of military troops and units as North Korea. The second one is to maintain the same level of military power as North Korea in terms of military force index. The third one is to maintain the same level of military power as North Korea, with the combination of the prompt mobilization force and the troops in active service under the system of technology-force combined military system. At present, it would be not possible for the ROK army to maintain such a large-size military force as North Korea (1,190,000 troops and 86 units). So it would be rational to maintain almost the same level of military force as North Korea with the combination of the troops on the active list and the prompt mobilization forces. In other words, with the combination of the troops in active service (60%) and the prompt mobilization force (40%), the ROK army should develop the strategies to harmonize technology and forces. The Korean government should also be prepared for the strategic flexibility of USFK, the possibility of American policy change about the location of foreign army, radical unexpected changes in North Korea, the emergence of potential threat, surrounding countries' demand for Korean force for the maintenance of regional stability, and demand for international cooperation against terrorism. For this, it is necessary to develop new approaches toward the proper number and size of troops and units. For instance, to prepare for radical unexpected political or military changes in North Korea, the Republic of Korea should have plans to protect a large number of refugees, to control arms and people, to maintain social security, and to keep orders in North Korea. From the experiences of other countries, it is estimated that 115,000 to 230,000 troops, plus ten thousands of police are required to stabilize the North Korean society, in the case radical unexpected military or political change happens in North Korea. In addition, if the Republic of Korea should perform the release of hostages, control of mass destruction weapons, and suppress the internal wars in North Korea, it should send 460,000 troops to North Korea. Moreover, if the Republic of Korea wants to stop the attack of North Korea and flow of refugees in DMZ area, at least 600,000 troops would be required. In sum, even if the ROK army maintains 600,000 troops, it may need additional 460,000 troops to prepare for unexpected radical changes in North Korea. For this, it is necessary to establish the prompt mobilization force whose size and number are almost the same as the troops in active service. In case the ROK army keeps 650,000 troops, the proper number of the prompt mobilization force would be 460,000 to 500,000.

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