# problem solving in financial mathematics

## 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

## Pricing European options using multiperiod binomial trees

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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Practice Problem 1 – pricing 6-month European put
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$\displaystyle \begin{array}{lllll} \displaystyle \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\ 55.36122584 \\ \text{ } & \text{ } & \text{ } & \text{ } & C_{uu}=\ 0 \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & S_u=\ 47.05793274 & \text{ } & \text{ } \\ \text{ } & \text{ } & C_u=\ 2.116325081 & \text{ } & \text{ } \\ \text{ } & \text{ } & \Delta=-0.277893964 & \text{ } & \text{ } \\ \text{ } & \text{ } & B= \ 15.19344057 & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ S= \ 40 & \text{ } & \text{ } & \text{ } & S_{ud}=\ 41.01260482 \\ C=\ 6.051211415 & \text{ } & \text{ } & \text{ } & C_{ud}=\ 3.98739518 \\ \Delta=-0.611918665 & \text{ } & \text{ } & \text{ } & \text{ } \\ B= \ 30.52795801 & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & S_d=\ 34.861374 & \text{ } & \text{ } \\ \text{ } & \text{ } & C_d=\ 9.579627019 & \text{ } & \text{ } \\ \text{ } & \text{ } & \Delta=-1 & \text{ } & \text{ } \\ \text{ } & \text{ } & B= \ 44.44100102 & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\ 30.38288493 \\ \text{ } & \text{ } & \text{ } & \text{ } & C_{dd}=\ 14.61711507 \end{array}$
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Practice Problem 2 – pricing 6-month European call
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$\displaystyle \begin{array}{lllll} \displaystyle \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\ 82.627665586 \\ \text{ } & \text{ } & \text{ } & \text{ } & C_{uu}=\ 27.62766586 \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & S_u=\ 70.41065226 & \text{ } & \text{ } \\ \text{ } & \text{ } & C_u=\ 15.9579114 & \text{ } & \text{ } \\ \text{ } & \text{ } & \Delta=1 & \text{ } & \text{ } \\ \text{ } & \text{ } & B=- \ 54.45274086 & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ S= \ 60 & \text{ } & \text{ } & \text{ } & S_{ud}=\ 61.2120804 \\ C=\ 8.821942361 & \text{ } & \text{ } & \text{ } & C_{ud}=\ 6.2120804 \\ \Delta=0.718552622 & \text{ } & \text{ } & \text{ } & \text{ } \\ B=- \ 34.291215 & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & S_d=\ 52.16149412 & \text{ } & \text{ } \\ \text{ } & \text{ } & C_d=\ 2.844930962 & \text{ } & \text{ } \\ \text{ } & \text{ } & \Delta=0.391557423 & \text{ } & \text{ } \\ \text{ } & \text{ } & B=- \ 17.57928923 & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\ 45.34702449 \\ \text{ } & \text{ } & \text{ } & \text{ } & C_{dd}=\ 0 \end{array}$
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Practice Problem 3 – pricing 3-month European put
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$\displaystyle \begin{array}{llll} \displaystyle \text{Initial Price} & \text{Period 1} & \text{Period 2} & \text{Period 3} \\ \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{uuu}=\ 52.51963372 \\ \text{ } & \text{ } & \text{ } & C_{uuu}=\ 0 \\ \text{ } & \text{ } & S_{uu}=\ 47.96242387 & \text{ } \\ \text{ } & \text{ } & C_{uu}=\ 0.432623115 & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{uud}=\ 44.16718067 \\ \text{ } & \text{ } & \text{ } & C_{uud}=\ 0.83281933 \\ \text{ } & S_u=\ 43.80065016 & \text{ } & \text{ } \\ \text{ } & C_u=\ 2.532353895 & \text{ } & \text{ } \\ S=\ 40 & \text{ } & S_{ud}=S_{du}=\ 40.33472609 & \text{ } \\ C=\ 5.253907227 & \text{ } & C_{ud}=\ 4.478163995 & \text{ } \\ \text{ } & S_d=\ 36.83481952 \text{ } & \text{ } \\ \text{ } & C_d=\ 7.79173865 \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{udd}=\ 37.1430589 \\ \text{ } & \text{ } & \text{ } & C_{udd}=\ 7.856941105 \\ \text{ } & \text{ } & S_{dd}=\ 33.92009822 & \text{ } \\ \text{ } & \text{ } & C_{dd}=\ 10.89279186 & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{ddd}=\ 31.2360174 \\ \text{ } & \text{ } & \text{ } & C_{ddd}=\ 13.7639826 \\ \end{array}$

