1

Multi-asset Options

1 In this lecture. . .

q how to model the behaviour of many assets simultaneously

q estimating correlation between asset price movements

q how to value and hedge options on many underlying assets in theBlack–Scholes framework

q the pricing formula for European non-path-dependent options ondividend-paying assets

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2 IntroductionIn this lecture we see the idea of higher dimensionality by examiningthe Black–Scholes theory for options on more than one underlyingasset. This theory is perfectly straightforward; the only new idea isthat of correlated random walks and the corresponding multifactorversion of Itô’s lemma.

Although the modeling and mathematics is easy, the final step ofthe pricing and hedging, the ‘solution,’ can be extremely hard in-deed. We see what makes a problem easy, and what makes it hard,from the numerical analysis point of view.

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3 Multi-dimensional lognormal randomwalksThe basic building block for option pricing with one underlying isthe lognormal random walk

G 6 { 6 G W � � 6 G ; �

This is readily extended to a world containing many assets via mod-els for each underlying

G 6 L {L6 L G W � � L6 L G ; �

Here 6 L is the price of theLth asset,L ������G , and{L

and � L

are the drift and volatility of that asset respectively andG ; L is theincrement of a Wiener process.

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We can still continue to think ofG ; L as a random number drawnfrom a Normal distribution with mean zero and standard deviationG W� � so that

( >G ; L@ � and ( >G ; �

L @ G W

but the random numbersG ; L andG ; M arecorrelated:

( >G ; L G ; M@ � LMG W�

here� LM is the correlation coefficient between theLth andM th randomwalks.

q The symmetric matrix with� LM as the entry in theLth row andM thcolumn is called thecorrelation matrix .

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For example, if we have seven underlyingsG � and the correla-tion matrix will look like this:

i

�%%%%%%%%#

� � �� � �� � �� � �� � � � � ��

� � � � � �� � �� � �� � � � � ��

� � � � �� � � �� � �� � � � � ��

� � � � �� � �� � � �� � � � � ��

� � � � �� � �� � �� � � � � � ��

� � � � �� � �� � �� � �� � � ��

� � � � �� � �� � �� � �� � � � �

�&&&&&&&&$

Note that�LL

� and�LM � ML. The correlation matrix is positive

definite, so thaty7i y w � . Thecovariance matrix is simply

Mi M�

whereM is the matrix with the� L along the diagonal and zeros every-where else.

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To be able to manipulate functions of many random variables weneed a multidimensional version of Itô’s lemma. If we have a func-tion of the variables6 � ,. . . ,6 G andW, 9 �6 ������6 G �W�, then

G 9

�# # 9

# W�

�

�

G;

L �

G;

M �

� L� M� LM6 L6 M

# �9

# 6 L# 6 M

�$ G W �

G;

L �

# 9

# 6 L

G 6 L�

We can get to this same result by using Taylor series and the rulesof thumb:

G ; �

L G W and G ; LG ; M � LM G W�

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4 Measuring correlationsIf you have time series data at intervals ofsW for all G assets youcan calculate the correlation between the returns as follows. First,take the price series for each asset and calculate the return over eachperiod. The return on theLth asset at theN th data point in the timeseries is simply

5 L�WN � 6 L�WN � sW� b 6 L�WN �

6 L�WN ��

The historical volatility of theLth asset is

� L

YXXW �

s W�0 b ��

0;

N �

�5 L�WN � b

{5 L��

where0 is the number of data points in the return series and{5 L isthe mean of all the returns in the series.

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The covariance between the returns on assetsL andM is given by

�

s W�0 b ��

0;

N �

�5 L�WN � b

{5 L��5 M�WN � b

{5 M��

The correlation is then

�

s W�0 b ��� L� M

0;

N �

�5 L�WN � b

{5 L��5 M�WN � b

{5 M��

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Correlations measured from financial time series data are notori-ously unstable.

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2t

1.A correlation time series.

The other possibility is to back out animplied correlation from thequoted price of an instrument. The idea behind that approach is thesame as with implied volatility, it gives an estimate of the market’sperception of correlation.

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5 Options on many underlyingsOptions with many underlyings are calledbasket options, optionson basketsor rainbow options. The theoretical side of pricing andhedging is straightforward, following the Black–Scholes argumentsbut now in higher dimensions.

