# problem solving in financial mathematics

## a companion blog for a blog on option pricing models # Pricing American options using binomial tree

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. $\text{ }$
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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 $\text{ }$ $\displaystyle \begin{array}{llll} \displaystyle \text{Initial Price} & \text{Period 1} & \text{Period 2} & \text{Period 3} \\ \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \Delta=0 & \text{ } \\ \text{ } & \text{ } & B=\ 0 & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \Delta=-0.40620893 & \text{ } & \text{ } \\ \text{ } & B=\ 20.93189592 & \text{ } & \text{ } \\ \Delta=-0.696829775 & \text{ } & \Delta=-0.865342936 & \text{ } \\ B=\ 33.89762491 & \text{ } & B=\ 39.78107178 & \text{ } \\ \text{ } & \Delta=-0.970815976 \text{ } & \text{ } \\ \text{ } & B=\ 43.70516647 \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \Delta=-1 & \text{ } \\ \text{ } & \text{ } & B=\ 44.62655817 & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \end{array}$ $\text{ }$

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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 $\text{ }$ $\displaystyle \begin{array}{llll} \displaystyle \text{Initial Price} & \text{Period 1} & \text{Period 2} & \text{Period 3} \\ \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \Delta=0.970445534 & \text{ } \\ \text{ } & \text{ } & B=-\ 53.91092703 & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \Delta=0.630179416 & \text{ } & \text{ } \\ \text{ } & B=-\ 29.92303685 & \text{ } & \text{ } \\ \Delta=0.370921823 & \text{ } & \Delta=0.163465681 & \text{ } \\ B=-\ 14.97237748 & \text{ } & B=-\ 6.713323251 & \text{ } \\ \text{ } & \Delta=0.086309792 \text{ } & \text{ } \\ \text{ } & B=-\ 3.00013532 \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \Delta=0 & \text{ } \\ \text{ } & \text{ } & B=\ 0 & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \end{array}$ $\text{ }$

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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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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.

_____________________________________________________________________________________ $\copyright \ 2015 \text{ by Dan Ma}$

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