# Finance 350: Problem Set 8 Alternative Solutions

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1 Finance 35: Problem Set 8 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. All payoff diagrams include the initial cost of the security, however they do not take into account the time value of this cost by computing the future value. Each payoff diagram is drawn with the future price of the security on the horizontal axis and the future payoff to the position on the vertical axes. Strictly speaking, when computing the future payoff to any position, we should calculate the future value of the money paid (or received) today when factoring this into the net position. This is not done here 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: P ayoff = max {S T K, } (1) where S T is the price of the underlying security at expiration of the contract (time T ) and K is the strike price. 2. Payoff to a short position in a call option: P ayoff = max {S T K, } = min {K S T, } (2) 3. Payoff to a long position in a put option: P ayoff = max {K S T, } (3) 1

2 4. Payoff to a short position in a put option: 5. Put-Call Parity (PCP): P ayoff = max {K S T, } = min {S T K, } (4) C = P + Se dt Ke rt, (5) where C is the price of a call option, P is the price of an otherwise identical put option, S is the price of the underlying security, d is the dividend yield, T is the time to maturity (in years), K is the strike price and r is the risk-free rate. II. Problems 1. 1.a The payoff at maturity (net of the initial cost of the option) to the call buyer is given by max[, S T K] F V (C), (6) where S T is the price of the underlying stock at maturity, K is the strike price and F V (C) is the future value of the call premium (price), C. The net payoff at maturity (net of the initial cost of the option) to the buyer of the put option at is, mathematically, max[, K S T ] F V (P ), (7) where F V (P ) is the future value of the put premium, P. Graphically, these payoffs 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 seller is given by (max[, S T K] F V (C)) = min(, K S T ) + F V (C), (8) where S T is the price of the underlying stock at maturity, K is the strike price and F V (C) is the future value of the call premium (price), C. The payoff at maturity (net of the initial cost of the option) to the put buyer is given by (max[, K S T ] F V (P )) = min(, S T K) + F V (P ), (9) where F V (P ) is the future value of the put premium, P. Graphically, these payoffs are depicted in Figures 3 and 4 below. 2

3 Figure 1: Net Payoff to a Long Call Position with \$5 Strike Price sition with \$5 Strike Figure 2: Net Payoff to a Long Put Po- Price c All of these graphs were done in Excel. The names of each position are provided upon request. You need not know them. 3

4 Figure 3: Net Payoff to a Short Position Figure 4: Net Payoff to a Short Position in a Call Option with \$5 Strike Price in a Put Option with \$5 Strike Price Figure 1: (Protective Put): Buy one share and a put with strike price = \$ Buy Share Buy Put(5) Combined 4

5 Figure 2: (Bear Cylinder): Buy a put with strike = \$5 and write (i.e. sell) a call with strike = \$ Buy Put(5) Write Call(7) Combined Figure 3: (Strangle I): Buy a call with strike = \$7 and buy a put with strike = \$ Buy Call(7) Buy Put(5) Combined 5

6 Figure 4: (Strangle II): Buy a call with strike = \$5 and buy a put with strike = \$ Buy Call(5) Buy Put(7) Combined

7 Figure 5: Short one share and buy a call with strike = \$ Short Share Buy Call(7) Combined

8 Figure 6: (Butterfly Spread I): Buy call with strike = \$5, buy call with strike = \$7 and sell 2 calls with strike =\$ Buy Call(5) Buy Call(7) Combined Sell 2xCall(6) 8

9 Figure 7: (Butterfly Spread II): Buy put with strike = \$5, buy put with strike = \$7 and sell 2 puts with strike =\$ Buy Put(5) Buy Put(7) Combined Sell 2xPut(6) 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 price of a put implied by put-call parity, using the price of the call. It makes no difference. To avoid a redundant calculation, the time to maturity in years for all options is 71/365 =.195. Table (1) below computes the relevant comparisons. The first column is simply the different strike prices. The second column repeats the market value of each call from the table on the problem set. The third column reports the price implied by put-call parity. That is, the price of the call option using equation (5). The fourth column reports the difference between the 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 likely due to market frictions such as: 1. The closing times for the options on the exchange might be different from those of the stock 9

