An adaptive and explicit fourth order Runge–Kutta–Fehlberg method coupled with compact finite differencing for pricing American put options View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2021-06-12

AUTHORS

Chinonso Nwankwo, Weizhong Dai

ABSTRACT

We propose an adaptive and explicit Runge–Kutta–Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. Furthermore, we implement an extrapolation method to ensure that at least, a third-order accuracy in space is maintained at the boundary point when computing the optimal exercise boundary from its derivative. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge–Kutta–Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in the numerical experiment, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution. More... »

PAGES

921-946

References to SciGraph publications

  • 2009-04-14. Numerically optimal Runge–Kutta pairs with interpolants in NUMERICAL ALGORITHMS
  • 2016. A Positive, Stable and Consistent Front-Fixing Numerical Scheme for American Options in PROGRESS IN INDUSTRIAL MATHEMATICS AT ECMI 2014
  • 2018-11-21. Fast and accurate calculation of American option prices in DECISIONS IN ECONOMICS AND FINANCE
  • 2017-06-30. Optimal exercise boundary via intermediate function with jump risk in JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
  • 2002-12. Global Error Estimators for Order 7, 8 Runge–Kutta Pairs in NUMERICAL ALGORITHMS
  • 2013-11-20. Efficient High-Order Numerical Methods for Pricing of Options in COMPUTATIONAL ECONOMICS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s13160-021-00470-2

    DOI

    http://dx.doi.org/10.1007/s13160-021-00470-2

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1138792270


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0103", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Numerical and Computational Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 60607, Chicago, IL, USA", 
              "id": "http://www.grid.ac/institutes/grid.185648.6", 
              "name": [
                "Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 60607, Chicago, IL, USA"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Nwankwo", 
            "givenName": "Chinonso", 
            "id": "sg:person.014363237745.59", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014363237745.59"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Department of Mathematics and Statistics, Louisiana Tech University, 71272, Ruston, LA, USA", 
              "id": "http://www.grid.ac/institutes/grid.259237.8", 
              "name": [
                "Department of Mathematics and Statistics, Louisiana Tech University, 71272, Ruston, LA, USA"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Dai", 
            "givenName": "Weizhong", 
            "id": "sg:person.014076405020.50", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014076405020.50"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/s10203-018-0224-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1110065100", 
              "https://doi.org/10.1007/s10203-018-0224-1"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10614-013-9405-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1034772769", 
              "https://doi.org/10.1007/s10614-013-9405-8"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1023/a:1021190918665", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1029279662", 
              "https://doi.org/10.1023/a:1021190918665"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-319-23413-7_10", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1091476142", 
              "https://doi.org/10.1007/978-3-319-23413-7_10"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s11075-009-9290-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1040005424", 
              "https://doi.org/10.1007/s11075-009-9290-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s13160-017-0261-0", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1090318547", 
              "https://doi.org/10.1007/s13160-017-0261-0"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2021-06-12", 
        "datePublishedReg": "2021-06-12", 
        "description": "We propose an adaptive and explicit Runge\u2013Kutta\u2013Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. Furthermore, we implement an extrapolation method to ensure that at least, a third-order accuracy in space is maintained at the boundary point when computing the optimal exercise boundary from its derivative. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge\u2013Kutta\u2013Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in the numerical experiment, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s13160-021-00470-2", 
        "inLanguage": "en", 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1041814", 
            "issn": [
              "0916-7005", 
              "1868-937X"
            ], 
            "name": "Japan Journal of Industrial and Applied Mathematics", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "3", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "38"
          }
        ], 
        "keywords": [
          "Fehlberg method", 
          "Runge-Kutta", 
          "fourth-order compact scheme", 
          "partial differential equations", 
          "American put option problem", 
          "third-order accuracy", 
          "fourth-order Runge-Kutta", 
          "optimal exercise boundary", 
          "explicit Runge\u2013Kutta", 
          "order Runge-Kutta", 
          "optimal exercise", 
          "American put option", 
          "Dirichlet boundary conditions", 
          "free boundary problem", 
          "differential equations", 
          "compact scheme", 
          "temporal discretization", 
          "option Greeks", 
          "numerical solution", 
          "exercise boundary", 
          "option problems", 
          "accurate solutions", 
          "boundary problem", 
          "numerical experiments", 
          "time step", 
          "computational speed", 
          "free boundary", 
          "boundary conditions", 
          "boundary points", 
          "quadratic formula", 
          "error control", 
          "put option", 
          "extrapolation method", 
          "present method", 
          "intermediate function", 
          "certain threshold", 
          "discretization", 
          "logarithm transformation", 
          "equations", 
          "finite", 
          "problem", 
          "solution", 
          "better performance", 
          "additional derivatives", 
          "boundaries", 
          "formula", 
          "scheme", 
          "space", 
          "velocity", 
          "error", 
          "accuracy", 
          "derivatives", 
          "terms", 
          "point", 
          "speed", 
          "function", 
          "system", 
          "transformation", 
          "performance", 
          "advantages", 
          "domain", 
          "step", 
          "conditions", 
          "experiments", 
          "control", 
          "threshold", 
          "addition", 
          "options", 
          "adjustment", 
          "Greek", 
          "method", 
          "exercise", 
          "put options problem", 
          "asset option", 
          "explicit fourth order Runge\u2013Kutta", 
          "compact finite"
        ], 
        "name": "An adaptive and explicit fourth order Runge\u2013Kutta\u2013Fehlberg method coupled with compact finite differencing for pricing American put options", 
        "pagination": "921-946", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1138792270"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s13160-021-00470-2"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s13160-021-00470-2", 
          "https://app.dimensions.ai/details/publication/pub.1138792270"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-01-01T19:00", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20220101/entities/gbq_results/article/article_881.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s13160-021-00470-2"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s13160-021-00470-2'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s13160-021-00470-2'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s13160-021-00470-2'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s13160-021-00470-2'


