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(3) Consider the following table of Call/Put premia for maturity T=1 year:
\b...
May 4, 2024
a Solution
a
Profit Diagram for Long Stock + Long Put (K=95): The profit for being long on a stock and long on a put option is the sum of the individual profits. For the stock, the profit is ST−S0. For the put option with strike price K=95, the profit is max(K−ST,0)−Put Premium. The total profit is thus ST−S0+max(95−ST,0)−2.64
b
Profit Diagram for Long Stock + Long Put (K=100): Similarly, for a put option with strike price K=100, the profit is max(K−ST,0)−Put Premium. The total profit is ST−S0+max(100−ST,0)−4.76
a Answer
The profit diagrams for long stock combined with long put options at different strike prices would reflect the stock's price gain or loss adjusted by the protective put's payoff minus the respective put premiums.
b Solution
a
Profit Diagram for Long Stock + Short Call (K=95): When you are long on the stock and short on a call option, the profit from the stock is ST−S0, and the profit from the short call is Call Premium−max(ST−K,0). The total profit is ST−S0+12.56−max(ST−95,0)
b
Profit Diagram for Long Stock + Short Call (K=105): For the call option with strike price K=105, the profit is Call Premium−max(ST−K,0). The total profit is ST−S0+8.68−max(ST−105,0)
b Answer
The profit diagrams for long stock combined with short call options at different strike prices would reflect the stock's price gain or loss adjusted by the income from the call premium and the potential loss if the stock price exceeds the strike price.
c Solution
a
Expected Profit Calculation: The expected profit for a uniform distribution is the average value of the profit over the range. For each strategy, integrate the profit function over the range [90,125] and divide by the range's length (35)
c Answer
The strategy with the highest average profit will be the one with the highest integral value of the profit function over the specified range, divided by the length of the range.
d Solution
a
Variance of Profit: Variance is calculated as the expected value of the squared deviation from the mean profit. For each strategy, calculate the integral of the squared profit function over the range [90,125] and divide by the range's length (35), then subtract the square of the expected profit
d Answer
The strategy with the lowest variance of profit is considered the least risky. It is found by comparing the variances calculated for each strategy.
Key Concept
Hedging and Risk Management
Explanation
The use of options in combination with stock positions can alter the risk-return profile of an investment, providing opportunities for hedging and risk management.
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