Question 33: Assuming A One-Year Call Option With An Exercis

22question33 Assuming That A One Year Call Option With An Exercise Pr

Assuming that a one-year call option with an exercise price of $38 is available for the stock of the DEW Corp., consider the following price tree for DEW stock over the next year: figure a. If the sequence of stock prices that DEW stock follows over the year is $40.00, $42.00, $40.32, and $38.71, describe the composition of the initial riskless portfolio of stock and options you would form and all the subsequent adjustments you would have to make to keep this portfolio riskless. Assume the one-year risk-free rate is 6 percent. b. Given the initial DEW price of $40, what are the probabilities of observing each of the four terminal stock prices in one year? (Hint: In arriving at your answer, it will be useful to consider (1) the number of different ways that a particular terminal price could be achieved and (2) the probability of an up or down movement.) c. Use the binomial option model to calculate the present value of this call option. d. Calculate the value of a one-year put option on DEW stock having an exercise price of $38; be sure your answer is consistent with the correct response to Part c.

Paper For Above instruction

The task involves a comprehensive analysis of a one-year call option on DEW Corp.'s stock. This analysis incorporates constructing a riskless portfolio, calculating probabilities of different terminal stock prices, applying the binomial model to estimate the option's present value, and assessing the value of a corresponding put option. This process combines concepts of risk-neutral valuation, binomial option pricing, and risk management strategies in derivatives.

Firstly, understanding the price tree of DEW stock is fundamental. The given sequence of stock prices—$40.00, $42.00, $40.32, and $38.71—depicts possible movements within the year. To form a riskless hedge, an investor would replicate the option by creating a portfolio consisting of holding a specific number of shares and units of options that offset each other's risks. The initial composition would involve solving for the change in option value relative to changes in the stock price to find the hedge ratio (delta). The initial portfolio combines \(\Delta\) shares of stock and a short or long position in the option, based on the calculated hedge ratio.

Adjustments to maintain the riskless nature of the portfolio over subsequent periods involve rebalancing, which requires computing the delta at each node of the price tree. These adjustments ensure the portfolio's value is insensitive to small changes in the underlying stock's price, enabling riskless arbitrage and discounting the future payoffs at the risk-free rate.

The probability assessment involves risk-neutral probabilities, calculated by comparing possible up and down movements relative to the riskless growth rate. Given the initial stock price of $40, the likelihoods of ending at each terminal stock price are derived using combinatorial methods—considering the number of paths leading to each outcome—and the derived risk-neutral probabilities that make the expected growth of the stock consistent with the risk-free rate.

Applying the binomial model involves discounting the expected payoff of the call option under these risk-neutral probabilities to the present value. This calculation accounts for the possible ending stock prices, the corresponding payoffs of the call, and the appropriate discounting factor considering the risk-free rate.

Finally, estimating the value of a one-year put option with the same strike price involves using put-call parity for European options or directly applying the binomial model to compute its price, ensuring consistency with the previously calculated call option value.

This comprehensive approach integrates theoretical models with practical computations, demonstrating the fundamental principles of option pricing, hedging strategies, and probabilistic valuation within financial markets.

References

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