How to calculate implied correlation via observed market price (Margrabe option) The 2019 Stack Overflow Developer Survey Results Are InCan the Heston model be shown to reduce to the original Black Scholes model if appropriate parameters are chosen?Calculate volatility from call option priceImplied Correlation using market quotesImplied Vol vs. Calibrated VolInterpretation of CorrelationPricing of Black-Scholes with dividendHow do they calculate stocks implied volatility?Implied correlationEuropean option Vega with respect to expiry and implied volatilityIs American option price lower than European option price?

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How to calculate implied correlation via observed market price (Margrabe option)



The 2019 Stack Overflow Developer Survey Results Are InCan the Heston model be shown to reduce to the original Black Scholes model if appropriate parameters are chosen?Calculate volatility from call option priceImplied Correlation using market quotesImplied Vol vs. Calibrated VolInterpretation of CorrelationPricing of Black-Scholes with dividendHow do they calculate stocks implied volatility?Implied correlationEuropean option Vega with respect to expiry and implied volatilityIs American option price lower than European option price?










3












$begingroup$


I can't seem to figure out how to do the following: compute the implied correlation $ρ_imp$ by using the observed market price $M_quote$ of a Margrabe option, and solving the non-linear equation shown below:



$$M_quote = e^−q_0Ttimes S_0(0)times N(d_+)−e^−q_1Ttimes S_1(0)times N(d_−)$$



where:



$$beginalign
& d_pm = fraclogfracS_0(0)S_1(0)+(q_1 − q_0 ±σ^2/2)TsigmasqrtT
\[4pt]
& sigma = sqrtsigma^2_0 + sigma^2_1 − 2rho_impsigma_0 sigma_1
endalign$$



Note that $d_− = d_+ − σsqrtT$.










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  • 1




    $begingroup$
    Bear in mind that what you're calculating is the margrabe option implied correlation, it's not necessarily the correct correlation to use for pricing other options, it's important to be aware of that.
    $endgroup$
    – will
    yesterday















3












$begingroup$


I can't seem to figure out how to do the following: compute the implied correlation $ρ_imp$ by using the observed market price $M_quote$ of a Margrabe option, and solving the non-linear equation shown below:



$$M_quote = e^−q_0Ttimes S_0(0)times N(d_+)−e^−q_1Ttimes S_1(0)times N(d_−)$$



where:



$$beginalign
& d_pm = fraclogfracS_0(0)S_1(0)+(q_1 − q_0 ±σ^2/2)TsigmasqrtT
\[4pt]
& sigma = sqrtsigma^2_0 + sigma^2_1 − 2rho_impsigma_0 sigma_1
endalign$$



Note that $d_− = d_+ − σsqrtT$.










share|improve this question









New contributor




Tara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$







  • 1




    $begingroup$
    Bear in mind that what you're calculating is the margrabe option implied correlation, it's not necessarily the correct correlation to use for pricing other options, it's important to be aware of that.
    $endgroup$
    – will
    yesterday













3












3








3


1



$begingroup$


I can't seem to figure out how to do the following: compute the implied correlation $ρ_imp$ by using the observed market price $M_quote$ of a Margrabe option, and solving the non-linear equation shown below:



$$M_quote = e^−q_0Ttimes S_0(0)times N(d_+)−e^−q_1Ttimes S_1(0)times N(d_−)$$



where:



$$beginalign
& d_pm = fraclogfracS_0(0)S_1(0)+(q_1 − q_0 ±σ^2/2)TsigmasqrtT
\[4pt]
& sigma = sqrtsigma^2_0 + sigma^2_1 − 2rho_impsigma_0 sigma_1
endalign$$



Note that $d_− = d_+ − σsqrtT$.










share|improve this question









New contributor




Tara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




I can't seem to figure out how to do the following: compute the implied correlation $ρ_imp$ by using the observed market price $M_quote$ of a Margrabe option, and solving the non-linear equation shown below:



$$M_quote = e^−q_0Ttimes S_0(0)times N(d_+)−e^−q_1Ttimes S_1(0)times N(d_−)$$



where:



$$beginalign
& d_pm = fraclogfracS_0(0)S_1(0)+(q_1 − q_0 ±σ^2/2)TsigmasqrtT
\[4pt]
& sigma = sqrtsigma^2_0 + sigma^2_1 − 2rho_impsigma_0 sigma_1
endalign$$



