AMS 511.01 - Foundations Class 11A

Robert J. Frey Research Professor Stony Brook University, Applied Mathematics and Statistics [email protected] http://www.ams.sunysb.edu/~frey/

In this lecture we will cover the pricing and use of derivative securities, covering Chapters 10 and 12 in Luenberger’s text. April, 2007

1. The Binomial Option Pricing Model

1.1 – General Single Step Solution

The geometric binomial model has many advantages. First, over a reasonable number of steps it represents a surprisingly realistic model of price dynamics. Second, the state price equations at each step can be expressed in a form indpendent of S(t) and those equations are simple enough to solve in closed form.

1+r D-1 u 1 1 + r D 1 + r D y y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = u fl u = 1+r D u-1 u 1+r-u 1 u 1 u yd yd ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1+r D ÅÅÅÅÅÅÅÅ1 uêÅÅÅÅ-ÅÅÅÅu ÅÅ H L H ê L i y As we will see shortlyH weL Hwill solveL the general problemj by solving a zsequence of single step problems on the lattice. That K O K O K O K O j H L H ê L z sequence solutions can be efficientlyê computed because wej only have to zsolve for the state prices once. k {

1.2 – Valuing an Option with One Period to Expiration

Let the current value of a stock be S(t) = 105 and let there be a call option with unknown price C(t) on the stock with a strike price of 100 that expires the next three month period. We’ll use a single binomial step, so D = 0.25 months. For the period of t to t+D the risky return is u = 110%; the binomial probability is Pu = 0.50, and the annual risk free rate is r = 0.04. 1 1 + r D 1 + r D 1 1.01 1.01 y y S t = u S t S t u u fl 105 = 115.50 95.45 u y y C t max u S t - K, 0 max S t u - K, 0 d C t 15.5 0 d H L H L ji zy ji zy ji zy ji zy j z j z j z j z We can solvej H L thez firstj two equationsH L for the stateH pricesL ê usingz Kthe closedO formj solutionz j z K O j z j z j z j z k H L { k 1+r D-@ 1 u H L D 1+0.04µ@0.25H L-ê1 1.10 D { k H L { y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.523 k { u = 1+r D u-1 u = 1+0.04µ0.25 1.10-1 1.10 = 1+r D-u 1+0.04µ0.25-1.10 yd 0.467 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1+r D ÅÅÅÅÅÅÅÅ1 uêÅÅÅÅ-ÅÅÅÅu ÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1+0.04µ0.25 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 1.10ê -ÅÅÅÅÅÅÅÅ1.10ÅÅÅÅÅ H L H ê L H L H ê L i y i y If we take these valuesj and substitutez j them into the last rowz we realize a call premium of C(t) > 5.23. K O j H L H ê L z j H L H ê L z K O j è z j z The risk neutral kmeasure P is { k { 2 ams-511-lec-11A-p.nb

è Pu 1 yu 1 0.673 0.529 è = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = Pd 1 + r D yd 1 + 0.01 0.288 0.471

It is remarkablei y but, other than meeting the requirement that 0 < P < 1, the ordinary probability measure does not play a role j z u in the solution.j z K O K O K O k {

1.3 – The General Binomial Option Pricing Model

We’ll return to the example above but now assume that the option expires is six months, but we wish to model the stock price evolution using two lattice steps. The binomial lattice for the stock is then 127.05

115.50

105.00 105.00

95.45

86.78 As before, the risk neutral measure is è Pu 0.529 è = Pd 0.471

The “leaves”i y of the lattice above occur at expiry, so for each price node we can calculate the value of the call option. This j z gives us jthe z K O k { 27.05

5.00

0.00

è T T 1 Pu 27.05 1 0.529 27.05 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = 16.49 1 + r D Pd 5.00 1.01 0.471 5.00

i è yT T 1 j Pu z 5.00 1 0.529 5.00 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ j è z K O= ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ K O K =O 2.62 1 + r D k Pd { 0.00 1.01 0.471 0.00

i y j z j z K O K O K O k { ams-511-lec-11A-p.nb 3

We can now add this information to the lattice.

