Ps Options Solutions

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Finance 100 Problem SetOptions (Alternative Solutions)

Note: Where appropriate, the final answer for each problem

is given in bold italics for those not interested in the discussionof the solution. All payoff diagrams include the initial cost of thesecurity, however they do not take into account the time value ofthis cost by computing the future value. Each payoff diagram isdrawn with the future price of the security on the horizontal axisand the future payoff to the position on the vertical axes. Strictlyspeaking, when computing the future payoff to any position, weshould calculate the future value of the money paid (or received)today when factoring this into the net position. This is not donehere because the dollar impact is small and it adds little to the

problem.

I. Formulas

This section contains the formulas you might need for this homework set:

1. Payoff to a long position in a call option:

Payoff = max {ST K, 0} (1)

where ST is the price of the underlying security at expiration of thecontract (time T) and K is the strike price.

2. Payoff to a short position in a call option:

Payoff = max {ST K, 0} = min {K ST, 0} (2)

3. Payoff to a long position in a put option:

Payoff = max {K ST, 0} (3)

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4. Payoff to a short position in a put option:

Payoff = max {K ST, 0} = min {ST K, 0} (4)

5. Put-Call Parity (PCP):

C = P + SedT KerT, (5)

where C is the price of a call option, P is the price of an otherwiseidentical put option, S is the price of the underlying security, d is thedividend yield, T is the time to maturity (in years), K is the strikeprice and r is the risk-free rate.

II. Problems1.

1.a

The payoff at maturity (net of the initial cost of the option) to the call buyeris given by

max[0, ST K] F V(C), (6)

where ST is the price of the underlying stock at maturity, K is the strikeprice and F V(C) is the future value of the call premium (price), C. The netpayoff at maturity (net of the initial cost of the option) to the buyer of theput option at is, mathematically,

max[0, K ST] F V(P), (7)

where F V(P) is the future value of the put premium, P. Graphically, thesepayoffs are depicted in Figures 1 and 2 below.

1.b

The payoff at maturity (net of the initial cost of the option) to the call selleris given by

(max[0, ST K] F V(C)) = min(0, K ST) + F V(C), (8)

where ST is the price of the underlying stock at maturity, K is the strike priceand F V(C) is the future value of the call premium (price), C. The payoff atmaturity (net of the initial cost of the option) to the put buyer is given by

(max[0, K ST] F V(P)) = min(0, ST K) + F V(P), (9)

where F V(P) is the future value of the put premium, P. Graphically, thesepayoffs are depicted in Figures 3 and 4 below.

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Figure 1:Net Payoff to a Long Call Po-

sition with $50 Strike Price Figure 2:Net Payoff to a Long Put Po-

sition with $50 Strike Price

-10

0

10

20

30

40

50

0 20 40 60 80 100

StockPriceatMaturity

NetPayoffatMaturity

-10

0

10

20

30

40

50

0 20 40 60 80 100


NetPayoffatMaturity

1.c

All of these graphs were done in Excel. The names of each position areprovided upon request. You need not know them.

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Figure 3: Net Payoff to a Short Positionin a Call Option with $50 Strike Price

Figure 4: Net Payoff to a Short Positionin a Put Option with $50 Strike Price

-50

-40

-30

-20

-10

0

10

0 20 40 60 80 100


NetPayoffatMaturity

-50

-40

-30

-20

-10

0

10

0 20 40 60 80 100


NetPayoffatMaturity

Figure 1: (Protective Put): Buy one share and a put with strike price = $50

-60

-40

-20

0

20

40

60

0 20 40 60 80 100 120


NetPayoffatMaturity

BuyShare

BuyPut(50)

Combined

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Figure 2: (Bear Cylinder): Buy a put with strike = $50 and write (i.e. sell)a call with strike = $70

-40

-30

-20

-10

0

10

20

30

40

50

60

0 20 40 60 80 100 120


NetPayoffatMaturity

BuyPut(50)

WriteCall(70)

Combined

Figure 3: (Strangle I): Buy a call with strike = $70 and buy a put with strike

= $50

-10

0

10

20

30

40

50

60

0 20 40 60 80 100 120


N

etPayoffatMaturity

BuyCall(70)

BuyPut(50)

Combined

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Figure 4: (Strangle II): Buy a call with strike = $50 and buy a put withstrike = $70

-30

-20

-10

0

10

20

30

40

50

60

0 20 40 60 80 100 120


Net

PayoffatMaturity

BuyCall(50)

BuyPut(70)

Combined

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Figure 5: Short one share and buy a call with strike = $70

