An Iterative Method for American Put Option Pricing under a CEV Model

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Journal of the Korean Institute of Industrial Engineers http://dx.doi.org/10.7232/JKIIE.2012.38.4.244

Vol. 38, No. 4, pp. 244-248, December 2012. © 2012 KIIE

ISSN 1225-0988 | EISSN 2234-6457 <Research Letter>

수치적 반복 수렴 방법을 이용한 CEV 모형에서의 아메리칸 풋 옵션 가격 결정

이승규¹ 장봉규¹ 김인준²

1POSTECH 산업경영공학과 /²연세대학교 경영대학

An Iterative Method for American Put Option Pricing under a CEV Model

Seungkyu Lee¹ Bong-Gyu Jang¹ In Joon Kim²

1Department of Industrial and Management Engineering, Pohang University of Science and Technology

2Yonsei School of Business

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.

Keywords: American Put Options, CEV Model, Numerical Algorithm, Iterative Method

1. 서 론

(American put options) (optimal exercise boundary)

(European put options) . McKean and Merton

(free-boundary

problem) ,

(McKean, 1965; Merton, 1973).

. Huang Ju

(Huang and Subrahmanyam, 1996; Ju, 1998).

-

. (fractal

Brownian motions), CEV(constant elasticity of variance) (stochastic volatility models), (jump models)

(Mandelbrot, 1963; White, 1980; Cox and Ross, 1976; Merton, 1976).

. Kim and Yu

(analytic

이 논문은 2012년도 정부(교육과학기술부)의 재원으로 한국연구재단의 지원을 받아 수행된 기초연구사업임(No. 2012R1A1A2038735).

이 논문은 2011년도 정부(교육과학기술부)의 재원으로 한국연구재단의 지원을 받아 수행된 교육인력양성사업임(No. 2011-0008628).

연락저자 장봉규 교수, 790-784 경북 포항시 남구 효자동 포항공과대학교 산업경영공학과, Tel : 054-279-2372, Fax : 054-279-2870, E-mail : [email protected]

2012년 10월 31일 접수; 2012년 11월 12일 수정본 접수; 2012년 11월 13일 게재 확정.

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수치적 반복 수렴 방법을 이용한 CEV 모형에서의 아메리칸 풋 옵션 가격 결정 245

valuation formula) (Kim and Yu, 1996).

CEV (mean-reverting process)

,

. Nunes CEV

, ,

(Nune, 2009).

Kim and Yu

(operator)

(iterative method) (Kim

and Yu, 1996). CEV

. Kim -

(Black-Scholes model)

(Kim et al., 2012), CEV

.

2. 이론적 배경

2.1 아메리칸 풋 옵션 공식 유도

 

.

(perfect hedging) (continuous

trading) ^

 .

Carr ⋅

(Carr et al., 1992).

⋅ t ( ^{ }^^{ }

)  ,

 ⋅   

⋅ ≤ ∞    ≤ .

(risk neutral probability)

  ′

(transition density function) ^^^{′ }^^ . Kim and Yu

(Kim and Yu, 1996).

⋅  ,

  ⋅   

, ^

 ⋅  

 ,

:

 ⋅





_^^^{   }^^^^   ⋅









^{ }  ⋅

(1)

 

(portfolio) .

:

 





_^^^{ }



_^∞^^^{ }^^{′ }^^{′ }^^{}^^′





_^∞^^{ }^^^{ }^^^^{ }^^{ }^^

(2)

^⋅ (self-

financing) :

, ^

.

, .



′(^{  ′}) ,

:

^





_′^^^{ }



_^^^^^^{ }^^′^^{′ }^^^{}^^′





_^^^′^    ′    ′^





_′

∞

    ′^′ ′^⋅′   ′^^′

(3)



 ⋅ (payoff)  

, :





_^ ^^{ }^^^{ }^^^^{ }^^{}^







∞

^{ }′ ⋅′ ′ 

(4)

′   , (implicit

formula) :

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246 Seungkyu Lee Bong-Gyu Jang In Joon Kim

^





_^^^{ }



_^^^^^^{ }^^{′ }^^{′ }^^^{}^^′





_^^^{ }^^^{ }^^^^{ }^^^{ }^^

(5)

(live American put option) :



_ 



^{ }  ^⋅





_^^^{ }



_^^^^{ }^^^^{ }^^^{}^^{ }^^^⋅^^





_^^^{ }^^^{ }^^^^{ }^^^^

(6)

  ^⋅ 0 ( ) ^{  }

^ 

. (6)



. (6) (

) (Bayesian Rule)

.