27.05

16.49

5.00

2.62

0.00

è T T 1 Pu 16.49 1 0.529 16.49 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = 9.85 1 + r D Pd 2.62 1.01 0.471 2.62

This allows us toi completey the lattice. The value at the “root”, 9.85, is the value of the call option. j z j z K O K O K O k { 27.05

16.49

9.85 5.00

2.62

0.00

1.4 – Calibrating the Model to Market Data

For a given stepsize D we have two free parameters in the geometric binomial model: Pu and u. As was shown earlier, with a sufficient number of steps the log price tends towards a Normal distribution. The calibration of the geometric binomial model at stepsize D, therefore, involves selecting a set of parameters Pu and u so that the mean and standard deviation of the log price are matched. S t + 1 m = E log ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ S t

+ 2 H S t L 1 s = VarC Clog ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅGGÅÅÅÅÅÅÅÅ H LS t

1 m H L Pu = ÅÅÅÅÅ 1C + CÅÅÅÅÅÅÅ D GG 2 s H L

s D u = e è!!!! J N è!!!! 4 ams-511-lec-11A-p.nb

1.5 – Mathematica Code

This code is not the most efficient; it’s meant to illustrate the computations in the simplest terms. Each step or level in the lattice is represented by a list. The entire lattice is a list of such lists, e.g., {{x11},. {x21, x22}, {x31, x32, x33}, ...}. The function fLatticeForwardStep takes one level of the lattice and produces the successor level using the supplied nUpFactor. fLatticeForwardStep vnOneLevel_, nUpFactor_ := Append vnOneLevel nUpFactor, Last vnOneLevel nUpFactor ; è The function fRiskNeutralMeasure calculates the value of Pu assuming a geometric binomial model. @ D fRiskNeutralMeasure nRiskFree_, DTime_, nUpFactor_ := @ @ D ê D -1 + nUpFactor + nRiskFree nUpFactor DTime ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ; -1 + nUpFactor2 @ D These functions are used at the leaves of the undelying’s lattice to compute the leaves of the option’s lattice. fCallAtExpiry nNowPrice_, nStrikePrice_ := Max nNowPrice - nStrikePrice, 0 ; fPutAtExpiry nNowPrice_, nStrikePrice_ := Max nStrikePrice@ - nNowPrice, 0 ; D @ D fLatticeBackStep vnOneLevel_, nRiskFree_,@ DTime_, nRiskNeutralProb_D := Module i@, D Table @ nRiskNeutralProb vnOneLevel i + 1D- nRiskNeutralProb@ 8 < vnOneLevel i + 1 1 + nRiskFree DTime , i, 1@, Length vnOneLevel - 1 H P T H L ; P TL ê H L 8 @ D < D D ams-511-lec-11A-p.nb 5

This is the top-level function. It returns the risk neutral measure, the underlying’s “forward” lattice, and the option’s “backward” lattice. Note that the forward and backward lattices, whose variables I’ve somewhat incorrectly called trees in the code, are computed by using NestList to recursively apply fLatticeForwardStep from the root to the leaves and then fLatticeBackStep from the leaves to the root. The last parameter is a character with “c” for a call and “p” for a put. fGeometricBinomialOption nNowPrice_, nUpFactor_, nStrikePrice_, nRiskFree_, iIntervals_, DTime_, cPutCall_ := Module vvnBackwardTree, @ vnExpiryPrice, vvnForwardTree, nRiskNeutralProb , vvnForwardTree = NestListD @ 8 fLatticeForwardStep #, nUpFactor &, nNowPrice , < iIntervals @ ; @ D nRiskNeutralProb8 < = fRiskNeutralMeasure nRiskFree, DTime, nUpFactor ; vnExpiryPriceD = If cPutCall == c, fCallAtExpiry #, nStrikePrice & ü Last vvnForwardTree , fPutAtExpiry #, nStrikePrice@ & ü Last vvnForwardTreeD ; @ vvnBackwardTree@= NestList D ê @ D fLatticeBackStep@ #, nRiskFreeD, DêTime, nRiskNeutralProb@ D &, DvnExpiryPrice, iIntervals @ ; @ D nRiskNeutralProb, vvnForwardTree, vvnBackwardTree ; D 8 < D 6 ams-511-lec-11A-p.nb

