# Question 1

An investor believes that therewill be a big jump in a stock price, but is uncertain as to thedirection. Identify the different strategies (about 4) the investorcan follow and briefly explain the differences among them. (Hint:There are potentially six different strategies)

The possible strategies are:

In periods of large movements ofstock price, all the above strategies provide positive profits.Though a strangle is less costly compared to a straddle, in order toprovide a positive profit, it requires a larger change in the stockprice. Straps and strips are more costly than straddles but incertain circumstances, they result in bigger profits. A large fall instock prices will result in a bigger strip profit whereas a largerise in stock price results in a bigger strap profit. For straddles,strangles, straps, and strips, there is a profit increase that isdirectly proportional to the corresponding stock price movements.Contrastingly, a reverse butterfly spread and reverse calendarspread, regardless of the magnitude of stock price movement, thepotential profit has a maximum limit that cannot be surpassed.

Question 2

Three call options on a stockhave the same expiration date and strike prices of \$55, \$60, and \$65.The market prices are \$8, \$5, and \$3, respectively. Explain how abutterfly spread can be created. Construct a table showing the profitfrom the strategy. For what range of stock prices would the butterflyspread lead to a loss? Plot a graph of the final profits (afterexcluding cost) from the strategy (Y axis) for a closing stock pricerange of \$0 to \$100 in increments of \$5, on the X-axis.

We can develop butterfly spreadby selling the \$65 call, selling the \$55 call, and buying two of the\$60 calls. The cost will be 8+3-2×5= \$1 initially. The strategy willdevelop the following profit/ loss table of outcomes.

 Stock Price Payoff Profit ST < 55 0 -1 55 ≤ ST < 60 ST-55 ST-56 60 ≤ ST < 65 65-ST 64-ST 65 ≤ ST 0 -1

When the final stock price ishigher than \$65 or lower than \$55, the butterfly spread results in aloss.

Question 3

Suppose that c1, c2, and c3 arethe prices of European call options with strike prices K1, K2, andK3, respectively, where K3 &gt K2 &gt K1 and K3 – K2 = K2 – K1.All options have the same maturity. Assuming no arbitrage, what isthe relation between c1, c2, and c3?

C1+ k1e-rt=P1+S

C1+ k1e-rt=P1+S

C2+ k2e-rt=P2+S

C3+ k3e-rt=P3+S

Hence

C1+C3-2C2+(K1+K2+2K2)e-rT=P1+P3-2P2

SinceK2-K1=K3-K2,then K1+K3-2K2=0

And C1+C3-2C2=P1+P3-2P2

This implies that European callsdevelop a butterfly spread whose cost exactly matches the cost ofEuropean puts butterfly spreads.

Question 4

The price of a stock is \$40. Theprice of a one-year European put option on the stock with a strikeprice of \$30 is quoted as \$7 and the price of a one-year Europeancall option on the stock with a strike price of \$50 is quoted as \$5.Suppose that an investor buys 100 shares, shorts 100 call options,and buys 100 put options. Draw a diagram (to scale) illustrating howthe investor’s profit or loss varies with the stock price over thenext year. How does the answer change if the investor buys 100shares, shorts 200 call options, and buys 200 put options? Clearlyshow the payoffs using different graphs for each of the securities(for example — for calls, ___for stock, … for puts, and ___for the overallpayoff). Note: I expect is 2 diagrams, one for each scenario.

Question 5

You are given the followinginformation from the WSJ and the Cumulative normal distributiontables. The option matures in 0.5 years and is at the money. Thecurrent stock price of the underlying stock is \$90.00, and theannualized 365 day t-bill rate is 2.0%. Compute the following:

a) Use the stock price and strikeprice information from above. Assume that the stock price can eithergo up by 10% or go down by 10% each period. Assume that each periodlasts 0.5 years. Set up a replicating portfolio of the stock and arisk free bond and use a two-period binomial model. Assume that therisk free rate is 1.0% per period.

= 18.9-(-0.9) = 1

108.9-89.1

Unit the stock, and

B= Cu,uSu,u=18.9-1 X 108.9= – 90 = – \$89.1

1+ rf1.01 1.01

Units of the riskless asset. Thevalue of the call at node u is therefore:

Cu= 1 X 99-90 = \$9

At node d, the replicatingportfolio contains

• = -0.9-0 = -0.9 = -0.05

89.1-72.9 16.2

units of the stock and

B = CD,USd,u = 18.9–( -0.05)X 89.1=-\$23.61

1+ rf 1.01

The call is worth:

Cd = 0.05 x 81- 23.61= \$27.66

At node O, the replicatingportfolio contains

• = 9- 0 = 0.5 Units of the stock

99-81

b) Show CLEARLY the payoffs forthe Stock, Bond and the Call for both periods.

Payoff for call

Bond Payoff

c) Calculate the Number of Stocksand Bonds in both periods required to replicate the call.

e) Compute the probability(implied) that the stock price will go up.

Where:

N() is the normal distributionfunction, and is equivalent to Excel’s NORMSDIST() function

S is the stock price= 90

X is the strike price= 90

v is the implied volatility= 15%

T is the time to expiry= 180 days

The

Impliedprobability of the stock price reaching 110 is 50% while that ofbeing the target price is 49.9%.

f) Compute the annualizedvariance of the stock.

We calculate the daily averageprice change of the stock’s closing prices. Then calculate thestandard deviation of all the closing prices for this purpose we use2.2.

Then square the standarddeviation to get the daily variance by finding the square root.Assuming there are 250 trading days in a year, we then multiply bythe variance of 0.05 to get an annualized variance of 0.12.

g) Compute the Black- Scholes-theoretical option price for a European Call option on the stockusing the above

d1= ln(90/90) + (r+ 902/2)0.5

90 X 0.5

d1= 83.04

d2= 83.04 – 63

= 20.04

h) Using Put – Call Parity,compute the price of a European Put option on a share of the stock.

C+ PV(x) = P + S

Where:

C = price of the European calloption

PV(x) = the present value of thestrike price (x), discounted&nbspfrom the value on the expirationdate at the risk-free rate

P = price of the European put

S = spotprice, the&nbspcurrent market value of the underlying asset

Solution

0+ 89.1= P+90

P= 89.1-90= -\$0.9