10 Table 1: Call Prices Implied from Put-Call Parity vs. Actual Market Prices Market Call Implied PCP Price Strike (K) Price (P + S Ke rt ) Difference Put-Call parity is for European options. These are American options. 3. The difference for the options at 5 are less than the transaction costs for the arbitrage. 4. Put-call parity relation above applies to non-dividend paying options only. (we assumed d=) Thus, if 3Com was paying a dividend, we would have to account for this by using a non-zero value for d (which we would have to estimate). 1.e The price of portfolio iii from part 1.c is: Call(7) + P ut(5) = = (1) The price of portfolio iv from part 1.c is: Call(5) + P ut(7) = = (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 time and for any maturity. Intuitively, this result must be true for the following reason. Portfolio iv s assets (the 5 call and 7 put) are more likely to finish in-the-money than portfolio iii s (7 call and 5 put). A call is more likely to finish in-the-money, the lower the strike price, and a put is more likely to finish in-the-money the higher the strike price. So you are more likely to make money with portfolio iv than portfolio iii. Hence, the higher cost. We will now prove this result more formally. We want to show that the cost of portfolio iv, Call(5) + Put(7), is always greater than the cost of 1

11 portfolio iii, Call(7) + Put(5). Mathematically, we need to show: which is equivalent to showing Call(5) + P ut(7) > Call(7) + P ut(5), Call(5) + P ut(7) Call(7) P ut(5) >. (12) Put-call parity allows us to write the value of the call options in terms of put options, the stock and cash: Call(5) = P ut(5) + Se dt 5e rt (13) Call(7) = P ut(7) + Se dt 7e rt, (14) We now plug in the results from equations (13) and (14), into equation (12). This yields, [P ut(5)+se dt 5e rt ]+P ut(7) [P ut(7)+se dt 7e rt ] P ut(5) > Simplification yields, 7e rt 5e rt >, 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 same price. However, we see that the cost of portfolio vi is: Call(5) + Call(7) 2 Call(6) = 2 (if you want to think of this in terms of cash flows, just reverse the signs on each position). The cost of portfolio vii is: P ut(5) + P ut(7) 2 P ut(6) = We can show that these two portfolios should be equal by using put-call parity in an identical manner as used above in part 1.e. 11

12 1.g First, take a look at portfolio i. This portfolio is hedged in the sense that the downside risk associated with owning the share is hedged by buying the put. 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 the loss from owning the share. Now examine portfolio v. When we short-sell a share, we lose money as the price increases above the price at which we purchased the share. By buying the call, however, we make money as the share price increases above the strike. This offsets losses in our short-sell position. 1.h If we believe that the volatility of the underlying asset price is going to be high, 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 will move a lot! In this scenario, we want to be holding portfolio iii since this pays off when the stock price moves far from its current position of \$5.75. If we believe volatility will be falling, we are saying that we do not expect the stock price to move far from its current position. In this case, we want to be holding portfolio vi, since it pays off when the price is near its current position. 1.i Exactly the same logic applies. Greater volatility yields a desire to hold portfolio iv. Less volatility yields a desire to hold portfolio vii. 1.j Put-call parity begins with equation (5): C = P + S Ke rt, (15) ignoring dividends. We can rearrange this equation to obtain C P = S Ke rt. Thinking in terms of cash flows, the left-hand side of the equation represents a cash inflow of C (sell a call) and a cash outflow of P (buy a put). The cash flows on the right-hand side represent a cash inflow of S (buy the stock) and 12

13 the cash outflow of Ke rt (lend cash). Plugging in values for S, K, r and T yields, C P = e.5 71/365 = This is a cash outflow, or cost, of \$8.669, which differs from the market value, equal to = Again, rearranging the original put-call parity equation (5) yields, P C = Ke rt S (ignoring the dividend yield). The left hand side corresponds to cash inflow of P (sell a put) and a cash outflow of C (buy a call). The right hand side corresponds to a cash inflow of Ke rt (borrowing cash) and a cash outflow of S (buy the asset). Plugging in the numbers yields: P C = 65e.5 71/ = This is a cash inflow (or a negative cost). Note, this too differs from the market price of the portfolio: = 15.5 (likely due to market frictions, 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 4 days. The American firms concern is that the exchange rate (\$/SFR) will fall, meaning that each SFR 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. Since our exposure is 5 million SFR and each futures contract is for.125 million SFR, we need 5 /.125 = 4 contracts to hedge all of our exposure. 13