1.6 – Example: Pricing a Put Option

Consider a stock with current price S(0)=125. It’s log mean and variance are m = 0.10 and s2=0.01. There is a put option on the stock with expiry T=0.5 and a strike price K = 120. The annual risk free rate is 4%. Estimate the price of the put option using a 5-step geometric binomial option model. nTimeToExpiry = 0.5;

iIntervals = 10;

DTime = nTimeToExpiry iIntervals;

nUpFactor = Exp 0.01 DTime ; ê

nRiskNeutral, vvnStockLatticeè!!!!!!!!!!!!!!!!!!!!!!!!! , vvnOptionLattice = fGeometricBinomialOptionB F125, nUpFactor, 120, 0.04, iIntervals, DTime, p ; 8 < MatrixForm ü vvnStockLattice@ MatrixForm ü vvnOptionLattice D

ê ê ams-511-lec-11A-p.nb 7

ol o 136.696 o 133.673 o 130.717 130.717 o o 127.827 127.827 o 125 , , 125. , , 125. , o 122.236 122.236 o i y o 119.533 j 119.533 z o ji 116.89 zy j z o ji zy j z j 114.305 z o j z j z j z mH L K O j z j z j z o j z j z j z o j z j z j z o k { j z j z o k { j z o o k { o o o o o o n 156.322 149.485 152.866 149.485 146.18 142.948 142.948 146.18 142.948 139.787 139.787 139.787 136.696 136.696 ji 136.696 zyo| j zo 133.673 133.673 ji 133.673 zy j zo 130.717 ji 130.717 zy j z j 130.717 zo 127.827 i 127.827 y j z j z j zo i 125. y j z j 125. z j 127.827 z j 125. zo , j z, j z, j z, j z, j zo ji 122.236 zy j z j 122.236 z j z j 122.236 z j zo j z j 119.533 z j z j 119.533 z j z j 119.533 zo j 116.89 z j z j 116.89 z j z j z j zo j z j z j z j z j 116.89 z j zo j z j 114.305 z j z j 114.305 z j z j 114.305 zo j 111.778 z j z j 111.778 z j z j 111.778 z j zo j z j z j z j z j z j z} j z j 109.306 z j z j 109.306 z j z j 109.306 zo j z j z j 106.889 z j z j 106.889 z j zo j z j z j z j 104.525 z j z j 104.525 zo j z j z j z j z j 102.214 z j zo j z j z j z j z j z j zo j z j z j z j z j 99.9537 zo k { j z j z j z j z j zo k { j z j z j z j zo k { j z j z j zo j z j zo k { j zo k { j zo k {~ 8 ams-511-lec-11A-p.nb

0 0 0 0 0 0 0 0 0 0 0 olji 0 zy 0 0 oj z i 0 y 0 oj 0 z j z i 0 y 0.0209016 oj z j z j z oj z j 0 z j z ji 0.045443 zy oj 0 z, j z, j 0.0987994 z, j z, i 0.352685 y, oj z j 0.214804 z j z j z j z oj z j z j z j 0.713626 z j z oj 0.467013 z j z j 1.43595 z j z j 1.84289 z oj z j 2.87067 z j z j 3.17189 z j z oj 5.69492 z j z j 5.21636 z j z j 5.15875 z oj z j z j z j z j z j z j 7.98297 z j z j 7.50536 z j z omj 10.6942 z j z j 10.2156 z j z j 9.73896 z oj z j 12.8717 z j z j 12.3941 z j z oj z j z j z j z j z oj 15.4748 z j z j 14.9962 z j z j z oj z j 17.5466 z j z j z j z oj 20.0463 z j z j z j z j z oj z j z j z j z j z oj z j z j z j z oj z j z j z k { oj z j z j z k { oj z j z k { oj z k { nk {

