problem solving in financial mathematics

a companion blog for a blog on option pricing models

Pricing American options using binomial tree

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Practice problems in this post reinforce the following blog post on pricing American options using binomial trees:

found in this companion blog.

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Practice Problems

Practice Problem 1
The inputs to a binomial tree are:

  • The initial stock price S is $40.
  • The strike price K is $45.
  • The stock is non-dividend paying.
  • The annual standard deviation of the stock return is \sigma= 0.3.
  • The annual risk-free interest rate is r= 5%.
  • The binomial tree has 3 periods.
  • The time to expiration T= 0.5 (6 months).

Price an American put option using this binomial tree.

Practice Problem 2
The inputs to a binomial tree are:

  • The initial stock price S is $50.
  • The strike price K is $55.
  • The stock pays dividends at the annual continuous rate of \delta= 6%.
  • The annual standard deviation of the stock return is \sigma= 0.25.
  • The annual risk-free interest rate is r= 4%.
  • The binomial tree has 3 periods.
  • The time to expiration T= 1.5 (18 months).

Price an American call option using this binomial tree.

Practice Problem 3
The inputs to a binomial tree are:

  • The initial stock price S is $40.
  • The strike price K is $45.
  • The stock is non-dividend paying.
  • The annual standard deviation of the stock return is \sigma= 0.3.
  • The annual risk-free interest rate is r= 5%.
  • The binomial tree has 3 periods.
  • The time to expiration T= 0.25 (3 months).

Price both the American put option using this binomial tree. The European put option using this binomial tree is priced in Example 3 in this previous post.

Practice Problem 4
The inputs to a binomial tree are:

  • The initial stock price S is $50.
  • The strike price K is $60.
  • The stock pays dividends at the annual continuous rate of \delta= 5%.
  • The annual standard deviation of the stock return is \sigma= 0.3.
  • The annual risk-free interest rate is r= 2%.
  • The binomial tree has 3 periods.
  • The time to expiration T= 2 years.

Price both the American put option using this binomial tree. The European put option using this binomial tree is priced in Example 4 in this previous post.

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Answers

    \text{ }

    Practice Problem 1 – pricing 6-month American put
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 59.22258163 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 0 \\        \text{ } & \text{ } & S_{uu}=\$ 51.96108614   & \text{ } \\   \text{ } & \text{ } & C_{uu}=\$ 0   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 46.3561487 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 0 \\     \text{ } & S_u=\$ 45.58994896  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 2.41285153  & \text{ }    & \text{ } \\     S=\$ 40 &  \text{ } & S_{ud}=S_{du}=\$ 40.67225322    & \text{ } \\   C=\$ 6.024433917 &  \text{ } & C_{ud}=\$ 4.585624746    & \text{ } \\    \text{ } & S_d=\$ 35.68528077 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 9.314719233 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 36.28501939 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 8.714980615 \\      \text{ } & \text{ } & S_{dd}=\$ 31.83598158   & \text{ } \\     \text{ } & \text{ } & \mathbf{C_{dd}=\$ 13.16401842}   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 28.40189853 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 16.59810147 \\      \end{array}

    \text{ }

In the above tree, the option value in bold is a node where early exercise is optimal.

    \text{ }

    Practice Problem 1 – pricing 6-month American put – Replicating portfolios
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & \text{N/A} \\   \text{ } & \text{ } & \text{ }   & \text{N/A} \\        \text{ } & \text{ } & \Delta=0   & \text{ } \\   \text{ } & \text{ } & B=\$ 0   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  \text{N/A} \\  \text{ } & \text{ } & \text{ }   &  \text{N/A} \\     \text{ } & \Delta=-0.40620893  & \text{ }    & \text{ } \\   \text{ } & B=\$ 20.93189592  & \text{ }    & \text{ } \\     \Delta=-0.696829775 &  \text{ } & \Delta=-0.865342936    & \text{ } \\   B=\$ 33.89762491 &  \text{ } & B=\$ 39.78107178    & \text{ } \\    \text{ } & \Delta=-0.970815976 \text{ }   &  \text{ } \\   \text{ } & B=\$ 43.70516647 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  \text{N/A} \\   \text{ } & \text{ } & \text{ }   &  \text{N/A} \\      \text{ } & \text{ } & \Delta=-1   & \text{ } \\     \text{ } & \text{ } & B=\$ 44.62655817   & \text{ } \\       \text{ } & \text{ } & \text{ } & \text{N/A} \\  \text{ } & \text{ } & \text{ } & \text{N/A} \\      \end{array}