-60

-40

-20

0

20

40

60

0 20 40 60 80 100 120


N

etPayoffatMaturity

ShortShare

BuyCall(70)

Combined

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Figure 6: (Butterfly Spread I): Buy call with strike = $50, buy call withstrike = $70 and sell 2 calls with strike =$60

-100

-80

-60

-40

-20

0

20

40

60

0 20 40 60 80 100 120


Net

PayoffatMaturity

BuyCall(50)

BuyCall(70)

Combined

Sell2xCall(60)

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Figure 7: (Butterfly Spread II): Buy put with strike = $50, buy put withstrike = $70 and sell 2 puts with strike =$60

-120

-100

-80

-60

-40

-20

0

20

40

60

0 20 40 60 80 100 120


NetPayoffatMaturity

BuyPut(50)

BuyPut(70)

Combined

Sell2xPut(60)

1.d

We will compute the price of a call implied by put-call parity (equation (5)),

using the price of a put. We could have just as easily computed the priceof a put implied by put-call parity, using the price of the call. It makes nodifference. To avoid a redundant calculation, the time to maturity in yearsfor all options is 71/365 = 0.195.

Table (1) below computes the relevant comparisons. The first columnis simply the different strike prices. The second column repeats the marketvalue of each call from the table on the problem set. The third columnreports the price implied by put-call parity. That is, the price of the calloption using equation (5). The fourth column reports the difference betweenthe market price and the implied price. All units are in dollars, so the dollar

sign is excluded.Clearly, there are a number of discrepancies between the market values

and the put-call parity implied value. However, these discrepancies are likelydue to market frictions such as:

1. The closing times for the options on the exchange might be differentfrom those of the stock

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Table 1: Call Prices Implied from Put-Call Parity vs. Actual Market PricesMarket Call Implied PCP Price

Strike (K) Price (P + S KerT) Difference45 8.000 8.811 0.81150 5.875 6.234 0.35955 2.750 4.407 0.65760 1.875 3.831 1.95665 1.250 3.129 1.879

2. Put-Call parity is for European options. These are American options.

3. The difference for the options at 50 are less than the transaction costsfor the arbitrage.

4. Put-call parity relation above applies to non-dividend paying optionsonly. (we assumed d=0) Thus, if 3Com was paying a dividend, wewould have to account for this by using a non-zero value for d (whichwe would have to estimate).

1.e

The price of portfolio iii from part 1.c is:

Call(70) + P ut(50) = 0.0625 + 2.375 = 2.4375 (10)

The price of portfolio iv from part 1.c is:

Call(50) + P ut(70) = 2.8125 + 20.75 = 23.5625 (11)

Clearly, for the set of prices examined here, portfolio iv is more expensive.However, we must show that this is always the case, at any point in timeand for any maturity. Intuitively, this result must be true for the followingreason. Portfolio ivs assets (the 50 call and 70 put) are more likely to finish

in-the-money than portfolio iiis (70 call and 50 put). A call is more likelyto finish in-the-money, the lower the strike price, and a put is more likelyto finish in-the-money the higher the strike price. So you are more likely tomake money with portfolio iv than portfolio iii. Hence, the higher cost.

We will now prove this result more formally. We want to show that thecost of portfolio iv, Call(50) + Put(70), is always greater than the cost of

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portfolio iii, Call(70) + Put(50). Mathematically, we need to show:

Call(50) + P ut(70) > Call(70) + P ut(50),

which is equivalent to showing

Call(50) + P ut(70) Call(70) P ut(50) > 0. (12)

Put-call parity allows us to write the value of the call options in terms ofput options, the stock and cash:

Call(50) = P ut(50) + Sedt 50erT (13)

Call(70) = P ut(70) + Sedt 70erT, (14)

We now plug in the results from equations (13) and (14), into equation (12).This yields,

[P ut(50)+Sedt50erT]+P ut(70)[P ut(70)+Sedt70erT]P ut(50) > 0

Simplification yields,70erT 50erT > 0,

which is true for all values of r and T. The result is proved.

1.f

Since portfolios vi & vii have the same payoff, they should have the sameprice. However, we see that the cost of portfolio vi is:

Call(50) + Call(70) 2 Call(60) = 2

(if you want to think of this in terms of cash flows, just reverse the signs oneach position). The cost of portfolio vii is:

P ut(50) + P ut(70) 2 P ut(60) = 3.625

We can show that these two portfolios should be equal by using put-callparity in an identical manner as used above in part 1.e.

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1.g

First, take a look at portfolio i. This portfolio is hedged in the sense thatthe downside risk associated with owning the share is hedged by buying theput. As the share price falls, we lose money on the share we own. However,we make money from the put option as the price falls and this offsets theloss from owning the share.

