58 Wilmott magazine

1. Introduction

Equity basket options have long been important tools in structuring

financial products. They are widely used in portfolio management and as

retail products as they are typically more cost-effective than multiple sin-

gle underlying options. The components in a basket can be single stocks

(names), equity indices or funds. Historically, managing basket option

books has proved to be problematic and rather difficult for many banks.

Quite often, when the markets move, the basket options and their

hedges, which are supposed to offset the basket positions, do not move in

the directions as they are supposed to. In these circumstances the basket

pricing models used are partially responsible as they fail to capture cer-

tain type of fundamental basket risks. Among them, the correlation

skew and its structural change triggered by sudden market move are the

key contributors.

In the absence of a volatility smile/skew, a European option on a bas-

ket can be priced using various techniques (ref. 1–4), including moment

matching or geometric conditioning. In the presence of a volatility

smile/skew, pricing basket options becomes much more complicated. It

becomes important to understand how the basket skew behaves and how

the skewed correlation structure is related to it.

Interestingly, although the correlation skew has long been a topic of

study in the context of equity derivatives basket, and the copula tech-

nique was one of the proposed methods (ref. 5) to deal with it, it is in the

credit derivatives business, the copula technique becomes an industry

standard to deal with basket of names. In this paper, we will still focus on

equity derivatives basket. The key risks associated with the basket options

will be investigated. We will review a technique to extract historic basket

Pricing Basket

Options With Skew

volatility surface and analyse a few examples. We will develop further the

copula implied basket volatility construction technique (ref. 5) and exam-

ine the relationship between the basket skew and correlation skew. The

effect of the correlation skew and its structural change on the basket

skew will also be examined.

2. Basket Risks

A typical basket consists of several underlyings with associated weight-

ings. The basket spot is:

S

B

=

n

i=1

w

i

· S

i

Where

w

i

is the weightings and

S

i

the individual underlyings. The values

of European call and put options on the basket are calculated from:

E

Q

Max(S

B

− K, 0)

; Call

E

Q

Max(K − S

B

, 0)

; Put

A basket option book will generally consist of numerous basket calls

and puts, as defined above. In addition to the normal first order risks

(e.g. delta, gamma, vega) associated with individual basket underlying,

there are several other key basket specific risk exposures. A summary of

the basket specific risks is given in the table below.

It is evident that a basket (correlation) book is a complex entity

involving significant second order effects that needs to be understood

and modelled. In the following, we will analyse the basket volatility skew

risks and the associated correlation issues.

Dong Qu

Quantitative Products Group—Derivatives & Structured Products—Abbey

Wilmott magazine 59

3. Historic Basket Volatility Surfaces

It is possible to construct a historic basket volatility surface from the his-

toric time series of the individual basket components. One such

where

N

defines the time interval. The bucketed log-returns

R

T

create the

historic PDF denoted as

P

. An example of a historic PDF is shown in Fig. 1;

Step 2: Recover the risk neutral PDF (

Q

) from the historic PDF (

P

). This is

a key step and a numerical optimisation technique is used for this. The

penalty function for optimisation is based on a relative entropy, which is

a mathematical measure of the uncertainty of a distribution. The relative

entropy of

Q

to

P

is defined as follows:

S(P, Q ) = E

Q

(ln Q − ln P)

By minimising relative entropy

S(P, Q )

, one can solve for

Q

, subject to

the following constraints,

Q (S

T

) · S

T

· dS

T

= Forward

Q (S

T

) · dS

T

= 1

Additional optimisation constraint can also be applied, such as impos-

ing the ATM volatility. The optimisation solution for the problem posed

above is:

Q (S

T

) =

P(S

T

)

P(S) · exp(−λS) · dS

exp(−λS

T

)

Where

λ

is a constant that can be obtained by iterating constraints dur-

ing the numerical optimisation. Note that given

P(S

T

)

is always positive,

Q (S

T

)

is guaranteed to be positive. In Fig. 2, an example of risk neutral

PDF is shown and it is derived from the historic PDF shown in Fig. 1. The

mean drift is the expected drift of the forward given the historic yield

curve and dividends.

Step 3: Use the risk neutral PDF

Q

to obtain option prices with different

strikes

K

i

:

Call (K

i

, T) = DF ·

∞

K

(S

T

− K

i

) · Q (S

T

) · dS

T

Put (K

i

, T) = DF ·

K

0

(K

i

− S

T

) · Q (S

T

) · dS

T

^

TECHNICAL ARTICLE 2

Basket Specific Risks Comments

Cross Gamma Risks The cross Gamma is defined as

∂

2

P

∂S

i

∂S

j

,

i = j

,

where P is the basket option price, S

i

and S

j

are the underlying spots. The cross Gamma

specifies the delta change of

i

-th underlying

caused by the spot change in the j-th

underlying.

Basket Correlation Risks When large moves occur in the market,

the basket correlation tends to spike

and this will impact the pricing as well as risk

parameters. This is also referred to as the

skewed correlation structure risks.

Basket Volatility Skew Risks The basket volatility surface is largely driven

by the volatility characteristics of the

individual components. It is also heavily

affected by the skewed correlation structure.