0 0.00442187 | o 0.00961376 0.0414559 o 0.084958 0.164021 o o 0.173464 0.308109 0.434328 o , 0.570488 , , 0.752418 , , 0.905688 o ji 1.0374 zy 1.27544 1.46101 o j z ji 2.10562 zy 2.29627 o j 3.36435 z j z i y o j z j z j 3.5004 z o j z j 5.14721 z j z ji zy o j 7.25513 z j z j z j z o j z j z j z j z K O H Lo} j z j z j z j z o j z j z j z o j z j z j z k { o j z j z k { o j z k { o o k { o o The price is > 0.91 o o o ~ 2 – Hedging and Risk Control with Derivatives

2.1 – Basic Tools

We can construct a number of different trading strategies using the basic tools of a long or short position in the underling and buying or writing puts and calls on that underlying. Often we’ll build complex strategies by combining simpler ones. These strategies will be illustrated using net profit plots in the price of the underlying is plotted against the profit or loss of the total position at expiration. At this stage we won’t consider the effects of transaction costs or the impact of unwinding these positions before expiration. We’ll assume that a position is put on at time t = 0 and evaluated at t = T when the options expire. The price of a stock, put and call is S(t), P(t) and C(t), respectively. Where we need to indicate different strikes, we’ll use subscripts; e.g., Ci t ’s strike will be Ki . The net profit of a long stock position is S(T) – S(0): H L ams-511-lec-11A-p.nb 9

p Long Stock 40

20

S T 20 40 60 80

H L -20

-40

The net profit of a short stock position is S(0) – S(T):

p Short Stock 40

20

S T 20 40 60 80

H L -20

-40

The net profit of a long call is Max[S(T) – K, 0] – C(0): 10 ams-511-lec-11A-p.nb

p Long Call

15

10

5

S T 40 50 60 70

H L The net profit of a short call is –Max[S(T) – K, 0] + C(0):

p Short Call

S T 40 50 60 70

H L -5

-10

-15

The net profit of a long put is Max[K – S(T) , 0] – P(0): ams-511-lec-11A-p.nb 11

p Long Put

15

10

5

S T 40 50 60 70

H L The net profit of a short put is –Max[K – S(T) , 0] + P(0):

p Short Put

S T 40 50 60 70

H L -5

-10

-15

2.2 – Downside Protection

Investment involves risk. Sometimes the potential size of a loss associated with a position is unacceptable. A natural case in point is a short position in a stock. If you buy a share of stock for $50, then your maximum loss is your original $50 invest- ment. On the other hand if you short that share of stock, then your potential loss is infinite. Often the distressed securities that we short are also the most volatile. A $5 tech stock we are convinced is on its way out may suddenly announce a major contract and triple in price. One way to control the downside risk of a stock is to buy an out-of-the-money call. The further out of the money the call is the lower the premium we will pay, but the further out our downside protection is. This makes sense: the more insurance we have, the more we have to pay for it. We assume that the current price of the stock S(0) = 160 and we buy a call at a strike of K = 180, 12 ams-511-lec-11A-p.nb

The net profits at expiration T of the two components of this strategy are plotted below. We’ve bought an option that’s quite far outside the money to keep costs low. Thus, we have protected ourselves from catastrophic loses, but left ourselves exposed to small increases in stock price.

p Short Stock with Downside Protection 40

20

S T 140 160 180 200

H L -20

-40

Taken together they have the desired effect. Note, however, that the premium of the option costs us: the net profit line of the hedged position is slightly lower for prices below the strike price of the option.

p Short Stock with Downside Protection

30

20

10

S T 140 160 180 200

-10 H L

-20

A similar strategy can be applied to a long position by offsetting it with a long put. Although we don’t face the prospect of unlimited loses, an investor might want to take advantage of the expected superior return on a stock, but may have future uses for the capital that requires that he or she place a floor on losses.