Set up a portfolio consisting of one basket option and short a num-berd L of each of the assets6 L:

h 9 �6 ������6 G �W� b

G;

L �

d L6 L�

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The change in this portfolio is given by

G h

�# # 9

# W�

�

�

G;

L �

G;

M �

� L� M� LM6 L6 M

# �9

# 6 L# 6 M

�$ G W�

G;

L �

w# 9

# 6 L

b d L

xG 6 L�

If we choose

d L # 9

# 6 L

for eachL, then the portfolio is hedged, is risk-free. Setting the returnequal to the risk-free rate we arrive at

# 9

# W�

�

�

G;

L �

G;

M �

� L� M� LM6 L6 M

# �9

# 6 L# 6 M

� U

G;

L �

6 L

# 9

# 6 L

b U 9 �� (1)

This is the multidimensional version of the Black–Scholes equation.

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The modifications that need to be made for dividends are obvious.When there is a dividend yield of' L on theLth asset we have

# 9

# W�

�

�

G;

L �

G;

M �

� L� M� LM6 L6 M

# �9

# 6 L# 6 M

�

G;

L �

�U b ' L�6 L

# 9

# 6 L

b U9 �

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6 The pricing formula for Europeannon-path-dependent options ondividend-paying assetsBecause there is a Green’s function for this problem we can writedown the value of a European non-path-dependent option with pay-off of Payoff�6 ������6 G � at time7 :

9 Hb U�7 b W� ��~ �7 b W��b G � �Deti �b � ��� � ccc� G �

b �

=�

�

ccc

=�

�

Payoff�6 �

�ccc6

�

G�

6 �

�ccc6

�

G

H[ S

wb

�

�p

7i

b �p

xG 6

�

�cccG 6

�

G�

(2)

p L

�

� L�7 b W�� �

wORJ

w6 L

6 �

L

x�

wU b ' L b

��

L

�

x�7 b W�

x

This has included a constant continuous dividend yield of' L oneach asset.

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7 Exchanging one asset for another: asimilarity solutionAn exchange optiongives the holder the right to exchange one assetfor another, in some ratio. The payoff for this contract at expiry is

P D[ �T�6 � b T �6 �����

whereT � andT� are constants.

The partial differential equation satisfied by this option in a Black–Scholes world is

# 9

# W� �

�

�;

L �

�;

M �

� L� M� LM6 L6 M

# �9

# 6 L# 6 M

�

�;

L �

�U b ' L�6 L

# 9

# 6 L

b U 9 ��

A dividend yield has been included for both assets. Since there areonly two underlyings the summations in these only go up to two.

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This contract is special in that there is a similarity reduction. Let’spostulate that the solution takes the form

9 �6 ��6 ��W� T�6 �+ �} �W��

where the new variable is

} 6 �

6 �

�

If this is the case, then instead of finding a function9 of three vari-ables, we only need find a function+ of two variables, a much easiertask.

Changing variables from6 ��6 � to } we must use the following forthe derivatives.

#

# 6 �

�

6 �

#

# }�

#

# 6 �

b

}

6 �

#

# }�

# �

# 6 �

�

�

6 �

�

# �

# } ��

# �

# 6 �

�

} �

6 �

�

# �

# } ��

�}

6 �

�

#

# }�

# �

# 6 �# 6 �

b

}

6 �

�

# �

# } �b

�

6 �

�

#

# }�

The time derivative is unchanged.

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The partial differential equation now becomes

# +

# W�

�

�� ��} �

# �+

# } �� �' � b ' ��}

# +

# }b ' �+ ��

where

��

T�

�

�b ��

� �� �� � � � �

��

You will recognise this equation as being the Black–Scholes equationfor a single stock with' � in place ofU , ' � in place of the dividendyield on the single stock and with a volatility of� �.

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From this it follows, retracing our steps and writing the result inthe original variables, that

9 �6 ��6 ��W� T�6 �Hb ' ��7 b W�

1 �G �

�� b T �6 �H

b ' ��7 b W�1 �G �

��

where

G�

�

ORJ �T�6 � T�6 �� � �' � b ' � ��

��

����7 b W�

� �

S7 b W

and G�

� G

�

�b �

�

S7 b W�

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The next example is of basket equity swap. This rather complex,high-dimensional contract, is for a swap of interest payments basedon three-month LIBOR and the level of an index. The index is madeup of the weighted average of 20 pharmaceutical stocks. To makematters even more complex, the index uses a time averaging of thestock prices.

