# problem solving in financial mathematics

## a companion blog for a blog on option pricing models # 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. $\text{ }$
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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.

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

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