2.3 – Covered Call

A covered call is a hedged position in which an at-the-money call is written against a long position in the underlying. The the case below we have S(0) = 160, C(0) = 6.325 with K = 160. ams-511-lec-11A-p.nb 13

p Short Stock with Downside Protection 40

20

S T 140 160 180 200

H L -20

-40

We exchanged the upside potential of the stock in return for the premium on the call. A fund manager might want to do this if he or she were convinced that a stock is likely to hold to its current value or decrease slightly.

p Covered Call

S T 140 160 180 200

H L -10

-20

-30

2.4 – Straddles and Strangles

Sometimes we are convinced that a stock will not move much. It may be in a stable business and an investor’s assessment of economic conditions is that there is little that is likely to come along to change things. On the other hand an investor may be convinced that a stock’s price will be highly volatile, but isn’t sure if it will go up or down. An example would a company awaiting the resolution of a lawsuit; if the company wins the price will increase dramatically, but if it loses the price will plummet.

2.4.1 – Straddles A long straddle is a long call and a long put on the same underlying, both bought at the money. Here we have S(0) = 160, P(0) = 3.325, C(0) = 3.750: 14 ams-511-lec-11A-p.nb

p Long Straddle

15

10

5

S T 150 160 170 180

H L

Clearly, this strategy makes sense if the investor expects the stock to move dramatically but isn’t sure what direction it will move.

p Long Straddle

10

5

S T 150 160 170 180

H L -5

A short strangle simply reverses the trades.

2.4.2 – Strangles By buying the put and call at different strike prices it’s possible to implement a similar strategy called a strangle. This is a bit cheaper to implement but doesn't “kick in” unless the movement of the price is above some threshold. ams-511-lec-11A-p.nb 15

p Long Strangle

15

10

5

S T 145 150 155 160 165 170 175

H L

p Long Strangle 14

12

10

8

6

4

2

S T 145 150 155 160 165 170 175

H L 2.4.3 – Butterflies to Control the Downside Once we begin to build a “library” of hedge strategies we can think of combining them to produce more complex results. For example, consider the net profit plot for a short straddle: 16 ams-511-lec-11A-p.nb

p Short Straddle

5

S T 150 160 170 180

H L -5

-10

While an investor may have an opinion that a stock’s price will not change dramatically, he or she may hold that opinion with varying levels of confidence. A short straddle exposes it holder to very limited upside compared to potentially huge downside. One way to control this risk is to put on a long strangle on top of the short straddle. The strangle’s strikes are in the money and it premia are, therefore, less than those of the straddle whose strikes are at the money. The long straddle looks like: p Long Strangle

8

6

4

2

S T 150 160 170 180

H L Putting them together produces what is called a long butterfly: ams-511-lec-11A-p.nb 17

p Long Butterfly

4

2

S T 150 160 170 180

H L -2

-4

Viewing a butterfly as a combination of a straddle and strangle is far easier to visualize than viewing it in terms of its sim- plest components:

p Long Butterfly

7.5

5

2.5

S T 150 160 170 180

-2.5 H L

-5

-7.5

2.5 – Synthetic Positions

Synthetic positions, a portfolio of options whose pay-offs replicate those of the underlying, have many applications. Some stocks are difficult to short, so a synthetic short is a more practical was to take a short position than a “real” short. There are cases in which it would be illegal or not politic to sell a stock position; a synthetic short could accomplish the same thing. Investors frequently employ leverage to magnify returns. Often a suitable options position is the best way to achieve this leverage.

