Homework 3: Pine Street Capital (Part I)

FINA 4529: Derivatives (II) Professor Hengjie AiUndergrad, Spring 2013 O¢ ce: 3-127Carlson School of Management Phone: 612-626-7348University of Minnesota Email: hengjie@umn.edu

Homework 3 is due on April 16, Tuesday, at the beginning of the class.

I Introduction and Instructions

Please read the HBS case 9-201-071 Pine Street Capital carefully, and answer the following

questions.

II Case Summary

Pine Street Capital (PSC) is a technology hedge fund located in San Francisco, California.

PSC manages a fund that invests primarily in semiconductor companies that produce inte-

grated circuits for broadband communications including wide and local area networks, optical

components, and storage area networks. It is currently July, 2000 in the case, and the NAS-

DAQ market (where most of the stocks that PSC invests in are traded) index reached a peak

of over 5,000 a few months ago but has dropped considerably since then. In addition, the

NASDAQ has experienced a historically unprecedented amount of volatility over the last sev-

eral months, as shown in Exhibit 7 of the case. Due to the market drop and volatility, PSC�s

fund has experienced considerable problems in the last few months. Even though the fund

has had a policy of hedging market risk, the fund has nevertheless experienced large losses on

many days (sometimes on several consecutive days) in the last few months. This has created

problems for the fund because of the leverage employed by the fund. Some of the losses that

the fund had experienced had been large enough that the fund had come dangerously close

to having its prime broker, a prominent Wall Street �rm, liquidate the fund. Harold Yoon, a

partner in the fund, now must evaluate the fund�s market hedging program over the last few

months to determine why the fund had experienced such large losses on many days.

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III Derivatives and Leverage (10 Points)

Consider the following binomial tree of the price movement of a non-dividend-paying stock,

assume the time period is one year, and the e¤ective one-year interest rate is 20%.

% Su = 140

S0 = 100

& Sd = 90

1. Please calculate the return on the stock for both states of the world (Note that the

return on the stock depends on the states of the world. You need calculate the return

on the stock at both nodes, u and d.).

2. Consider an at-the-money European call option. Compute the � and B of the call

option. What is the return on a long position in the call option (Again, please calculate

the return on the call option at both nodes, u and d)?

3. Is the call option riskier than the stock? Why? Sometimes we say a call option is like

a leveraged position on the underlying stock, comment on this.

4. Consider a futures contract on one share of the stock. What is the no-arbitrage futures

price, F0;T ?

5. Assume a margin requirement of 8:33%. What is the return on a long position in

the futures contract (Again, the realized return will depend on the state of the world.

Also, note that balances on the margin account will earn risk-free return.)? What is

the expected return on the futures contract under the risk-neutral probability? Is the

futures contract riskier than the stock?

6. Can you invest in stocks and a risk-free bond to replicate the payo¤ of the futures

contact, how? (You need to take into account that investing in the futures contract

involves an initial investment in the margins account.) Is the futures contract in the

above example a leveraged position in the underlying stock?

IV Hedging with Put Options (15 Points)

1. Use the historical data on NASDAQ 100 index in the Excel �le NASDAQ Index_PSC.xls1 to estimate the volatility of the NASDAQ 100 index during the sample period.

1This can be downloaded from the course website.

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Hint: You need to �rst get the data series for ln�St+1St

�, and estimate the standard

deviation of the log return process (Recall that we assume the log return process is i.i.d.

normal). Since you are using monthly data, you need to transform the monthly estimate

into annual volatility. You can do so by using the following formula:

�Annual = �Monthly �p12

2. Suppose on Jan 31, 2007, An at-the-money European call option on the NASDAQ

100 index with maturity of 1 year is traded at $121.9639. Assume the continuously

compounded LIBOR is 5% per annum, and the continuously compounded dividend

yield is 0%. What is the implied volatility of the NASDAQ 100 index (from the Black-

Scholes model)?

3. Please �nd the data for the NASDAQ 100 index during Feb 2006 in the spread sheet

named "Feb 2006" in the Excel �le NASDAQ Index_PSC.xls. Consider an at-the-

money European put option on one NASDAQ 100 index on Jan 31, 2006. The maturity

of the option is one year (the option expires on Jan 31, 2008). Calculate the theoretical

Black-Scholes prices of this put option for each day in Feb 2006 and �ll in the column

"BS Put Price". Please use the continuously compounded annual interest of 5% per

year. Please use the implied volatility you obtained in part 2 when calculating the

Black-Scholes prices.

4. Suppose you invested 1 million US dollar in the NASDAQ 100 index 2 3 on Jan 31,

2006. Suppose you want to use the above put option on NASDAQ 100 index (That is,

the at-the-money put option on the index with one year maturity described in the last

question) to hedge the risks of your investment. To construct a ��neutral position,how many put option contracts do you need to purchase on Jan 31, 2006 (Note that

2Of course, trading the NASDAQ 100 index directly can be very costly. In practice, you will probablyinvest in a ETF that tracks the NASDAQ 100 index (the ticker symbol for the ETF that tracks the NASDAQ100 is QQQQ).

3A Short Note on ETF: You can think of an exchange-traded fund as a mutual fund that trades like astock. Just like an index fund, an ETF represents a basket of stocks that re�ect an index such as the S&P500. An ETF, however, isn�t a mutual fund; it trades just like any other company on a stock exchange. Unlikea mutual fund that has its net-asset value (NAV) calculated at the end of each trading day, an ETF�s pricechanges throughout the day, �uctuating with supply and demand. It is important to remember that whileETFs attempt to replicate the return on indexes, there is no guarantee that they will do so exactly. It is notuncommon to see a 1% or more di¤erence between the actual index�s year-end return and that of an ETF.By owning an ETF, you get the diversi�cation of an index fund plus the �exibility of a stock. Another

advantage is that the expense ratios of most ETFs are lower than that of the average mutual fund. Whenbuying and selling ETFs, you pay your broker the same commission that you�d pay on any regular trade.

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each put option contract on the index contains 100 put options on the index.)?

5. Suppose you rebalance your position in the put option everyday to maintain a �neutral

position during Feb 2006. In the column "Number of Contracts", calculate the number

of put option contracts you need to maintain in your portfolio on each day in Feb

2006. What is the theoretical return of your hedged portfolio (Again, please assume

a continuously compounded risk free rate of 5% per year, and a dividend yield of 0.

Please use the implied volatility you obtained in question 2 in your calculation. Assume

also all assumptions of the Black-Scholes model hold.)?

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