FX Correlation Risks In a Quanto or Compo basket, the FX

correlation with each individual basket

component will impact the price.

Contractual Risks If merge and acquisition activity happens

to the basket member, the basket

composition will change and so will

its volatility structure.

Jump Risks In a large single stock basket, it is more

likely that one or some of the basket

underlying default. The corresponding

underlying price may jump to zero.

approach is to minimise the relative entropy (ref. 6) in which a risk neu-

tral probability density function (PDF) for the basket can be built. Subject

to constraints of the forward and ATM implied volatility, the risk neutral

PDF is used to construct a historic volatility surface for the basket.

Specifically, given the historic time series of the basket spot

S

i

the fol-

lowing steps are taken to build a historic basket volatility surface:

Step 1: Recover the historic PDF for the basket at time interval

T

by cal-

culating the log-return at appropriate time interval:

R

T

= ln

S

t+N

S

t

0.0%

0.5%

1.0%

1.5%

70% 90% 110% 130% 150%

Figure 1: Historic PDF (Mean Drift = 9%).

60 Wilmott magazine

Where

DF

is the discount factor. The option prices can then be used to

recover the implied volatility points at

K

i

. By repeating the above steps at

different time interval

T

, one can build an historic volatility surface for the

basket. In the following, we examine a few examples of the historic volatil-

ity surfaces. Fig. 3 shows a historic volatility surface for FTSE-100. The short

end skew is quite strong as expected. The historic volatility surface shown

in Fig. 4 is that of the FTSE-100 banking sector basket. Clearly there is a

smile at the short end and skew at the longer end. In Fig. 5, the historic

volatility surface for an equally weighted index basket of FTSE-100, S&P and

EUROSTOCK 50 is shown. Again the smile/skew is apparent in the figure.

While the historic basket volatility surfaces provide valuable smile/skew

information of the basket, it is relatively static. When the market moves, the

historic information changes little given that the recent contribution con-

stitutes a relatively small proportion of the time series. For day-to-day trad-

ing and hedging purposes, one requires different approaches to cater for

rapidly changing market conditions and the basket volatility surfaces

should incorporate the latest market information.

4. Implied Basket Volatility Surfaces

It is conceivable that one can construct an implied basket volatility sur-

face from the implied volatility surfaces of individual basket compo-

nents. One potential approach is to use the local volatility surfaces, in

which the stripped local volatility surfaces of the basket components

can be integrated with appropriate correlation structures to build an

implied basket volatility surface. The local volatility approach is a small

step operation and can be computationally expensive. A more efficient

0.0%

0.5%

1.0%

1.5%

70% 90% 110% 130% 150%

Figure 2: Risk Neutral PDF (Mean Drift = 6.75%).

50%

80%

110%

140%

170%

200%

0.08

2.0

7.0

0%

10%

20%

30%

40%

50%

Figure 3: FTSE Historic Vol Surface.

50%

80%

110%

140%

170%

200%

0.08

2.0

7.0

0%

10%

20%

30%

40%

50%

Figure 4: FT-Banks Historic Vol Surface.

70%

90%

110%

130%

0.08

0.75

3.0

0%

5%

10%

15%

20%

25%

Figure 5: Index Basket Historic Vol Surface.

^

Wilmott magazine 61

technique (ref. 5) of constructing an implied basket volatility surface is

to use a copula alike technique, in large steps dealing with the terminal

distributions.

4.1. The Copula Technique to Construct Basket

Volatility Surfaces

In order to build an implied basket volatility surface, one needs to obtain

the joint distribution of the basket components. In general, however,

knowing the marginal distributions of the individual component and

the correlation structure does not guarantee an unique joint distribu-

tion. The copula technique allows us to specify a joint distribution func-

tion for known marginal distributions with a dependence (correlation)

structure. Specifically, for given univariate marginal distributions

F

i

(S

i

)

,

the Copula links the univariate marginal distributions to their multi-

variate joint distribution

F(S

1

, S

2

, ···, S

n

)

:

F(S

1

, S

2

, ···, S

n

) = C(F

1

(S

1

), F

2

(S

2

), ···, F

n

(S

n

))

Given the implied volatility surfaces of each basket component, the

Cumulative Distribution Functions (CDF) at a chosen time slice for the

individual component can be built and these CDFs are the univariate

marginal distributions. If we impose correlation structures onto these

CDFs and link them up using a Gaussian copula, the joint (basket) distri-

bution can be created as follows:

F(S

1

, S

2

, ···, S

n

) = C(CDF

1

(S

1

), CDF

2

(S

2

), ···, CDF

n

(S

n

))

The joint distribution given above allows us to simulate the basket

spot path by sampling the correlated CDFs. Assuming there are

n

individ-

ual components in the basket, at a given time slice

T

, the following steps

can be taken to construct an implied basket volatility surface:

