problem solving in financial mathematics

a companion blog for a blog on option pricing models

Interpreting hedge ratio in one-period binomial option model

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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.

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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.

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\copyright \ 2015 \text{ by Dan Ma}

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