problem solving in financial mathematics

a companion blog for a blog on option pricing models

Pricing European options using multiperiod binomial trees

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Practice problems in this post reinforce the following blog post on multiperiod binomial option pricing calculation:

found in this companion blog.

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Practice Problems

Practice Problem 1
The following gives the information on a particular stock.

  • The current stock price is $40.
  • 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%.

Price a 6-month European put option on this stock using a 2-period binomial tree. The strike price of the option is $45. Include the replicating portfolio on each node in the binomial tree.

Practice Problem 2
Calculate the price of a 6-month European call option on a certain stock with the following characteristics:

  • The initial stock price is $60.
  • Strike price of the call option is $55.
  • 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= 4%.

Use a 2-period binomial tree. Include the replicating portfolio on each node in the binomial tree.

Practice Problem 3
The following gives the information on a 3-month European put option:

  • The initial stock price is $40.
  • Strike price of the call option 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%.

Price this put option using a 3-period binomial tree. Include the replicating portfolio on each node in the binomial tree.

Practice Problem 4
The following gives the information on a 2-year European call option:

  • The initial stock price is $50.
  • Strike price of the call option 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%.

Price this call option using a 3-period binomial tree. Include the replicating portfolio on each node in the binomial tree.

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Answers

    \text{ }

    Practice Problem 1 – pricing 6-month European put
    \text{ }
    \displaystyle \begin{array}{lllll} \displaystyle   \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\$ 55.36122584 \\   \text{ } & \text{ } & \text{ } & \text{ } & C_{uu}=\$ 0 \\   \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & S_u=\$ 47.05793274 & \text{ } & \text{ } \\  \text{ } & \text{ } & C_u=\$ 2.116325081 & \text{ } & \text{ } \\  \text{ } & \text{ } & \Delta=-0.277893964 & \text{ } & \text{ } \\  \text{ } & \text{ } & B= \$ 15.19344057 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  S= \$ 40 & \text{ } & \text{ } & \text{ } & S_{ud}=\$ 41.01260482 \\  C=\$ 6.051211415 & \text{ } & \text{ } & \text{ } & C_{ud}=\$ 3.98739518 \\  \Delta=-0.611918665 & \text{ } & \text{ } & \text{ } & \text{ } \\  B= \$ 30.52795801 & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\   \text{ } & \text{ } & S_d=\$ 34.861374 & \text{ } & \text{ } \\   \text{ } & \text{ } & C_d=\$ 9.579627019 & \text{ } & \text{ } \\   \text{ } & \text{ } & \Delta=-1 & \text{ } & \text{ } \\   \text{ } & \text{ } & B= \$ 44.44100102 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\$ 30.38288493  \\  \text{ } & \text{ } & \text{ } & \text{ } & C_{dd}=\$ 14.61711507 \end{array}
    \text{ }

    \text{ }

    Practice Problem 2 – pricing 6-month European call
    \text{ }
    \displaystyle \begin{array}{lllll} \displaystyle   \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\$ 82.627665586 \\   \text{ } & \text{ } & \text{ } & \text{ } & C_{uu}=\$ 27.62766586 \\   \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & S_u=\$ 70.41065226 & \text{ } & \text{ } \\  \text{ } & \text{ } & C_u=\$ 15.9579114 & \text{ } & \text{ } \\  \text{ } & \text{ } & \Delta=1 & \text{ } & \text{ } \\  \text{ } & \text{ } & B=- \$ 54.45274086 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  S= \$ 60 & \text{ } & \text{ } & \text{ } & S_{ud}=\$ 61.2120804 \\  C=\$ 8.821942361 & \text{ } & \text{ } & \text{ } & C_{ud}=\$ 6.2120804 \\  \Delta=0.718552622 & \text{ } & \text{ } & \text{ } & \text{ } \\  B=- \$ 34.291215 & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\   \text{ } & \text{ } & S_d=\$ 52.16149412 & \text{ } & \text{ } \\   \text{ } & \text{ } & C_d=\$ 2.844930962 & \text{ } & \text{ } \\   \text{ } & \text{ } & \Delta=0.391557423 & \text{ } & \text{ } \\   \text{ } & \text{ } & B=- \$ 17.57928923 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\$ 45.34702449  \\  \text{ } & \text{ } & \text{ } & \text{ } & C_{dd}=\$ 0 \end{array}
    \text{ }

    \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=\$ 2.532353895  & \text{ }    & \text{ } \\     S=\$ 40 &  \text{ } & S_{ud}=S_{du}=\$ 40.33472609    & \text{ } \\   C=\$ 5.253907227 &  \text{ } & C_{ud}=\$ 4.478163995    & \text{ } \\    \text{ } & S_d=\$ 36.83481952 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 7.79173865 \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{ } & C_{dd}=\$ 10.89279186   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 31.2360174 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 13.7639826 \\      \end{array}

    \text{ }

    \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=-0.530375088  & \text{ }    & \text{ } \\   \text{ } & B=\$ 25.76312758  & \text{ }    & \text{ } \\     \Delta=-0.755026216 &  \text{ } & \Delta=-1    & \text{ } \\   B=\$ 35.45495588 &  \text{ } & B=\$ 44.81289008    & \text{ } \\    \text{ } & \Delta=-1 \text{ }   &  \text{ } \\   \text{ } & B=\$ 44.62655817 \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{ }

    \text{ }

    Practice Problem 4 – pricing 2-year European 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{ } & C_{uu}=\$ 16.63179043   & \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=\$ 7.243606191  & \text{ }    & \text{ } \\     S=\$ 50 &  \text{ } & S_{ud}=S_{du}=\$ 48.03947196    & \text{ } \\   C=\$ 3.154705319 &  \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{ }

    \text{ }

    Practice Problem 4 – pricing 2-year European 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=0.527539397  & \text{ }    & \text{ } \\   \text{ } & B=-\$ 25.78718685  & \text{ }    & \text{ } \\     \Delta=0.287722745 &  \text{ } & \Delta=0.00655273    & \text{ } \\   B=-\$ 11.23143192 &  \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 4.

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\copyright \ 2015 \text{ by Dan Ma}

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