Now examine portfolio v. When we short-sell a share, we lose moneyas the price increases above the price at which we purchased the share. Bybuying the call, however, we make money as the share price increases abovethe strike. This offsets losses in our short-sell position.

1.hIf we believe that the volatility of the underlying asset price is going to behigh, what we are saying is that we expect large swings in the stock price.That is, the stock price may move up or down, but when it moves, it willmove a lot! In this scenario, we want to be holding portfolio iii since thispays off when the stock price moves far from its current position of $50.75.

If we believe volatility will be falling, we are saying that we do not expectthe stock price to move far from its current position. In this case, we wantto be holding portfolio vi, since it pays off when the price is near its currentposition.

1.i

Exactly the same logic applies. Greater volatility yields a desire to holdportfolio iv. Less volatility yields a desire to hold portfolio vii.

1.j

Put-call parity begins with equation (5):

C = P + S KerT, (15)

ignoring dividends. We can rearrange this equation to obtain

C P = S KerT.

Thinking in terms of cash flows, the left-hand side of the equation representsa cash inflow of C (sell a call) and a cash outflow of P (buy a put). The cashflows on the right-hand side represent a cash inflow of S (buy the stock) and

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the cash outflow of KerT (lend cash). Plugging in values for S, K, r and T

yields,C P = 50.75 60e0.0571/365 = 8.669

This is a cash outflow, or cost, of $8.669, which differs from the market value,equal to 12.5 - 1.875 = 10.625.

Again, rearranging the original put-call parity equation (5) yields,

P C = KerT S

(ignoring the dividend yield). The left hand side corresponds to cash inflowof P (sell a put) and a cash outflow of C (buy a call). The right hand side

corresponds to a cash inflow of KerT

(borrowing cash) and a cash outflowof S (buy the asset). Plugging in the numbers yields:

P C = 65e0.0571/365 50.75 = 13.6209

This is a cash inflow (or a negative cost). Note, this too differs from themarket price of the portfolio: 16.75 - 1.25 = 15.50 (likely due to marketfrictions, such as those mentioned in part 1.d.

1.k

See Excel Solutions

1.l

See Excel Solutions

2.

2.a

Our exposure (amount at risk) is 5 million SFR in 40 days. The Americanfirms concern is that the exchange rate ($/SFR) will fall, meaning that eachSFR can buy fewer $US. To hedge we will want to sell the 5 million SFR

forward (i.e. in the future) by selling futures contracts on the SFR. Sinceour exposure is 5 million SFR and each futures contract is for 0.125 millionSFR, we need 5 / 0.125 = 40 contracts to hedge all of our exposure.

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2.b

Scenario 1 assumes that the spot exchange rate, 40 days hence, is 0.65 $/SFR.This implies that we can sell the 5 million SFRs we receive from the saleof goods for 65 cents per SFR, netting us 3.25 million $US. However, ourcash balance from the short futures position is (0.7021 - 0.65) * 5 = 0.2605million $US, offsetting any loss from the transaction in the spot market. Ouraggregate gain is 3.25 + 0.2605 = 3.5105 million $US.

Scenario 2 assumes that the exchange rate will be 0.70 $/SFR in March.The company receives 0.70 * 5 = 3.5 million $US from sale of goods. Ourfutures position pays the company (0.7021 - 0.70) * 5 = 0.0105 million $US,for an aggregate position of 3.5 + 0.0105 = 3.5105 million $US.

Scenario 3 assumes the exchange rate will be 0.75 $/SFR, implying thatthe company receives 0.75 * 5 = 3.75 million $US from the sale of goods(after exchanging the SFRs at the spot rate). The futures position suffers aloss of (0.7021 - 0.75) * 5 = -0.2395 million $US. The aggregate position is:3.75 - 0.2395 = 3.5105 million $US.

Note, in all scenarios the aggregate position is unaffected by the spotexchange rate in March. This is the perfect hedge.

2.c

Since we want to sell Swiss Francs in the future, we want to buy put options,

which gives us the option to sell the underlying asset in the future. Since eachcontract is for 0.0625 million SFRs, we need to sell 5/0.0625 = 80 contracts.

Table (2) through (4) below looks at the effect of entering into 3 typesof put contracts differentiated by their strike price, when the spot exchangerate in March is 0.65, 0.70 and 0.75 $/SFR.