Preliminary and IndicativeFor Discussion Purposes Only

International Pharmaceutical Basket Equity Swap

Indicative termsTrade Date [ ]Initial Valuation Date [ ]Effective Date [ ]Final Valuation Date 26th September 2004Averaging Dates The monthly anniversaries of the Initial

Valuation Date commencing 26th March 2004and up to and including the Expiration Date

Notional Amount US$25,000,000

Counterparty floatingamounts (US$ LIBOR)Floating Rate Payer [ ]Floating Rate Index USD-LIBORDesignated Matrurity Three monthsSpread Minus 0.25%Day Count Fraction Actual/360Floating Rate PaymentDates

Each quarterly anniversary of the Effective Date

Initial Floating Rate Index [ ]

The Bank Fixed andFloating Amounts (Fee,Equity Option)Fixed Amount Payer XXXXFixed Amount 1.30% of Notional AmountFixed Amount PaymentDate

Effective Date

Basket A basket comprising 20 stocks and constructedas described in attached Appendix

Initial Basket Level Will be set at 100 on the Initial Valuation DateFloating Equity AmountPayer

XXXX

Floating Equity Amount Will be calculated according to the performanceof the basket of stocks in the following way:

where

And for each stock the Weight is given in theAppendixP_initial is the local currency price of each stockon the Initial Valuation DateP_average is the arithmetic average of the localcurrency price of each stock on each of theAveraging Dates

Floating Equity AmountPayment Date

Termination Date

AppendixEach of the following stocks are equally weighted (5%):Astra (Sweden), Glaxo Wellcome (UK), Smithkline Beecham (UK), Zeneca Group (UK),Novartis (Switzerland), Roche Holding Genus (Switzerland), Sanofi (France), Synthelabo(France), Bayer (Germany), Abbott Labs (US), Bristol Myers Squibb (US), AmericanHome Products (US), Amgen (US), Eli Lilly (US), Medtronic (US), Merck (US), Pfizer(US), Schering-Plough (US), Sankyo (Japan), Takeda Chemical (Japan).

This indicative termsheet is neither an offer to buy or sell securities or an OTC derivative product which includesoptions, swaps, forwards and structured notes having similar features to OTC derivative transactions, nor asolicitation to buy or sell securities or an OTC derivative product. The proposal contained in the foregoing is nota complete description of the terms of a particular transaction and is subject to change without limitation.

−100

100averageBASKET0,max * AmountNotional

∑

=

stocks initial

averageaverage P

PWeightBASKET

120

**100

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8 Realities of pricing basket optionsThe factors that determine the ease or difficulty of pricing and hedg-ing multi-asset options are

q existence of a closed-form solution

q number of underlying assets, the dimensionality

q path dependency

q early exercise

We have seen most of these in a single-asset setting.

The solution technique that we use will generally be one of

q finite-difference solution of a partial differential equation

q numerical integration

q Monte Carlo simulation

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8.1 Easy problems

If we have a closed-form solution then our work is done; we caneasily find values and hedge ratios. This is provided that the solu-tion is in terms of sufficiently simple functions for which there arespreadsheet functions or other libraries. If the contract is Europeanwith no path-dependency then there may be a solution in the formofa multiple integral. If this is the case, then we often have to do theintegration numerically.

8.2 Medium problems

If we have low dimensionality, less than three or four, say, the finite-difference methods are the obvious choice. They cope well withearly exercise and many path-dependent features can be incorpo-rated, though usually at the cost of an extra dimension.

For higher dimensions, Monte Carlo simulations are good. Theycope with all path-dependent features. Unfortunately, they are notvery efficient for American-style early exercise.

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8.3 Hard problems

The hardest problems to solve are those with both high dimension-ality, for which we would like to use Monte Carlo simulation, andwith early exercise, for which we would like to use finite-differencemethods. There is currently no numerical method that copes wellwith such a problem.

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9 Realities of hedging basket optionsEven if we can find option values and the greeks, they are often verysensitive to the level of the correlation. But the correlation is a verydifficult quantity to measure. So the hedge ratios are very likely to beinaccurate. If we are delta hedging then we need accurate estimatesof the deltas. This makes basket options very difficult to delta hedgesuccessfully.

When we have a contract that is difficult to delta hedge we can tryto reduce sensitivity to parameters, and the model, by hedging withother derivatives. This is the basis of vega hedging. We could try touse the same idea to reduce sensitivity to the correlation. Unfortu-nately, that is also difficult because there just aren’t enough contractstraded that depend on the right correlations.