AMS 511.01 - Foundationsfrey/Instruction/PriorSemesters/2006-07/Spri… · AMS 511.01 - Foundations Class 11A Robert J. Frey Research Professor Stony Brook University, Applied Mathematics

AMS 511.01 - Foundations Class 11A

Robert J. FreyResearch ProfessorStony Brook University, Applied Mathematics and [email protected]

http://www.ams.sunysb.edu/~frey/

In this lecture we will cover the pricing and use of derivative securities, covering Chapters10 and 12 in Luenberger’s text.April, 2007

1. The Binomial Option Pricing Model

1.1 – General Single Step SolutionThe geometric binomial model has many advantages. First, over a reasonable number of steps it represents a surprisinglyrealistic 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.

K 11 O = K H1 + r DL H1 + r DL

u 1 êu O Kyu

ydO fl Kyu

ydO =

ikjjjjjj

1+r D-1êuÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1+r DL Hu-1êuL1+r-uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1+r DL H1êu-uL

y{zzzzzz

As we will see shortly we will solve the general problem by solving a sequence of single step problems on the lattice. Thatsequence solutions can be efficiently computed because we only have to solve for the state prices once.

1.2 – Valuing an Option with One Period to ExpirationLet 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 strikeprice of 100 that expires the next three month period. We’ll use a single binomial step, so D = 0.25 months. For the period oft 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.

i

kjjjjjjj

1SHtLCHtL

y

{zzzzzzz =

i

kjjjjjjj

H1 + r DL H1 + r DLu SHtL SHtL ê u

max@u SHtL - K, 0D max@SHtL êu - K, 0D

y

{zzzzzzz Kyu

ydO fl

i

kjjjjjjjj

1105CHtL

y

{zzzzzzzz =

i

k

jjjjjjjj1.01 1.01

115.50 95.4515.5 0

y

{

zzzzzzzz Kyu

ydO

We can solve the first two equations for the state prices using the closed form solution

Kyu

ydO =

ikjjjjjj

1+r D-1êuÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1+r DL Hu-1êuL1+r D-uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1+r DL H1êu-uL

y{zzzzzz =

ikjjjjjj

1+0.04µ0.25-1ê1.10ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1+0.04µ0.25L H1.10-1ê1.10L1+0.04µ0.25-1.10ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1+0.04µ0.25L H1ê1.10-1.10L

y{zzzzzz = K 0.523

0.467O

If we take these values and substitute them into the last row we realize a call premium of C(t) > 5.23.

The risk neutral measure Pè is

ikjjjj

Pè u

Pè d

y{zzzz =

1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + r D

Kyu

ydO =

1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + 0.01

K0.6730.288

O = K 0.5290.471

O

It is remarkable but, other than meeting the requirement that 0 < Pu < 1, the ordinary probability measure does not play a rolein the solution.

1.3 – The General Binomial Option Pricing ModelWe’ll return to the example above but now assume that the option expires is six months, but we wish to model the stock priceevolution using two lattice steps. The binomial lattice for the stock is then

105.00

115.50

95.45

127.05

105.00

86.78

As before, the risk neutral measure is

ikjjjj

Pè u

Pè d

y{zzzz = K 0.529

0.471O

The “leaves” of the lattice above occur at expiry, so for each price node we can calculate the value of the call option. Thisgives us the

27.05

5.00

0.00


ikjjjj

Pè u

Pè d

y{zzzz

T

K27.055.00

O =1

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1.01

K0.5290.471

OT

K27.055.00

O = 16.49


ikjjjj

Pè u

Pè d

y{zzzz

T

K5.000.00

O =1


K 0.5290.471

OT

K 5.000.00

O = 2.62

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

We can now add this information to the lattice.

16.49

2.62

27.05

5.00

0.00


ikjjjj

Pè u

Pè d

y{zzzz

T

K16.492.62

O =1


K0.5290.471

OT

K16.492.62

O = 9.85

This allows us to complete the lattice. The value at the “root”, 9.85, is the value of the call option.