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    Practice Problem 2 – pricing 1.5-year American call
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 82.46327901 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 27.46327901 \\        \text{ } & \text{ } & S_{uu}=\$ 69.79597868   & \text{ } \\   \text{ } & \text{ } & \mathbf{C_{uu}=\$ 14.79597868}   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 57.9047663 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 2.904766301 \\     \text{ } & S_u=\$ 59.07452017  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 7.304509772  & \text{ }    & \text{ } \\     S=\$ 50 &  \text{ } & S_{ud}=S_{du}=\$ 49.00993367    & \text{ } \\   C=\$ 3.573713671 &  \text{ } & C_{ud}=\$ 1.298118927    & \text{ } \\    \text{ } & S_d=\$ 41.48144879 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 0.580119904 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 40.66006107 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 0 \\      \text{ } & \text{ } & S_{dd}=\$ 34.41421187   & \text{ } \\     \text{ } & \text{ } & C_{dd}=\$ 0   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 28.55102735 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 0 \\      \end{array}

    \text{ }

In the above tree, the option value in bold is a node where early exercise is optimal.

    \text{ }

    Practice Problem 2 – pricing 1.5-year American call – Replicating portfolios
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & \text{N/A} \\   \text{ } & \text{ } & \text{ }   & \text{N/A} \\        \text{ } & \text{ } & \Delta=0.970445534   & \text{ } \\   \text{ } & \text{ } & B=-\$ 53.91092703   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  \text{N/A} \\  \text{ } & \text{ } & \text{ }   &  \text{N/A} \\     \text{ } & \Delta=0.630179416  & \text{ }    & \text{ } \\   \text{ } & B=-\$ 29.92303685  & \text{ }    & \text{ } \\     \Delta=0.370921823 &  \text{ } & \Delta=0.163465681    & \text{ } \\   B=-\$ 14.97237748 &  \text{ } & B=-\$ 6.713323251    & \text{ } \\    \text{ } & \Delta=0.086309792 \text{ }   &  \text{ } \\   \text{ } & B=-\$ 3.00013532 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  \text{N/A} \\   \text{ } & \text{ } & \text{ }   &  \text{N/A} \\      \text{ } & \text{ } & \Delta=0   & \text{ } \\     \text{ } & \text{ } & B=\$ 0   & \text{ } \\       \text{ } & \text{ } & \text{ } & \text{N/A} \\  \text{ } & \text{ } & \text{ } & \text{N/A} \\      \end{array}

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    \text{ }

    Practice Problem 3 – pricing 3-month European put
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 52.51963372 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 0 \\        \text{ } & \text{ } & S_{uu}=\$ 47.96242387   & \text{ } \\   \text{ } & \text{ } & C_{uu}=\$ 0.432623115   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 44.16718067 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 0.83281933 \\     \text{ } & S_u=\$ 43.80065016  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 2.629551537  & \text{ }    & \text{ } \\     S=\$ 40 &  \text{ } & S_{ud}=S_{du}=\$ 40.33472609    & \text{ } \\   C=\$ 5.494200779 &  \text{ } & \mathbf{C_{ud}=\$ 4.665273912}    & \text{ } \\    \text{ } & S_d=\$ 36.83481952 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 8.165180481 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 37.1430589 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 7.856941105 \\      \text{ } & \text{ } & S_{dd}=\$ 33.92009822   & \text{ } \\     \text{ } & \text{ } & \mathbf{C_{dd}=\$ 11.07990178}   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 31.2360174 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 13.7639826 \\      \end{array}

    \text{ }

In the above tree, the option values in bold are nodes where early exercise is optimal.