2.5.1 – Straight Synthetic Positions A synthetic long is achieved by writing a put and buying a call, both usually (but not always) at the money: 18 ams-511-lec-11A-p.nb

p Synthetic Long Stock

5

S T 150 160 170 180

H L -5

-10

p Synthetic Long Stock

5

S T 155 160 165 170

H L -5

-10

If we compare the net profit of a long position to that of a synthetic, then we note that the net on the long position is slightly higher. However, for the synthetic position an investor only has to put the premia, not the full value of the position, and is, therefore, borrowing at a fairly low rate of return. Transaction costs on the options may also be much less that those of a direct stock purchase. ams-511-lec-11A-p.nb 19

Synthetic Long Stock Red Solid p Long Stock Black Dashed 10 H L H L 5

S T 155 160 165 170

H L -5

-10

2.5.2 – Boxes A box can be thought of two offsetting synthetic positions, one long and one short. The options in the long component are written at a strike below the underlying’s and the options of the short component are written at one higher:

p Box 30

20

10

S T 150 160 170 180

H L -10

20 ams-511-lec-11A-p.nb

p Box 40

30

20

10

S T 150 160 170 180

Again, it is easier to visualize a box as a combination of synthetic positions rather than in terms of theH baseL options positions. The net effect is to lock in a profit, with no upside or downside participation:

2.6 – Vertical Spreads

Vertical spreads are means of producing synthetic positons with both up- and downside protection.

2.6.1 – Bullish Vertical Spread Using Calls

p Bullish Vertical Spread Calls

7.5 H L

5

2.5

S T 75 80 85 90

-2.5 H L

-5

-7.5 ams-511-lec-11A-p.nb 21

p Bullish Vertical Spread Calls 2

1 H L

S T 75 80 85 90

H L -1

-2

-3

2.6.2 – Bullish Vertical Spreads Using Puts

p Bullish Vertical Spread Puts 4

H L 2

S T 75 80 85 90

H L -2

-4

-6

22 ams-511-lec-11A-p.nb

p Bullish Vertical Spread Puts

2 H L

1

S T 75 80 85 90

H L -1

-2

2.6.3 – Bearish Vertical Spreads

p Bearish Vertical Spread

2

1

S T 75 80 85 90

H L -1

-2

2.7 – Getting the Details Right

There’s still a lot of details to consider before anyone goes out and starts trading options.

2.7.1 – Transaction Costs Trades are entered into to either increase return or decrease risk–often both. As such an investor needs to accurately balance costs against benefits in order to decide whether or not a given strategy makes sense. The transaction costs—both the fees paid and the bid-ask spreads in the market—are important elements that we haven’t explored here. ams-511-lec-11A-p.nb 23

2.7.2 – Time Decay and the Potential for Early Exercise We evaluated the net profit assuming the trade is entered into at time t = 0 and completed at time t = T. Unfortunately, events sometimes intervene. Some of the options may be subject to early exercise. Other financial pressures may arise that demand immediate capital. The point is that part of the assessment of the risk of a strategy has to factor in the time decay of its options: C t Long Call Time Decay

H L 100

80

60

40

20

S t 50 100 150 200

p Time Decay in a Covered Call H L

20

S T 50 100 150 200

H L -20

-40

-60

2.7.3 – Dividends and Other Distributions Distributions, such as dividends, can materially affect the price of a stock. When a stock pays a dividend its price declines by a commensurate amount. Generally, the pricing function of an option is based on the raw price of the stock and does not take adjustements for divdends into account. 24 ams-511-lec-11A-p.nb

2.8 – More Complex Strategies

We’ve just scratched the surface of options strategies. For example, we haven’t consider strategies across different underly- ing securities, nor have we looked at strategies involving more than one expiration date.