Table 2: Scenario 1: Spot Exchange Rate in March = 0.65 $/SFR

Buy80contracts: Exchangerate=0.65

FutureSpotexchangerate 0.65 0.65 0.65

PutOptionStrikeprice 0.695 0.71 0.74

PremiumperSFR(Costperunit) 0.0107 0.0196 0.0442Totalpremium(costper80contracts) 53,500 98,000 221,000

Valuewhenexercised(K>ST) 225,000 300,000 450,000

Valuefromsaleof5mSFR 3,250,000 3,250,000 3,250,000

Netpayoff(disregarddiscounting) 3,421,500 3,452,000 3,479,000

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PremiumperSFR(Costperunit) 0.0107 0.0196 0.0442

Totalpremium(costper80contracts) 53,500 98,000 221,000

Valuewhenexercised(K>ST) - 50,000 200,000

Valuefromsaleof5mSFR 3,500,000 3,500,000 3,500,000Netpayoff(disregarddiscounting) 3,446,500 3,452,000 3,479,000

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PremiumperSFR(Costperunit) 0.0107 0.0196 0.0442

Totalpremium(costper80contracts) 53,500 98,000 221,000

Valuewhenexercised(K>ST) - - -

Valuefromsaleof5mSFR 3,750,000 3,750,000 3,750,000

Netpayoff(disregarddiscounting) 3,696,500 3,652,000 3,529,000

The conclusions to be made from the tables are as follows. Options do

not offer a perfect hedge in the sense that we do not know for certain whatour future payoff will be. They do offer a floor, or minimum amount, alongwith unlimited upside. Of course, the price of maintaining this upside is thecost of the option. Their is also a tradeoff between the higher strike price(more likely to finish in-the-money and the higher premium). The hedgetransaction itself has a zero NPV.

2.d

The futures hedge locks in an exchange rate. We know for certain whatour future payoff will be, regardless of what happens to the underlying spot

exchange rate. The option hedge is more uncertain. While we are guaranteeda minimum payoff, we do need a bit of upside movement in the exchange rateto recoup the initial cost of the option contracts (there is no cost to enteringfutures contracts, beyond transaction costs). Beyond recovering the initialcost of the option contracts, the upside gains are unlimited for the optionhedge. Assuming there is no private information, there is little reason tohedge using the options when the futures contracts accomplish this goal ata cheaper price and with no uncertainty.

2.e

The CEO is incorrect. Assuming the company has another buyer offering anamount in $US that the firm would otherwise get from the sale to the Swissfirm at 0.695 $/SFR, the firm should exit the sale and take the cash fromtheir futures position. Agreeing to the sale at that point would force the firmto take a loss on the sale of goods. Something they need not do under theexit clause.

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2.f

The company breaks even at an exchange rate of 0.695 $/SFR. That is, theyare indifferent between selling at 0.695 and not selling at all. So, the exitclause acts in the exact same manner as the put option. Therefore, it shouldhave the same value as the put, or $53,500 (see above).

This is an example of a real option.

3.

The parameters of the problem are:

1. Price of call option (C) = $3.10

2. Price of put option (P) = $0.40

3. Strike (K) = $45

4. Current Stock Price (S) = $46.60

5. Risk-free interest rate (r) = 5%

6. Time to expiration, in years (T) = 1/12 = 0.0833

Put-call parity is violated since

C = P + S KerT

= 0.4 + 46.6 45 e0.050.0833

= 2.19,

which is less than the market price of $3.10.We can use the put-call parity violation to tell us what arbitrage strategy

to undertake. The intuition is as follows. We know the market price of thecall is too high so that C > P+ SKerT. Now lets rearrange the equationso that we get an expression that is greater than zero. By doing this, andinterpreting the dollar amounts as cash flows, we are saying that the cashflows today will be greater than zero, implying an arbitrage since we knowthe cash flows in the future will be the same.

A little algebra reveals

C P S+ KerT > 0.

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Table 5: Arbitrage TableAt Maturity

Position Today St < K St > KSell Call 3.10 0 (ST 45)

Buy Put -0.40 45 ST 0Buy Stock -46.60 ST STBorrow Cash 44.81 -45 -45Net Cash Flows 0.91 0 0

Think of this equation in terms of the cash flows at time 0 (i.e. today). To

receive a positive cash flow of C today, we must sell the call option. But,this makes sense since the call option price in the market is too high. Thenegative cash flows of -P and -S correspond to buying the put and buyingthe stock. The positive cash flow of KerT reflects borrowed cash.

The arbitrage table is presented in (5).

These solutions are produced by Michael R. Roberts. Thanks go to Jen Rother forher excellent assistance, and to an anonymous TA. Any remaining errors are mine.

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Ps Options Solutions