    \text{ }

    Practice Problem 3 – pricing 3-month European put – Replicating portfolios
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & \text{N/A} \\   \text{ } & \text{ } & \text{ }   & \text{N/A} \\        \text{ } & \text{ } & \Delta=-0.099709549   & \text{ } \\   \text{ } & \text{ } & B=\$ 5.21493479   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  \text{N/A} \\  \text{ } & \text{ } & \text{ }   &  \text{N/A} \\     \text{ } & \Delta=-0.554905415  & \text{ }    & \text{ } \\   \text{ } & B=\$ 26.93476947  & \text{ }    & \text{ } \\     \Delta=-0.794683251 &  \text{ } & \Delta=-1    & \text{ } \\   B=\$ 37.28153083 &  \text{ } & B=\$ 44.81289008    & \text{ } \\    \text{ } & \Delta=-1 \text{ }   &  \text{ } \\   \text{ } & B=\$ 44.81289008 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  \text{N/A} \\   \text{ } & \text{ } & \text{ }   &  \text{N/A} \\      \text{ } & \text{ } & \Delta=-1   & \text{ } \\     \text{ } & \text{ } & B=\$ 44.81289008   & \text{ } \\       \text{ } & \text{ } & \text{ } & \text{N/A} \\  \text{ } & \text{ } & \text{ } & \text{N/A} \\      \end{array}

    \text{ }

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    \text{ }

    Practice Problem 4 – pricing 2-year American call
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 98.18661752 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 38.18661752 \\        \text{ } & \text{ } & S_{uu}=\$ 78.40760726   & \text{ } \\   \text{ } & \text{ } & \mathbf{C_{uu}=\$ 18.40760726}   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 60.15785233 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 0.15785233 \\     \text{ } & S_u=\$ 62.61294086  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 8.012981928  & \text{ }    & \text{ } \\     S=\$ 50 &  \text{ } & S_{ud}=S_{du}=\$ 48.03947196    & \text{ } \\   C=\$ 3.488038698 &  \text{ } & C_{ud}=\$ 0.068389797    & \text{ } \\    \text{ } & S_d=\$ 38.36225491 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 0.029629999 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 36.85804938 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 0 \\      \text{ } & \text{ } & S_{dd}=\$ 29.43325204   & \text{ } \\     \text{ } & \text{ } & C_{dd}=\$ 0   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 22.58251835 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 0 \\      \end{array}

    \text{ }

In the above tree, the option value in bold is a node where early exercise is optimal.

    \text{ }

    Practice Problem 4 – pricing 2-year American call – Replicating portfolios
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & \text{N/A} \\   \text{ } & \text{ } & \text{ }   & \text{N/A} \\        \text{ } & \text{ } & \Delta=0.9672161   & \text{ } \\   \text{ } & \text{ } & B=-\$ 59.20530971   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  \text{N/A} \\  \text{ } & \text{ } & \text{ }   &  \text{N/A} \\     \text{ } & \Delta=0.584098636  & \text{ }    & \text{ } \\   \text{ } & B=-\$ 28.5591514  & \text{ }    & \text{ } \\     \Delta=0.318408582 &  \text{ } & \Delta=0.00655273    & \text{ } \\   B=-\$ 12.43239039 &  \text{ } & B=-\$ 0.246399887    & \text{ } \\    \text{ } & \Delta=0.00355514 \text{ }   &  \text{ } \\   \text{ } & B=-\$ 0.106753181 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  \text{N/A} \\   \text{ } & \text{ } & \text{ }   &  \text{N/A} \\      \text{ } & \text{ } & \Delta=0   & \text{ } \\     \text{ } & \text{ } & B=\$ 0   & \text{ } \\       \text{ } & \text{ } & \text{ } & \text{N/A} \\  \text{ } & \text{ } & \text{ } & \text{N/A} \\      \end{array}

    \text{ }

For more information on how to calculate the option prices for these practice problems, refer to The binomial option pricing model, part 5.

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\copyright \ 2015 \text{ by Dan Ma}

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