     

    This table displays all metadata directly associated to this object as RDF triples.

    168 TRIPLES      22 PREDICATES      107 URIs      93 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s13160-021-00470-2 schema:about anzsrc-for:01
    2 anzsrc-for:0103
    3 schema:author N170dd7d42a4440459268949e8dbe62a7
    4 schema:citation sg:pub.10.1007/978-3-319-23413-7_10
    5 sg:pub.10.1007/s10203-018-0224-1
    6 sg:pub.10.1007/s10614-013-9405-8
    7 sg:pub.10.1007/s11075-009-9290-3
    8 sg:pub.10.1007/s13160-017-0261-0
    9 sg:pub.10.1023/a:1021190918665
    10 schema:datePublished 2021-06-12
    11 schema:datePublishedReg 2021-06-12
    12 schema:description We propose an adaptive and explicit Runge–Kutta–Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. Furthermore, we implement an extrapolation method to ensure that at least, a third-order accuracy in space is maintained at the boundary point when computing the optimal exercise boundary from its derivative. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge–Kutta–Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in the numerical experiment, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution.
    13 schema:genre article
    14 schema:inLanguage en
    15 schema:isAccessibleForFree true
    16 schema:isPartOf Nab45551ac20649ec8d9556ae928195ea
    17 Ncc3fb390b05b46ff940dfce5c8558b35
    18 sg:journal.1041814
    19 schema:keywords American put option
    20 American put option problem
    21 Dirichlet boundary conditions
    22 Fehlberg method
    23 Greek
    24 Runge-Kutta
    25 accuracy
    26 accurate solutions
    27 addition
    28 additional derivatives
    29 adjustment
    30 advantages
    31 asset option
    32 better performance
    33 boundaries
    34 boundary conditions
    35 boundary points
    36 boundary problem
    37 certain threshold
    38 compact finite
    39 compact scheme
    40 computational speed
    41 conditions
    42 control
    43 derivatives
    44 differential equations
    45 discretization
    46 domain
    47 equations
    48 error
    49 error control
    50 exercise
    51 exercise boundary
    52 experiments
    53 explicit Runge–Kutta
    54 explicit fourth order Runge–Kutta
    55 extrapolation method
    56 finite
    57 formula
    58 fourth-order Runge-Kutta
    59 fourth-order compact scheme
    60 free boundary
    61 free boundary problem
    62 function
    63 intermediate function
    64 logarithm transformation
    65 method
    66 numerical experiments
    67 numerical solution
    68 optimal exercise
    69 optimal exercise boundary
    70 option Greeks
    71 option problems
    72 options
    73 order Runge-Kutta
    74 partial differential equations
    75 performance
    76 point
    77 present method
    78 problem
    79 put option
    80 put options problem
    81 quadratic formula
    82 scheme
    83 solution
    84 space
    85 speed
    86 step
    87 system
    88 temporal discretization
    89 terms
    90 third-order accuracy
    91 threshold
    92 time step
    93 transformation
    94 velocity
    95 schema:name An adaptive and explicit fourth order Runge–Kutta–Fehlberg method coupled with compact finite differencing for pricing American put options
    96 schema:pagination 921-946
    97 schema:productId N0fa29f6d01e84676b4e04c09b4015701
    98 N77d9da61bd49462ea86232531c869747
    99 schema:sameAs https://app.dimensions.ai/details/publication/pub.1138792270
    100 https://doi.org/10.1007/s13160-021-00470-2
    101 schema:sdDatePublished 2022-01-01T19:00
    102 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    103 schema:sdPublisher Nc165f037cb53475abd635266f2f620b4
    104 schema:url https://doi.org/10.1007/s13160-021-00470-2