14 2.b Scenario 1 assumes that the spot exchange rate, 4 days hence, is.65 \$/SFR. This implies that we can sell the 5 million SFR s we receive from the sale of goods for 65 cents per SFR, netting us 3.25 million \$US. However, our cash balance from the short futures position is ( ) * 5 =.265 million \$US, offsetting any loss from the transaction in the spot market. Our aggregate gain is = million \$US. Scenario 2 assumes that the exchange rate will be.7 \$/SFR in March. The company receives.7 * 5 = 3.5 million \$US from sale of goods. Our futures position pays the company ( ) * 5 =.15 million \$US, for an aggregate position of = million \$US. Scenario 3 assumes the exchange rate will be.75 \$/SFR, implying that the company receives.75 * 5 = 3.75 million \$US from the sale of goods (after exchanging the SFR s at the spot rate). The futures position suffers a loss of ( ) * 5 = million \$US. The aggregate position is: = million \$US. Note, in all scenarios the aggregate position is unaffected by the spot exchange 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 each contract is for.625 million SFR s, we need to sell 5/.625 = 8 contracts. Table (2) through (4) below looks at the effect of entering into 3 types of put contracts differentiated by their strike price, when the spot exchange rate in March is.65,.7 and.75 \$/SFR. Table 2: Scenario 1: Spot Exchange Rate in March =.65 \$/SFR Buy 8 contracts: Exchange rate =.65 Future Spot exchange rate Put Option Strike price Premium per SFR (Cost per unit) Total premium (cost per 8 contracts) 53,5 98, 221, Value when exercised (K > S T ) 225, 3, 45, Value from sale of 5m SFR 3,25, 3,25, 3,25, Net payoff (disregard discounting) 3,421,5 3,452, 3,479, 14

15 Table 3: Scenario 2: Spot Exchange Rate in March =.7 \$/SFR Buy 8 contracts: Exchange rate =.7 Future Spot exchange rate Put Option Strike price Premium per SFR (Cost per unit) Total premium (cost per 8 contracts) 53,5 98, 221, Value when exercised (K > S T ) - 5, 2, Value from sale of 5m SFR 3,5, 3,5, 3,5, Net payoff (disregard discounting) 3,446,5 3,452, 3,479, 15

16 Table 4: Scenario 3: Spot Exchange Rate in March =.75 \$/SFR Buy 8 contracts: Exchange rate =.75 Future Spot exchange rate Put Option Strike price Premium per SFR (Cost per unit) Total premium (cost per 8 contracts) 53,5 98, 221, Value when exercised (K > S T ) Value from sale of 5m SFR 3,75, 3,75, 3,75, Net payoff (disregard discounting) 3,696,5 3,652, 3,529, 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 what our future payoff will be. They do offer a floor, or minimum amount, along with unlimited upside. Of course, the price of maintaining this upside is the cost 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 hedge transaction itself has a zero NPV. 2.d The futures hedge locks in an exchange rate. We know for certain what our future payoff will be, regardless of what happens to the underlying spot exchange rate. The option hedge is more uncertain. While we are guaranteed a minimum payoff, we do need a bit of upside movement in the exchange rate to recoup the initial cost of the option contracts (there is no cost to entering futures contracts, beyond transaction costs). Beyond recovering the initial cost of the option contracts, the upside gains are unlimited for the option hedge. Assuming there is no private information, there is little reason to hedge using the options when the futures contracts accomplish this goal at a cheaper price and with no uncertainty. 2.e The CEO is incorrect. Assuming the company has another buyer offering an amount in \$US that the firm would otherwise get from the sale to the Swiss firm at.695 \$/SFR, the firm should exit the sale and take the cash from their futures position. Agreeing to the sale at that point would force the firm to take a loss on the sale of goods. Something they need not do under the exit clause. 16

17 2.f The company breaks even at an exchange rate of.695 \$/SFR. That is, they are indifferent between selling at.695 and not selling at all. So, the exit clause acts in the exact same manner as the put option. Therefore, it should have the same value as the put, or \$53,5 (see above). This is an example of a real option. 3. The parameters of the problem are: 1. Price of call option (C) = \$ Price of put option (P) = \$.4 3. Strike (K) = \$45 4. Current Stock Price (S) = \$ Risk-free interest rate (r) = 5% 6. Time to expiration, in years (T) = 1/12 =.833 Put-call parity is violated since C = P + S Ke rt = e = 2.19, which is less than the market price of \$3.1. 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 the call is too high so that C > P +S Ke rt. Now let s rearrange the equation so that we get an expression that is greater than zero. By doing this, and interpreting the dollar amounts as cash flows, we are saying that the cash flows today will be greater than zero, implying an arbitrage since we know the cash flows in the future will be the same. A little algebra reveals C P S + Ke rt >. 17

18 Table 5: Arbitrage Table At Maturity Position Today S t < K S t > K Sell Call 3.1 (S T 45) Buy Put S T Buy Stock S T S T Borrow Cash Net Cash Flows.91 Think of this equation in terms of the cash flows at time (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. The negative cash flows of -P and -S correspond to buying the put and buying the stock. The positive cash flow of Ke rt reflects borrowed cash. The arbitrage table is presented in (5). These solutions are produced by Michael R. Roberts. Thanks go to Jen Rother for her excellent assistance, and to an anonymous TA. Any remaining errors are mine. 18

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