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Practice Problem 3 – pricing 3-month European put – Replicating portfolios
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$\displaystyle \begin{array}{llll} \displaystyle \text{Initial Price} & \text{Period 1} & \text{Period 2} & \text{Period 3} \\ \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \Delta=-0.099709549 & \text{ } \\ \text{ } & \text{ } & B=\ 5.21493479 & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \Delta=-0.530375088 & \text{ } & \text{ } \\ \text{ } & B=\ 25.76312758 & \text{ } & \text{ } \\ \Delta=-0.755026216 & \text{ } & \Delta=-1 & \text{ } \\ B=\ 35.45495588 & \text{ } & B=\ 44.81289008 & \text{ } \\ \text{ } & \Delta=-1 \text{ } & \text{ } \\ \text{ } & B=\ 44.62655817 \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \Delta=-1 & \text{ } \\ \text{ } & \text{ } & B=\ 44.81289008 & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \end{array}$

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Practice Problem 4 – pricing 2-year European call
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$\displaystyle \begin{array}{llll} \displaystyle \text{Initial Price} & \text{Period 1} & \text{Period 2} & \text{Period 3} \\ \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{uuu}=\ 98.18661752 \\ \text{ } & \text{ } & \text{ } & C_{uuu}=\ 38.18661752 \\ \text{ } & \text{ } & S_{uu}=\ 78.40760726 & \text{ } \\ \text{ } & \text{ } & C_{uu}=\ 16.63179043 & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{uud}=\ 60.15785233 \\ \text{ } & \text{ } & \text{ } & C_{uud}=\ 0.15785233 \\ \text{ } & S_u=\ 62.61294086 & \text{ } & \text{ } \\ \text{ } & C_u=\ 7.243606191 & \text{ } & \text{ } \\ S=\ 50 & \text{ } & S_{ud}=S_{du}=\ 48.03947196 & \text{ } \\ C=\ 3.154705319 & \text{ } & C_{ud}=\ 0.068389797 & \text{ } \\ \text{ } & S_d=\ 38.36225491 \text{ } & \text{ } \\ \text{ } & C_d=\ 0.029629999 \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{udd}=\ 36.85804938 \\ \text{ } & \text{ } & \text{ } & C_{udd}=\ 0 \\ \text{ } & \text{ } & S_{dd}=\ 29.43325204 & \text{ } \\ \text{ } & \text{ } & C_{dd}=\ 0 & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{ddd}=\ 22.58251835 \\ \text{ } & \text{ } & \text{ } & C_{ddd}=\ 0 \\ \end{array}$

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Practice Problem 4 – pricing 2-year European call – Replicating portfolios
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$\displaystyle \begin{array}{llll} \displaystyle \text{Initial Price} & \text{Period 1} & \text{Period 2} & \text{Period 3} \\ \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \Delta=0.9672161 & \text{ } \\ \text{ } & \text{ } & B=-\ 59.20530971 & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \Delta=0.527539397 & \text{ } & \text{ } \\ \text{ } & B=-\ 25.78718685 & \text{ } & \text{ } \\ \Delta=0.287722745 & \text{ } & \Delta=0.00655273 & \text{ } \\ B=-\ 11.23143192 & \text{ } & B=-\ 0.246399887 & \text{ } \\ \text{ } & \Delta=0.00355514 \text{ } & \text{ } \\ \text{ } & B=-\ 0.106753181 \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \Delta=0 & \text{ } \\ \text{ } & \text{ } & B=\ 0 & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \text{ } & \text{ } & \text{ } & \text{N/A} \\ \end{array}$

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

## Interpreting hedge ratio in one-period binomial option model

This practice problem reinforces the concept of the hedge ratio discussed in the following post

found in this companion blog.

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Practice Problems

The stock prices in the following problems are modeled by a 1-year binomial tree with $u=$ 1.2 and $d=$ 0.8. The current stock price is $50. The stock is non-dividend paying. The annual risk-free interest rate is 5%. Practice Problem 1 A market maker has just sold a 1-year call option with strike price$55. Determine the replicating portfolio that has the same payoff as this call option. What is the price of this call option?

Practice Problem 2
Repeat Problem 1 for the initial stock prices $55,$60, $65, and$70. What is the pattern of the hedge ratio $\Delta$ as initial stock price goes from $50 to$70? Explain this pattern.