9.85

16.49

2.62

27.05

5.00

0.00

1.4 – Calibrating the Model to Market DataFor a given stepsize D we have two free parameters in the geometric binomial model: Pu and u. As was shown earlier, with asufficient number of steps the log price tends towards a Normal distribution. The calibration of the geometric binomial modelat stepsize D, therefore, involves selecting a set of parameters Pu and u so that the mean and standard deviation of the logprice are matched.

m = EClogC SHt + 1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

SHtL GG

s2 = VarClogC SHt + 1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

SHtL GG

Pu =1ÅÅÅÅÅ2

J1 +m

ÅÅÅÅÅÅÅs

è!!!!

D N

u = es è!!!!D

ams-511-lec-11A-p.nb 3

1.5 – Mathematica CodeThis code is not the most efficient; it’s meant to illustrate the computations in the simplest terms. Each step or level in thelattice 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 suppliednUpFactor.

fLatticeForwardStep@vnOneLevel_, nUpFactor_D :=Append@vnOneLevel nUpFactor, Last@vnOneLevelD ê nUpFactorD;

The function fRiskNeutralMeasure calculates the value of Pè u assuming a geometric binomial model.

fRiskNeutralMeasure@nRiskFree_, DTime_, nUpFactor_D :=-1 + nUpFactor + nRiskFree nUpFactor DTimeÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

-1 + nUpFactor2;

These functions are used at the leaves of the undelying’s lattice to compute the leaves of the option’s lattice.

fCallAtExpiry@nNowPrice_, nStrikePrice_D :=Max@nNowPrice - nStrikePrice, 0D;

fPutAtExpiry@nNowPrice_, nStrikePrice_D :=Max@nStrikePrice - nNowPrice, 0D;

fLatticeBackStep@vnOneLevel_,nRiskFree_, DTime_, nRiskNeutralProb_D := Module@8i<,Table@HnRiskNeutralProb vnOneLevelPiT + H1 - nRiskNeutralProbL

vnOneLevelPi + 1TLê H1 + nRiskFree DTimeL,8i, 1, Length@vnOneLevelD - 1<

DD;


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 inthe code, are computed by using NestList to recursively apply fLatticeForwardStep from the root to the leaves and thenfLatticeBackStep 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_D := Module@8vvnBackwardTree,vnExpiryPrice, vvnForwardTree, nRiskNeutralProb<,vvnForwardTree = NestList@

fLatticeForwardStep@#, nUpFactorD &,8nNowPrice<,iIntervals

D;nRiskNeutralProb =fRiskNeutralMeasure@nRiskFree, DTime, nUpFactorD;vnExpiryPrice = If@cPutCall == c,

fCallAtExpiry@#, nStrikePriceD & êü Last@vvnForwardTreeD,fPutAtExpiry@#, nStrikePriceD & êü Last@vvnForwardTreeD

D;vvnBackwardTree = NestList@

fLatticeBackStep@#, nRiskFree, DTime, nRiskNeutralProbD &,vnExpiryPrice,iIntervals

D;8nRiskNeutralProb, vvnForwardTree, vvnBackwardTree<

D;


1.6 – Example: Pricing a Put OptionConsider 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 onthe 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 optionusing a 5-step geometric binomial option model.

nTimeToExpiry = 0.5;

iIntervals = 10;

DTime = nTimeToExpiry ê iIntervals;

nUpFactor = ExpBè!!!!!!!!!!!!!!!!!!!!!!!!!0.01 DTime F;

8nRiskNeutral, vvnStockLattice, vvnOptionLattice< =fGeometricBinomialOption@125, nUpFactor,120, 0.04, iIntervals, DTime, pD;