Note that $d_− = d_+ − σsqrtT$.







black-scholes correlation european-options implied nonlinear






share|improve this question









New contributor




Tara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|improve this question









New contributor




Tara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|improve this question




share|improve this question








edited yesterday









Daneel Olivaw

3,0981629




3,0981629






New contributor




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Check out our Code of Conduct.









asked 2 days ago









TaraTara

186




186




New contributor




Tara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor





Tara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Tara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







  • 1




    $begingroup$
    Bear in mind that what you're calculating is the margrabe option implied correlation, it's not necessarily the correct correlation to use for pricing other options, it's important to be aware of that.
    $endgroup$
    – will
    yesterday












  • 1




    $begingroup$
    Bear in mind that what you're calculating is the margrabe option implied correlation, it's not necessarily the correct correlation to use for pricing other options, it's important to be aware of that.
    $endgroup$
    – will
    yesterday







1




1




$begingroup$
Bear in mind that what you're calculating is the margrabe option implied correlation, it's not necessarily the correct correlation to use for pricing other options, it's important to be aware of that.
$endgroup$
– will
yesterday




$begingroup$
Bear in mind that what you're calculating is the margrabe option implied correlation, it's not necessarily the correct correlation to use for pricing other options, it's important to be aware of that.
$endgroup$
– will
yesterday










2 Answers
2






active

oldest

votes


















2












$begingroup$

Let $rhotriangleqrho_imp$. Note that:
$$fracpartial sigmapartial rho(rho)=-fracsigma_0sigma_1sigma(rho)<0$$
Therefore $sigma$ is monotonic in implied correlation. In addition, the Margrabe pricing function $M(cdot)$ is also monotonic in volatility $sigma$ thus you can find an unique solution to the equation:
$$tag1M_textquote=M(rho)$$
where:
$$M(rho)=e^−q_0TS_0(0)N(d_+)−e^−q_1TS_1(0)N(d_−)$$
and $d_pm$ as defined in your question, with $M_textquote$ the observed market price. In practice, this can be restated as:
$$beginalign
&min_rholeft(M(rho)-M_textquoteright)^2tag2
\
& texts.t. rho in [-1,1]
endalign$$

because $(M(rho)-M_textquote)^2geq0$. This is an optimization problem which can be solved through traditional techniques:



  • The solution suggested by @Alex C will give you a quick, approximate answer;

  • If you want arbitrary precision, you can use a simple Newton algorithm on either $(1)$ or $(2)$ with root value $rho=0$, this is quick to program in Excel VBA, or you can maybe even find an online tool that does it. This PDF explains the method for a vanilla call in a Black-Scholes framework to find the implied volatility, but the set-up is very similar. Another alternative is gradient descent but this would probably take longer to program and you have to do it on $(2)$;

  • You can also use Excel's Solver to find a solution to $(1)$ directly. I have tried with $S_0(0)=$101$, $S_1(0)=$113.5$, $sigma_0=45%$, $sigma_1=37%$, $T=1text year$ and $q_0=q_1=0$ and it has worked just fine.





share|improve this answer











$endgroup$




















    3












    $begingroup$

    We know that $-1lerho_imple 1$ so perhaps the simplest approach is to try the possible values $rho_imp=-1,-0.9,-0.8,cdots,0.8,0.9,+1$, to calculate resulting $sigma$ values, d± values, and $M_quote$ values, then see which of these is closest to the observed market price. If desired you can then search a finer grid between two adjacent assumed correlations to pin it down more precisely. It is a manual but relatively simple method.






    share|improve this answer









    $endgroup$













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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      Let $rhotriangleqrho_imp$. Note that:
      $$fracpartial sigmapartial rho(rho)=-fracsigma_0sigma_1sigma(rho)<0$$
      Therefore $sigma$ is monotonic in implied correlation. In addition, the Margrabe pricing function $M(cdot)$ is also monotonic in volatility $sigma$ thus you can find an unique solution to the equation:
      $$tag1M_textquote=M(rho)$$
      where:
      $$M(rho)=e^−q_0TS_0(0)N(d_+)−e^−q_1TS_1(0)N(d_−)$$
      and $d_pm$ as defined in your question, with $M_textquote$ the observed market price. In practice, this can be restated as:
      $$beginalign
      &min_rholeft(M(rho)-M_textquoteright)^2tag2
      \
      & texts.t. rho in [-1,1]
      endalign$$

      because $(M(rho)-M_textquote)^2geq0$. This is an optimization problem which can be solved through traditional techniques:



      • The solution suggested by @Alex C will give you a quick, approximate answer;

      • If you want arbitrary precision, you can use a simple Newton algorithm on either $(1)$ or $(2)$ with root value $rho=0$, this is quick to program in Excel VBA, or you can maybe even find an online tool that does it. This PDF explains the method for a vanilla call in a Black-Scholes framework to find the implied volatility, but the set-up is very similar. Another alternative is gradient descent but this would probably take longer to program and you have to do it on $(2)$;

      • You can also use Excel's Solver to find a solution to $(1)$ directly. I have tried with $S_0(0)=$101$, $S_1(0)=$113.5$, $sigma_0=45%$, $sigma_1=37%$, $T=1text year$ and $q_0=q_1=0$ and it has worked just fine.





      share|improve this answer











      $endgroup$

















        2












        $begingroup$

        Let $rhotriangleqrho_imp$. Note that:
        $$fracpartial sigmapartial rho(rho)=-fracsigma_0sigma_1sigma(rho)<0$$
        Therefore $sigma$ is monotonic in implied correlation. In addition, the Margrabe pricing function $M(cdot)$ is also monotonic in volatility $sigma$ thus you can find an unique solution to the equation:
        $$tag1M_textquote=M(rho)$$
        where:
        $$M(rho)=e^−q_0TS_0(0)N(d_+)−e^−q_1TS_1(0)N(d_−)$$
        and $d_pm$ as defined in your question, with $M_textquote$ the observed market price. In practice, this can be restated as:
        $$beginalign
        &min_rholeft(M(rho)-M_textquoteright)^2tag2
        \
        & texts.t. rho in [-1,1]
        endalign$$

        because $(M(rho)-M_textquote)^2geq0$. This is an optimization problem which can be solved through traditional techniques:



        • The solution suggested by @Alex C will give you a quick, approximate answer;

        • If you want arbitrary precision, you can use a simple Newton algorithm on either $(1)$ or $(2)$ with root value $rho=0$, this is quick to program in Excel VBA, or you can maybe even find an online tool that does it. This PDF explains the method for a vanilla call in a Black-Scholes framework to find the implied volatility, but the set-up is very similar. Another alternative is gradient descent but this would probably take longer to program and you have to do it on $(2)$;

        • You can also use Excel's Solver to find a solution to $(1)$ directly. I have tried with $S_0(0)=$101$, $S_1(0)=$113.5$, $sigma_0=45%$, $sigma_1=37%$, $T=1text year$ and $q_0=q_1=0$ and it has worked just fine.





        share|improve this answer











        $endgroup$















          2












          2








          2





          $begingroup$

          Let $rhotriangleqrho_imp$. Note that:
          $$fracpartial sigmapartial rho(rho)=-fracsigma_0sigma_1sigma(rho)<0$$
          Therefore $sigma$ is monotonic in implied correlation. In addition, the Margrabe pricing function $M(cdot)$ is also monotonic in volatility $sigma$ thus you can find an unique solution to the equation:
          $$tag1M_textquote=M(rho)$$
          where:
          $$M(rho)=e^−q_0TS_0(0)N(d_+)−e^−q_1TS_1(0)N(d_−)$$
          and $d_pm$ as defined in your question, with $M_textquote$ the observed market price. In practice, this can be restated as:
          $$beginalign
          &min_rholeft(M(rho)-M_textquoteright)^2tag2
          \
          & texts.t. rho in [-1,1]
          endalign$$

          because $(M(rho)-M_textquote)^2geq0$. This is an optimization problem which can be solved through traditional techniques:



          • The solution suggested by @Alex C will give you a quick, approximate answer;

          • If you want arbitrary precision, you can use a simple Newton algorithm on either $(1)$ or $(2)$ with root value $rho=0$, this is quick to program in Excel VBA, or you can maybe even find an online tool that does it. This PDF explains the method for a vanilla call in a Black-Scholes framework to find the implied volatility, but the set-up is very similar. Another alternative is gradient descent but this would probably take longer to program and you have to do it on $(2)$;