Practice Problem 3
A market maker has just sold a 1-year put option with strike price $45. Determine the replicating portfolio that has the same payoff as this call option. What is the price of this call option? Practice Problem 4 Perform the same calculation for Problem 3 for the initial stock prices$45, $40,$35, and $30. What is the pattern of the hedge ratio $\Delta$ as initial stock price increases? Explain this pattern. $\text{ }$ _____________________________________________________________________________________ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ _____________________________________________________________________________________ Answers Practice Problem 2 $\text{ }$ Call option hedge ratio when initial stock prices are increasing $\left[\begin{array}{lllllllll} \text{Call Option} \\ \text{Strike Price} & \text{ } & \text{Initial Stock Price} & \text{ } & \text{Option Price} & \text{ } & \text{Hedge Ratio } \Delta & \text{ } & \text{Borrowing} \\ \text{ } & \text{ } \\ \ 55 & \text{ } & \ 50 & \text{ } & \ 2.987705755 & \text{ } & 0.25 & \text{ } & -\ 9.512294245 \\ \ 55 & \text{ } & \ 55 & \text{ } & \ 6.572952661 & \text{ } & 0.5 & \text{ } & -\ 20.92704734 \\ \ 55 & \text{ } & \ 60 & \text{ } & \ 10.15819957 & \text{ } & 0.708333333 & \text{ } & -\ 32.34180043 \\ \ 55 & \text{ } & \ 65 & \text{ } & \ 13.74344647 & \text{ } & 0.884615385 & \text{ } & -\ 43.75655353 \\ \ 55 & \text{ } & \ 70 & \text{ } & \ 17.68238165 & \text{ } & 1.000 & \text{ } & -\ 52.317618359 \\ \end{array}\right]$ $\text{ }$ Practice Problem 4 $\text{ }$ Put option hedge ratio when initial stock prices are decreasing $\left[\begin{array}{lllllllll} \text{Put Option} \\ \text{Strike Price} & \text{ } & \text{Initial Stock Price} & \text{ } & \text{Option Price} & \text{ } & \text{Hedge Ratio } \Delta & \text{ } & \text{Lending} \\ \text{ } & \text{ } \\ \ 45 & \text{ } & \ 50 & \text{ } & \ 1.768441368 & \text{ } & -0.25 & \text{ } & \ 14.26844137 \\ \ 45 & \text{ } & \ 45 & \text{ } & \ 3.183194462 & \text{ } & -0.5 & \text{ } & \ 25.68319446 \\ \ 45 & \text{ } & \ 40 & \text{ } & \ 4.597947556 & \text{ } & -0.8125 & \text{ } & \ 37.09794756 \\ \ 45 & \text{ } & \ 35 & \text{ } & \ 7.805324103 & \text{ } & -1.0000 & \text{ } & \ 42.8053241 \\ \ 45 & \text{ } & \ 30 & \text{ } & \ 12.8053241 & \text{ } & -1.0000 & \text{ } & \ 42.8053241 \\ \end{array}\right]$ $\text{ }$ Refer to The binomial option pricing model, part 3 for the explanation. _____________________________________________________________________________________ $\copyright \ 2015 \text{ by Dan Ma}$ ## Basic practice problem set 2 – one-period binomial pricing model This practice problem reinforces the concept of binomial option pricing model discussed in the following two posts found in this companion blog. The practice problems are basic and straightforward problem to price an option using a binomial tree based on a volatility factor $\sigma$. _____________________________________________________________________________________ Practice Problems Practice Problem 1 A 6-month call option on a stock is modeled as a binomial tree. Given the following information: • The initial stock price is$65.
• The strike price is $70. • $r=$ 0.05. • $\delta=$ 0. • $\sigma=$ 0.3. find the replicating portfolio and the price of this 6-month call option. Practice Problem 2 This problem uses the same information as in Problem 1, except that the stock pays annual continuous dividends at the rate of $\delta=0.02$. Practice Problem 3 A 6-month put option on a stock is modeled as a binomial tree. Given the following information: • The initial stock price is$65.
• The strike price is $60. • $r=$ 0.05. • $\delta=$ 0.02. • $\sigma=$ 0.3. use risk-neutral probabilities to price this 6-month put option. $\text{ }$ _____________________________________________________________________________________ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ _____________________________________________________________________________________ Answers 1. Replicating portfolio: $\Delta=$ 0.43508237 shares (long), $B=$ -$22.8747876 (borrowing).
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Call option price = $\Delta S + B$ = $5.405566446 $\text{ }$ 2. Replicating portfolio: $\Delta=$ 0.406303779 shares (long), $B=$ -$21.36173125 (borrowing).
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Call option price = $\Delta S + B$ = $5.048014419 $\text{ }$ • $\text{ }$ $p^*=$ 0.447164974 $1-p^*=$ 0.552835026 $uS=$ =$81.57471137 (up stock price)

$dS=$ =$53.37034398 (down stock price) Put option price = $p^*$ (0) + $(1-p^*)$ (6.629656019) =$3.574614265
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$\copyright \ 2015 \text{ by Dan Ma}$

## Basic practice problem set 1 – one-period binomial pricing model

These practice problems reinforces the concept of binomial option pricing model discussed in the following post

found in this companion blog. The practice problems are basic and straightforward problem to price an option using a binomial tree with arbitrary up factor $u$ and down factor $d$.