    105 sgo:license sg:explorer/license/
    106 sgo:sdDataset articles
    107 rdf:type schema:ScholarlyArticle
    108 N0fa29f6d01e84676b4e04c09b4015701 schema:name dimensions_id
    109 schema:value pub.1138792270
    110 rdf:type schema:PropertyValue
    111 N170dd7d42a4440459268949e8dbe62a7 rdf:first sg:person.014363237745.59
    112 rdf:rest N94837e493aab49a09498b9aa8fd1e81f
    113 N77d9da61bd49462ea86232531c869747 schema:name doi
    114 schema:value 10.1007/s13160-021-00470-2
    115 rdf:type schema:PropertyValue
    116 N94837e493aab49a09498b9aa8fd1e81f rdf:first sg:person.014076405020.50
    117 rdf:rest rdf:nil
    118 Nab45551ac20649ec8d9556ae928195ea schema:issueNumber 3
    119 rdf:type schema:PublicationIssue
    120 Nc165f037cb53475abd635266f2f620b4 schema:name Springer Nature - SN SciGraph project
    121 rdf:type schema:Organization
    122 Ncc3fb390b05b46ff940dfce5c8558b35 schema:volumeNumber 38
    123 rdf:type schema:PublicationVolume
    124 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    125 schema:name Mathematical Sciences
    126 rdf:type schema:DefinedTerm
    127 anzsrc-for:0103 schema:inDefinedTermSet anzsrc-for:
    128 schema:name Numerical and Computational Mathematics
    129 rdf:type schema:DefinedTerm
    130 sg:journal.1041814 schema:issn 0916-7005
    131 1868-937X
    132 schema:name Japan Journal of Industrial and Applied Mathematics
    133 schema:publisher Springer Nature
    134 rdf:type schema:Periodical
    135 sg:person.014076405020.50 schema:affiliation grid-institutes:grid.259237.8
    136 schema:familyName Dai
    137 schema:givenName Weizhong
    138 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014076405020.50
    139 rdf:type schema:Person
    140 sg:person.014363237745.59 schema:affiliation grid-institutes:grid.185648.6
    141 schema:familyName Nwankwo
    142 schema:givenName Chinonso
    143 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014363237745.59
    144 rdf:type schema:Person
    145 sg:pub.10.1007/978-3-319-23413-7_10 schema:sameAs https://app.dimensions.ai/details/publication/pub.1091476142
    146 https://doi.org/10.1007/978-3-319-23413-7_10
    147 rdf:type schema:CreativeWork
    148 sg:pub.10.1007/s10203-018-0224-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1110065100
    149 https://doi.org/10.1007/s10203-018-0224-1
    150 rdf:type schema:CreativeWork
    151 sg:pub.10.1007/s10614-013-9405-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1034772769
    152 https://doi.org/10.1007/s10614-013-9405-8
    153 rdf:type schema:CreativeWork
    154 sg:pub.10.1007/s11075-009-9290-3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1040005424
    155 https://doi.org/10.1007/s11075-009-9290-3
    156 rdf:type schema:CreativeWork
    157 sg:pub.10.1007/s13160-017-0261-0 schema:sameAs https://app.dimensions.ai/details/publication/pub.1090318547
    158 https://doi.org/10.1007/s13160-017-0261-0
    159 rdf:type schema:CreativeWork
    160 sg:pub.10.1023/a:1021190918665 schema:sameAs https://app.dimensions.ai/details/publication/pub.1029279662
    161 https://doi.org/10.1023/a:1021190918665
    162 rdf:type schema:CreativeWork
    163 grid-institutes:grid.185648.6 schema:alternateName Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 60607, Chicago, IL, USA
    164 schema:name Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 60607, Chicago, IL, USA
    165 rdf:type schema:Organization
    166 grid-institutes:grid.259237.8 schema:alternateName Department of Mathematics and Statistics, Louisiana Tech University, 71272, Ruston, LA, USA
    167 schema:name Department of Mathematics and Statistics, Louisiana Tech University, 71272, Ruston, LA, USA
    168 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...