          • You can also use Excel's Solver to find a solution to $(1)$ directly. I have tried with $S_0(0)=$101$, $S_1(0)=$113.5$, $sigma_0=45%$, $sigma_1=37%$, $T=1text year$ and $q_0=q_1=0$ and it has worked just fine.





          share|improve this answer











          $endgroup$



          Let $rhotriangleqrho_imp$. Note that:
          $$fracpartial sigmapartial rho(rho)=-fracsigma_0sigma_1sigma(rho)<0$$
          Therefore $sigma$ is monotonic in implied correlation. In addition, the Margrabe pricing function $M(cdot)$ is also monotonic in volatility $sigma$ thus you can find an unique solution to the equation:
          $$tag1M_textquote=M(rho)$$
          where:
          $$M(rho)=e^−q_0TS_0(0)N(d_+)−e^−q_1TS_1(0)N(d_−)$$
          and $d_pm$ as defined in your question, with $M_textquote$ the observed market price. In practice, this can be restated as:
          $$beginalign
          &min_rholeft(M(rho)-M_textquoteright)^2tag2
          \
          & texts.t. rho in [-1,1]
          endalign$$

          because $(M(rho)-M_textquote)^2geq0$. This is an optimization problem which can be solved through traditional techniques:



          • The solution suggested by @Alex C will give you a quick, approximate answer;

          • If you want arbitrary precision, you can use a simple Newton algorithm on either $(1)$ or $(2)$ with root value $rho=0$, this is quick to program in Excel VBA, or you can maybe even find an online tool that does it. This PDF explains the method for a vanilla call in a Black-Scholes framework to find the implied volatility, but the set-up is very similar. Another alternative is gradient descent but this would probably take longer to program and you have to do it on $(2)$;

          • You can also use Excel's Solver to find a solution to $(1)$ directly. I have tried with $S_0(0)=$101$, $S_1(0)=$113.5$, $sigma_0=45%$, $sigma_1=37%$, $T=1text year$ and $q_0=q_1=0$ and it has worked just fine.






          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited 8 hours ago

























          answered yesterday









          Daneel OlivawDaneel Olivaw

          3,0981629




          3,0981629





















              3












              $begingroup$

              We know that $-1lerho_imple 1$ so perhaps the simplest approach is to try the possible values $rho_imp=-1,-0.9,-0.8,cdots,0.8,0.9,+1$, to calculate resulting $sigma$ values, d± values, and $M_quote$ values, then see which of these is closest to the observed market price. If desired you can then search a finer grid between two adjacent assumed correlations to pin it down more precisely. It is a manual but relatively simple method.






              share|improve this answer









              $endgroup$

















                3












                $begingroup$

                We know that $-1lerho_imple 1$ so perhaps the simplest approach is to try the possible values $rho_imp=-1,-0.9,-0.8,cdots,0.8,0.9,+1$, to calculate resulting $sigma$ values, d± values, and $M_quote$ values, then see which of these is closest to the observed market price. If desired you can then search a finer grid between two adjacent assumed correlations to pin it down more precisely. It is a manual but relatively simple method.






                share|improve this answer









                $endgroup$















                  3












                  3








                  3





                  $begingroup$

                  We know that $-1lerho_imple 1$ so perhaps the simplest approach is to try the possible values $rho_imp=-1,-0.9,-0.8,cdots,0.8,0.9,+1$, to calculate resulting $sigma$ values, d± values, and $M_quote$ values, then see which of these is closest to the observed market price. If desired you can then search a finer grid between two adjacent assumed correlations to pin it down more precisely. It is a manual but relatively simple method.






                  share|improve this answer









                  $endgroup$



                  We know that $-1lerho_imple 1$ so perhaps the simplest approach is to try the possible values $rho_imp=-1,-0.9,-0.8,cdots,0.8,0.9,+1$, to calculate resulting $sigma$ values, d± values, and $M_quote$ values, then see which of these is closest to the observed market price. If desired you can then search a finer grid between two adjacent assumed correlations to pin it down more precisely. It is a manual but relatively simple method.







                  share|improve this answer












                  share|improve this answer



                  share|improve this answer










                  answered 2 days ago









                  Alex CAlex C

                  6,65611123




                  6,65611123




















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