MatrixForm êü vvnStockLatticeMatrixForm êü vvnOptionLattice


looooooooooooooooooooooooom

n

ooooooooooooooooooooooooo

H 125 L, K 127.827122.236 O,

i

k

jjjjjjjj130.717125.119.533

y

{

zzzzzzzz,i

k

jjjjjjjjjjjj

133.673127.827122.236116.89

y

{

zzzzzzzzzzzz,

i

k

jjjjjjjjjjjjjjjjjjj

136.696130.717125.119.533114.305

y

{

zzzzzzzzzzzzzzzzzzz,

i

k

jjjjjjjjjjjjjjjjjjjjjjj

139.787133.673127.827122.236116.89111.778

y

{

zzzzzzzzzzzzzzzzzzzzzzz

,

i

k

jjjjjjjjjjjjjjjjjjjjjjjjjjjjj

142.948136.696130.717125.119.533114.305109.306

y

{

zzzzzzzzzzzzzzzzzzzzzzzzzzzzz

,

i

k

jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

146.18139.787133.673127.827122.236116.89111.778106.889

y

{

zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

,

i

k

jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

149.485142.948136.696130.717125.119.533114.305109.306104.525

y

{

zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

,

i

k

jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

152.866146.18139.787133.673127.827122.236116.89111.778106.889102.214

y

{

zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

,

i

k

jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

156.322149.485142.948136.696130.717125.119.533114.305109.306104.52599.9537

y

{

zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

|ooooooooooooooooooooooooo}

~

ooooooooooooooooooooooooo


loooooooooooooooooooooooom

n

oooooooooooooooooooooooo

i

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jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

0000000.4670135.6949210.694215.474820.0463

y

{

zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

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jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

000000.2148042.870677.9829712.871717.5466

y

{

zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

,

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k

jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

00000.09879941.435955.2163610.215614.9962

y

{

zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

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0000.0454430.7136263.171897.5053612.3941

y

{

zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

,

i

k

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000.02090160.3526851.842895.158759.73896

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00.009613760.1734641.03743.364357.25513

y

{

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0.004421870.0849580.5704882.105625.14721

y

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i

k

jjjjjjjjjjjjj

0.04145590.3081091.275443.5004

y

{

zzzzzzzzzzzzz,i

kjjjjjjjj

0.1640210.7524182.29627

y

{zzzzzzzz, K

0.4343281.46101 O, H 0.905688 L

|oooooooooooooooooooooooo}

~

oooooooooooooooooooooooo

The price is > 0.91

2 – Hedging and Risk Control with Derivatives

2.1 – Basic ToolsWe can construct a number of different trading strategies using the basic tools of a long or short position in the underling andbuying 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 ofthe total position at expiration. At this stage we won’t consider the effects of transaction costs or the impact of unwindingthese 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, putand call is S(t), P(t) and C(t), respectively. Where we need to indicate different strikes, we’ll use subscripts; e.g., CiHtL ’sstrike will be Ki .

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


20 40 60 80SHTL

-40

-20

20

40

p Long Stock

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

20 40 60 80SHTL

-40

-20

20

40

p Short Stock

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


40 50 60 70SHTL

5

10

15

p Long Call

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

40 50 60 70SHTL

-15

-10

-5

p Short Call

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


40 50 60 70SHTL

5

10

15

p Long Put

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

40 50 60 70SHTL

-15

-10

-5

p Short Put

2.2 – Downside ProtectionInvestment involves risk. Sometimes the potential size of a loss associated with a position is unacceptable. A natural case inpoint 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 outmay 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 isthe lower the premium we will pay, but the further out our downside protection is. This makes sense: the more insurance wehave, 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,


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

140 160 180 200SHTL

-40

-20

20

40

p Short Stock with Downside Protection

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

140 160 180 200SHTL

-20

-10

10

20

30


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

2.3 – Covered CallA covered call is a hedged position in which an at-the-money call is written against a long position in the underlying. The thecase below we have S(0) = 160, C(0) = 6.325 with K = 160.


140 160 180 200SHTL

-40

-20

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 ifhe or she were convinced that a stock is likely to hold to its current value or decrease slightly.

140 160 180 200SHTL

-30

-20

-10

p Covered Call

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

2.4.1 – StraddlesA 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:


150 160 170 180SHTL

5

10

15

p Long Straddle

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

150 160 170 180SHTL

-5

5

10

p Long Straddle

A short strangle simply reverses the trades.

