# Derivatives Question and Answers

Essay by Amine Arezki • April 10, 2016 • Exam • 6,152 Words (25 Pages) • 1,021 Views

**Page 1 of 25**

## Payoff Diagrams (16p)

Draw the gross payoff (not net-payoff/profit) diagram as a function of MLM stock for the following portfolios consisting of: (Strike values are given in parentheses)

- one long position in the stock and two short positions in the same put option (K).
- two long positions in the stock, two short call options (2K), and one long position ina put (K).
- two short positions in the stock, two long call options (2K), and one short put option(3K).
- one long position in the stock, two short call options (2K), two short call options (3K),and one short put option (2K). For this question, take into account the net profit only for the stock. You can assume that the stock is purchased at a price of K.

## Binomial Model (27p)

Chevron Corporation has stakes in various oil development projects in Yemen and is considering to bid on a development of a new oil platform in Yemen in the future. However, due to ongoing civil war in Yemen and the instability of the Yemeni Currency (YER), Chevron wants to be hedged against two possible scenarios:

- First scenario: The country becomes stable in the future and Chevron decides to go on with the bid and the bid is accepted, in that case Chevron would need Yemeni Rial (YER).
- Second scenario: The country’s situation deteriorates and Chevron is forced to liquidate its current assets, in that case Chevron would want to exchange Yemeni Rial for USD and would want to be hedged against a possible depreciation of the currency.

Chevron approaches J.P Morgan and asks for a derivative on the Yemeni Rial currency exchange rate that would hedge the company’s interests in both scenarios. J.P. Morgan suggests an exotic option on YER/USD. More specifically, the bank suggests a three-year chooser option. The chooser option is a special type of option contract that allows the purchaser to decide during a predetermined period of time whether the derivative will be a European put or a European call option. This way, Chevron can choose later, after two years, if the option is a call or put after, when it has more certainty about the future scenario. Assume that the constant risk-free rate is 4% (using continuous compounding) and that the estimated monthly sample standard deviation of the underlying YER/USD currency exchange rate is 6.35%. Using a three-step binomial tree with a time step of 1 year, calculate the following:

- Calculate u, d (the up and down multipliers in the binomial tree), and the risk neutralprobability of an upward move, p.
- Assume that the current YER/USD exchange rate is 224 (S0 = 224 ), and draw the binomial tree of the underlying asset of the option.
- Use the binomial tree to find the value of a simple three-year European put option onthe YER/USD exchange rate at t=0, with the strike price of K=235. How would your answer change if the option was American?
- Assume that the strike prices of the chooser call and put options are 230 and 235,respectively. Calculate the value of the chooser option at time zero, considering that Chevron has to make the decision on whether the option is call or a put at t=2.
- Given the high volatility of the underlying YER/USD currency exchange rate, Chevronbelieves that the option is expensive. In addition, there is some uncertainty for scenario 1, i.e., if Chevron’s bid on the new oil project would not be accepted, then the money spent on the option would be wasted. In order to address this concern, J.P.

Morgan suggests a compound option instead.^{[1]} More precisely, J.P. Morgan suggests that Chevron buys a second chooser option on the value of the original chooser option, for which the price was calculated in part (d). For the second chooser option, the decison that the option is a call or a put is made at t=1, and the strike prices for the call and the put options are 100 and 54 respectively. Using a binomial tree, calculate the value of the compound option at t=0.

- What would be the benefits or downsides of using the compound option given thedifferent possible scenarios? Explain briefly!

## Black-Scholes-Merton Model (32p)

Go to the website of the Montreal Exchange (http://www.m-x.ca/accueil en.php) and find the historical prices of traded options on Suncor Energy (symbol: SU) on February 10, 2016. Consider the March 2016 contract, i.e., the one-month option. The three month LIBOR rate is 0.62% (continuously compounded), a good proxy for the risk-free rate.

- What are the standardized contract terms of this equity option contract (contract size,expiration day, contract type, American or European, clearing corporation)?
- What is the range of available strike prices offered by TMX on February 10, 2016?Which option had the highest open interest and which option had the greatest trading volume on that day? Consider call and put options separately.
- Calculate the percentage bid-ask spreads for the call options with strike prices equalto $22, $29, and $42. Using the percentage bid-ask spread as a measure of illiquidity, which option is the most illiquid among the in-the-money call, the at-the-money call, and the out-of-money call? How does the liquidity of the options compare to the liquidity of the underlying stock? Hint: The percentage bid-ask spread is defined as

BS = 2|PriceBid−PriceAsk|/(PriceBid+PriceAsk). TMX’s website also provides price information and historical volatility for the underlying.

- Use the call and put option prices with a strike price of $30 to derive the implieddividend yield of the underlying stock? Use the last price as the market price for call and put options. Assume that the time-to-maturity is 38 days.
- Use an Excel sheet (or any other software you prefer) to compute the historical volatility of the underlying stock return that you could use to price another option using the Black-Scholes framework. To answer this question, use daily stock returns from February 11, 2015 to February 10, 2016 for the Suncore Energy stock, available from the Yahoo Finance web page.
^{[2]} - The TMX website also provides closing prices and the past 30-day historical volatilityof the underlying stock. Use this information and the BSM formula to calculate the price of a European call and a European put option with strike prices equal to $29, and an expiration date in March.
- Repeat the process in (f) for call and put options with strike prices equal to $44 and$22, and an expiration date in March. Compare the BSM prices to the market prices for in-the-money and out-of-money options? What do you observe? If your calculated price differs from the market price, what would be a potential explanation for this discrepancy? List at least two reasons. Is this an arbitrage opportunity?
- You are analyzing financial markets and conclude that you are neutral about the direction and the volatility of future price movements. To implement your view in a trading strategy, you could purchase a butterfly call spread on Suncor Energy, using options at strike prices of $24, $29, and $35. What is the cost of establishing such a trading position? What is the portfolio’s delta, gamma, and vega? Hint: You can rely on the Black-Scholes formula to calculate the Greeks.
- Suppose you wrote a covered call on February 10, 2016. Thus, you purchased one shareof the Suncor Energy stock, and you shorted one March-18 call option with a strike price of $34. After one month, on March 10, 2016, the stock price has increased to $34.28. You decide to “roll up and out,” using the July-17 call contract with a strike price of $38.
^{[3]}Describe the meaning of “rolling up and out,” and in what situation this would be a good strategy. Calculate the net payoff of this rollover strategy (ignoring the time value of money) and explain its benefits and risks.

## Greeks (25p)

Consider a one-year European call option on a stock with time-to-maturity of 6-months when the stock price is $48, the strike price is $50, the risk-free rate is 5% (measured with continuous compounding), and the volatility is 30% per annum. The underlying stock pays no dividends.

- What is the call option premium, the delta, the gamma, and the vega, according tothe Black-Scholes-Merton (BSM) model?
- Given your answers in (a), compute the delta-approximated option price when thestock price is equal to $45, $47, $49, and $51, respectively.
- How does your answer in (b) change if you compute the delta-gamma approximationof the predicted option price change when the stock price is equal to $45, $47, $49, and $51, respectively? Calculate and compare the relative prediction error (in %) of the delta and the delta-gamma approximations obtained in answers (b) and (c). How is the relative prediction error related to the change of the underlying stock price? Does

the delta approximation tend to predict a greater or a lower price change than the actual change in option prices? Explain! Hint: The relative prediction error is defined as |PriceApproximation − PriceBS|/PriceBS.

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