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Chapter 22: Problem 2

Describe the payoff from a portfolio consisting of a lookback call and a lookback put with the same maturity.

Short Answer

Expert verified
The portfolio payoff is \(\max(S_{max} - K, 0) + \max(K - S_{min}, 0)\).

Step by step solution

01

Understand Lookback Option Payoffs

A lookback call option has a payoff based on the difference between the highest price of the underlying asset during the life of the option and the strike price. Conversely, a lookback put option's payoff is based on the difference between the strike price and the lowest price of the underlying asset during the life of the option.
02

Define Payoff Function for Lookback Call

The payoff for a lookback call option can be represented as \( ext{max}(S_{max} - K, 0)\), where \(S_{max}\) is the maximum asset price during the option's life and \(K\) is the strike price.
03

Define Payoff Function for Lookback Put

The payoff for a lookback put option is given by \( ext{max}(K - S_{min}, 0)\), where \(S_{min}\) is the minimum asset price during the option's life.
04

Combine the Payoffs from Both Options

Since the portfolio consists of both a lookback call and a put with the same maturity, the total payoff is the sum of both: \[ ext{Total Payoff} = ext{max}(S_{max} - K, 0) + ext{max}(K - S_{min}, 0).\]
05

Analyze the Portfolio Payoff

The combined payoff expression reflects that the investor receives a payoff based on both the maximum rise above the strike price and the minimum decline below it. This ensures double benefits: gaining from the highest upside beyond the strike price and the greatest downside below the strike price.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Option Payoffs
Options are financial instruments that offer unique payoffs. The payoff is the amount earned when the option is exercised. - **Call Options**: Provide a payoff when the market price exceeds the strike price. - **Put Options**: Offer a benefit if the market price is below the strike price. Both are used to hedge risks or speculate in financial markets. By understanding the payoff, investors can better predict the potential profits or losses.
Lookback Call Option
A lookback call option allows you to "look back" over the life of the option to determine its payoff. This option's value is tied to the highest price reached by the underlying asset before the option expires.The payoff formula is:\[\text{max}(S_{max} - K, 0)\]- **\(S_{max}\)**: The maximum asset price over the option duration.- **\(K\)**: The strike price.This allows investors to benefit from the maximum rise in asset price, making it a potent tool during volatile market conditions.
Lookback Put Option
A lookback put option provides a payoff based on the lowest price of the underlying asset during the option's life. This flexibility means the option considers the most favorable point in time to exercise.The payoff for a lookback put is:\[\text{max}(K - S_{min}, 0)\]- **\(S_{min}\)**: The lowest asset price during the option's life.- **\(K\)**: The strike price.This option ensures you can capitalize on the minimum price drop, adding strategic value in uncertain markets.
Portfolio Payoff Analysis
Combining a lookback call and lookback put option into one portfolio provides compelling benefits. The overall payoff combines both upside and downside protections.The combined expression is:\[\text{Total Payoff} = \text{max}(S_{max} - K, 0) + \text{max}(K - S_{min}, 0)\]- **Upside Capture**: Benefits from the highest price above the strike.- **Downside Protection**: Gains from the lowest reverse move below the strike.This dual feature ensures that investors can maximize profits or minimize losses, making it a versatile tool in various market conditions.

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Most popular questions from this chapter

What is the value of a derivative that pays off \(\$ 100\) in 6 months if the S&P 500 index is greater than 1,000 and zero otherwise? Assume that the current level of the index is 960 the risk-free rate is \(8 \%\) per annum, the dividend yield on the index is \(3 \%\) per annum, and the volatility of the index is \(20 \%\)

Does a down-and-out call become more valuable or less valuable as we increase the frequency with which we observe the asset price in determining whether the barrier has been crossed? What is the answer to the same question for a down- and-in call?

Calculate the price of a 1 -year European option to give up 100 ounces of silver in exchange for 1 ounce of gold. The current prices of gold and silver are \(\$ 380\) and \(\$ 4\) respectively; the risk-free interest rate is \(10 \%\) per annum; the volatility of each commodity price is \(20 \% ;\) and the correlation between the two prices is 0.7 . Ignore storage costs.

What is the value in dollars of a derivative that pays off \(拢 10,000\) in 1 year provided that the dollar/sterling exchange rate is greater than 1.5000 at that time? The current exchange rate is \(1.4800 .\) The dollar and sterling interest rates are \(4 \%\) and \(8 \%\) per annum, respectively. The volatility of the exchange rate is \(12 \%\) per annum.

Consider a chooser option where the holder has the right to choose between a European call and a European put at any time during a 2 -year period. The maturity dates and strike prices for the calls and puts are the same regardless of when the choice is made. Is it ever optimal to make the choice before the end of the 2 -year period? Explain your answer.
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