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Practice Problems

Practice Problem 1
Suppose that the future prices for a stock are modeled with a one-period binomial tree with $u=1.2$ and $d=0.7$ and having a period of 6 months. The current price of the stock is $80. The stock pays no dividends. The annual risk-free interest rate is $r=$ 4%. • Determine the replicating portfolio and the price of a European 70-strike call option on this stock that will expire in 6 months. Practice Problem 2 The stock is as in Practice Problem 1. Determine the replicating portfolio and the price of a European 85-strike put option on this stock that will expire in 6 months. Practice Problem 3 • Suppose that the price of the call option in Practice Problem 1 is observed to be$16.00. Describe the arbitrage.

Practice Problem 4

• Suppose that the price of the call option in Practice Problem 1 is observed to be $16.90. Describe the arbitrage. Practice Problem 5 • Suppose that the price of the put option in Practice Problem 2 is observed to be$9.80. Describe the arbitrage.

Practice Problem 6

• Suppose that the price of the put option in Practice Problem 2 is observed to be $10.60. Describe the arbitrage. $\text{ }$ _____________________________________________________________________________________ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ _____________________________________________________________________________________ Answers 1. Replicating portfolio: $\Delta=$ 0.65 shares (long), $B=$ -$35.67923171 (borrowing).
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Call option price = $\Delta S + B$ = $16.32076829 $\text{ }$ 2. Replicating portfolio: $\Delta=$ -0.725 shares (short), $B=$$68.221827665 (lending).
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Put option price = $\Delta S + B$ = $10.22182766 $\text{ }$ 3. Buy low (the option at$16.00) and sell the synthetic option at the theoretical price of $16.32076829. The following table shows the cash flows at expiration. $\text{ }$ $\left[\begin{array}{llll} \text{Expiration Cash Flows} & \text{ } & \text{Share Price = } \ 56 & \text{Share Price = } \ 96 \\ \text{ } & \text{ } \\ \text{Sell synthetic call} & \text{ } & \text{ } & \text{ } \\ \ \ \ \ \text{Short 0.65 shares} & \text{ } & - \ 36.4 & - \ 62.4 \\ \ \ \ \ \text{Lend } \ 35.67923 & \text{ } & + \ 36.4 & + \ 36.4 \\ \text{ } & \text{ } \\ \text{Buy call } & \text{ } & \ \ \ 0 & \ \ \ 26 \\ \text{ } & \text{ } \\ \text{Total payoff} & \text{ } & \text{ } \ \ 0 & \ \ \ 0 \end{array}\right]$ $\text{ }$ The above table shows that the buy low sell high strategy produces no loss at expiration of the option regardless of the share prices at the end of the option period. But the payoff at time 0 is certain:$16.32076829 – $16.00 =$0.32076829.

4. Buy low (the synthetic call option at $16.32076829) and sell high (the call option at the observed price of$16.90). The following table shows the cash flows at expiration.
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$\left[\begin{array}{llll} \text{Expiration Cash Flows} & \text{ } & \text{Share Price = } \ 56 & \text{Share Price = } \ 96 \\ \text{ } & \text{ } \\ \text{Buy synthetic call} & \text{ } & \text{ } & \text{ } \\ \ \ \ \ \text{Long 0.65 shares} & \text{ } & + \ 36.4 & + \ 62.4 \\ \ \ \ \ \text{Borrow } \ 35.67923 & \text{ } & - \ 36.4 & - \ 36.4 \\ \text{ } & \text{ } \\ \text{Buy call } & \text{ } & \ \ \ 0 & - \ 26 \\ \text{ } & \text{ } \\ \text{Total payoff} & \text{ } & \text{ } \ \ 0 & \ \ \ 0 \end{array}\right]$

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The above table shows that the buy low sell high strategy produces no loss at expiration of the option regardless of the share prices at the end of the option period. But the payoff at time 0 is certain: $16.90 –$16.32076829 = \$0.57923171.

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