2.4.2 – StranglesBy buying the put and call at different strike prices it’s possible to implement a similar strategy called a strangle. This is a bitcheaper to implement but doesn't “kick in” unless the movement of the price is above some threshold.


145 150 155 160 165 170 175SHTL

5

10

15

p Long Strangle

145 150 155 160 165 170 175

SHTL

2

4

6

8

10

12

14

p Long Strangle

2.4.3 – Butterflies to Control the DownsideOnce we begin to build a “library” of hedge strategies we can think of combining them to produce more complex results. Forexample, consider the net profit plot for a short straddle:


150 160 170 180SHTL

-10

-5

5

p Short Straddle

While an investor may have an opinion that a stock’s price will not change dramatically, he or she may hold that opinion withvarying 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 moneyand it premia are, therefore, less than those of the straddle whose strikes are at the money. The long straddle looks like:

150 160 170 180SHTL

2

4

6

8

p Long Strangle

Putting them together produces what is called a long butterfly:


150 160 170 180SHTL

-4

-2

2

4

p Long Butterfly

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:

150 160 170 180SHTL

-7.5

-5

-2.5

2.5

5

7.5

p Long Butterfly

2.5 – Synthetic PositionsSynthetic positions, a portfolio of options whose pay-offs replicate those of the underlying, have many applications. Somestocks are difficult to short, so a synthetic short is a more practical was to take a short position than a “real” short. There arecases 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 thisleverage.

2.5.1 – Straight Synthetic PositionsA synthetic long is achieved by writing a put and buying a call, both usually (but not always) at the money:


150 160 170 180SHTL

-10

-5

5

p Synthetic Long Stock

155 160 165 170SHTL

-10

-5

5

p Synthetic Long Stock

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 slightlyhigher. 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 adirect stock purchase.


155 160 165 170SHTL

-10

-5

5

10

pSynthetic Long Stock HRed SolidL

Long Stock HBlack DashedL

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

150 160 170 180SHTL

-10

10

20

30

p Box


150 160 170 180SHTL

10

20

30

40

p Box

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

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

2.6.1 – Bullish Vertical Spread Using Calls

75 80 85 90SHTL

-7.5

-5

-2.5

2.5

5

7.5

p Bullish Vertical Spread HCallsL


75 80 85 90SHTL

-3

-2

-1

1

2

p Bullish Vertical Spread HCallsL

2.6.2 – Bullish Vertical Spreads Using Puts

75 80 85 90SHTL

-6

-4

-2

2

4

p Bullish Vertical Spread HPutsL


75 80 85 90SHTL

-2

-1

1

2

p Bullish Vertical Spread HPutsL

2.6.3 – Bearish Vertical Spreads

75 80 85 90SHTL

-2

-1

1

2

p Bearish Vertical Spread

2.7 – Getting the Details RightThere’s still a lot of details to consider before anyone goes out and starts trading options.

2.7.1 – Transaction CostsTrades are entered into to either increase return or decrease risk–often both. As such an investor needs to accurately balancecosts against benefits in order to decide whether or not a given strategy makes sense. The transaction costs—both the feespaid and the bid-ask spreads in the market—are important elements that we haven’t explored here.


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, eventssometimes intervene. Some of the options may be subject to early exercise. Other financial pressures may arise that demandimmediate capital. The point is that part of the assessment of the risk of a strategy has to factor in the time decay of itsoptions:

50 100 150 200SHtL

20

40

60

80

100

CHtL Long Call Time Decay

50 100 150 200SHTL

-60

-40

-20

20

p Time Decay in a Covered Call

2.7.3 – Dividends and Other DistributionsDistributions, such as dividends, can materially affect the price of a stock. When a stock pays a dividend its price declines bya commensurate amount. Generally, the pricing function of an option is based on the raw price of the stock and does not takeadjustements for divdends into account.


2.8 – More Complex StrategiesWe’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.


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AMS 511.01 - Foundationsfrey/Instruction/PriorSemesters/2006-07/Spri… · AMS 511.01 - Foundations Class 11A Robert J. Frey Research Professor Stony Brook University, Applied Mathematics