(5) (1) , (6)

^ ^⋅ (5)

^⋅

:

 ^⋅





_^^^{   }



_^^^^^^{ }^^{′ }^^{′   }^^^^′





_^^^{ }^^^{ }^^^^{  }^^^^

(7)

(6) (double integral)

, (5) (1)

(triple integral) (7) .

(7) (right hand side)

(

). (7)

(early exercise

premium) (integral representation) .

Carr (Carr et al., 1992).

3.2 CEV 모형에서의 아메리칸 풋 옵션 가격

(section) CEV

. _

(brownian motion) _ (the elasticity of variance) ^{  }

(stochastic differential equation) :

_   _  _^_

- ^{  } CEV

. ^

0 2 .

(volatility clustering)

 ≤    . Cox

, Schroder

(Cox, 1975; Schroder, 1989).

Cox _ CEV

(risk-neutral world) ^  (Cox, 1975):

 

   _^{   }  ^{  }   

×     _{   }^   

_,  ,   :

_  ^         

  



   _^{  }      

   _^{  }

_  1 (the modified Bessel function

of the first kind of order ^) :

_ 



  

∞

      





^





^{ }^

Cox ,

(Cox, 1975):

    





_^^      _{  }

  

  ^  

      

×^       

    

(8)

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An Iterative Method for American Put Option Pricing under a CEV Model 247

   _^

(the noncentral Chi-square random variable with  degrees of freedom and non-centrality parameter ^)

. ^^^{ }

:

  ^{ } _{  }

  

   

 ^{ }      

   

3.3 수치적 해법 및 결과

(5) ,

.

 0 .

(8) ^ , ^ ^⋅ 

^ , :

^

  ^{ } ^_{  }

  

  





_^^^{ }^{ }^^^^     

  ^  

÷       

  ^ 

⋅

:

 

  ^{ } _{  }

  

   





_^^^{ }^{ }^^^      

     

÷ _{   }

  

   

⋅



^^^{ }^  ∀ ∊ 

_{ }  _  ∀ ∊    …

(function sequence) ^⋅_{  }^∞

, ^⋅ .

:

Step 1 : _ _  .

Step 2 : _{ } ⋅

_ _{ }  _ . Step 3 : ^ 1 , ^_{ } _ ≤ 

Step 2 .

<Figure 1> (parameter) Step

2 6 . ^

, ^ . <Figure

1> , 4

.

Figure 1. The convergency of optimal exercise boundary. The parameters are  ,  ,   ,   ,

  , and ^^{  } .

(8) . - CEV

. <Table 1>

.

Table 1. Early exercise premium with various values of the constant elasticity of variance ^. The parameters are

  ,  ,   ,  , and ^{  } 

The early exercise premium, the difference between American option price and European option price, rises with increasing the constant elasticity of variance ^

 0.4 0.8 1.3 1.7

ITM ( )

European 6.71497 6.76183 6.82243 6.87262 American 6.94477 6.99324 7.05641 7.10916 Premium 0.2298 0.23141 0.23398 0.23654 ATM

( )

European 3.97596 3.97167 3.96819 3.96673 American 4.09503 4.09316 4.093 4.09442 Premium 0.11907 0.12149 0.12481 0.12769 OTM

( )

European 2.07055 2.01746 1.95377 1.90474 American 2.13379 2.08168 2.01933 1.97142 Premium 0.06324 0.06422 0.06556 0.06668

(5)

248 이승규 장봉규 김인준

4. 결 론

.

CEV

. Kim and Yu

_^ (module)

(Kim and Yu, 1996).

.

참고문헌

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Cox, J. C. and Ross, S. A. (1976), The Valuation of Options for Alternative

Stochastic Processes, Journal of Financial Economics, 3, 145-166.

Huang, J., Subrahmanyam, M. G., and Yu, G. G. (1996), Pricing and Hedging American Options : A Recursive Integration Method, Review of Financial Studies, 9, 277-300.

Ju, N. (1998), Pricing American Option by Approximating its Early Exercise Boundary as a Multipiece Exponential Function, Review of Financial Studies, 11, 627-646.

Kim, I. J. and Yu, G. G. (1996), An Alternative Approach to the Valuation of American Options and Applications, Review of Derivatives Re- search, 1, 61-85.

Kim, I. J., Jang, B-G., and Kim, K. T. (2012), A Simple Iterative Method for the Valuation of American Options, Quantitative Finance, forth- coming.

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