•

Build CDFs for all basket components at a chosen time slice

T

. This

can be achieved by calculating digital calls or puts at the appropriate

strikes since a undiscounted digital option specifies the terminal

probability for the underlying spot to be above (call) or under (put)

the strike. The CDFs are obviously distributed between 0 and 1 and a

typical CDF is shown in Fig. 6. The CDFs can be used later on to sam-

ple the individual component spot path given a uniformly distrib-

uted random number;

•

Generate

n

independent random Gaussian numbers (

g

i

) and impose a

correlation structure among them. This is a key step in which one

can either use a simple correlation number or skewed correlation

structure. If Cholesky decomposition technique is used, given the

decomposed correlation matrix

C

ij

, the correlated Gaussian number

is given by

G

i

=

j

C

ij

· g

j

;

•

Convert the correlated Gussian numbers back to uniformly distrib-

uted numbers by reversing the Wiener process

U

i

= W

−1

(G

i

)

;

•

Use the correlated uniform numbers (

U

i

)to sample individual CDFs

to obtain underlying spots (

S

i

), and in turn calculate the basket spot

S

B

=

n

i=1

w

i

· S

i

;

•

Calculate either a call or put on the basket for a given strike

K

:

Call = E

Max(S

B

− K, 0)

Put = E

Max(K − S

B

, 0)

•

Calculate the basket implied volatility from either the call or put by

reversing the Black-Scholes formula (

σ

B

(K, T) = BS

−1

(K, T))

.

The above steps can be repeated for different strikes (

K

) and maturities

(

T

) to build an entire basket implied volatility surface. It is important to

note that in order to obtain a high quality basket implied volatility sur-

face using this technique, some numerical smoothing techniques are

needed in the process.

The constructed basket implied volatility surface could be used in a

variety of ways. Firstly, it allows one to quantify the basket skew and

hence analyse the skew exposures. Secondly, the basket can be treated as

a new single underlying and other basket payoffs may be priced and risk

managed efficiently in a consistent manner using the same basket

implied volatility surface.

4.2. Basket Skew Under A Constant Correlation

In the following, we examine a basket of three underlyings. Their

volatility surfaces are shown in Fig. 7, Fig. 8 and Fig. 9 respectively. The

volatility surface for the underlying 1 has a relatively low ATM volatility

but strong skew. The volatility surface for the underlying 2 has a medi-

um ATM volatility with a medium skew. The volatility surface for the

underlying 3 has a high ATM volatility with medium skew. The underly-

ing volatility surfaces are chosen to be representative to cover a wide

range of possible market implied volatility surfaces. Fig. 10 shows the

constructed implied basket volatility surface using a constant correla-

tion coefficient of 40

%

. As can be seen in figure, the basket skew is quite

pronounced. The implied volatility points in the near right corner of the

basket volatility surface for the short maturities and high strikes are not

fully recovered. In practice, this is not a problem given that the missing

points are in the very low probability region and extrapolation into this

region is relatively straightforward.

TECHNICAL ARTICLE 2

0.0

0.5

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure 6: Sampling CDF.

62 Wilmott magazine

more negative) as the correlation increases. This is because the larger cor-

relation makes the constituents of the basket move together more often

and increases the probability of the basket spot reaching large extreme

values. The resulting probability density function can only be matched

by that from an implied volatility curve with a larger skew. When corre-

lation approaches 100

%

, the basket skew reaches maximum of

−9.8%

.

This is substantially less skewed than the most skewed underlying in the

basket which has a skew of

−23.4%

. Actually the basket skew is even less

than the least skewed underlying which has a skew of

−11.6%

.

Let us now examine the full skew term structures. Fig. 13 plots the

skew term structures of two basket underlyings and those of basket for

50%

70%

90%

110%

130%

150%

170%

190%

0.1

2.0

7.0

0%

10%

20%

30%

40%

50%

Strike

Expir

y

Figure 7: Implied Vol Surface (Underlying 1).

50%

70%

90%

110%

130%

150%

170%

190%

0.1

2.0

7.0

0%

20%

40%

60%

Strike

Expir

y

Figure 8: Implied Vol Surface (Underlying 2).

The correlation effect on the basket volatility is shown in Fig.11. The

implied basket volatility curves (volatility versus strike) at the time slice

year 1 for different correlation coefficients (

−

20

%

, 0

%

, 50

%

and 90

%

) are

plotted in the figure. It is apparent that the higher the correlation, the

higher the basket volatility level. This is a well understood fact as the

higher correlations indicate that the basket components move more

coherently, resulting in the higher basket volatility.

Fig. 11 also shows that the slope of the volatility curve changes with

the correlation. This indicates that the skew is also a function of the cor-

relation. Fig. 12 plots the basket skew versus correlation at the time slice

year 1. The basket skew is defined as the volatility slope at the strike of

100

%

. As can be seen in Fig. 12, the basket skew increases (slope becomes

50%

70%

90%

110%

130%

150%

170%

190%

0.1

2.0

7.0

0%

25%

50%

75%

Strike

Expir

y

Figure 9: Implied Vol Surface (Underlying 3).

50%

80%

110%

140%

170%

200%

0.1

2.0

7.

0

0%

10%

20%

30%

40%

Strike

Expiry

Figure 10: Basket Implied Vol Surface (Cor = 40%).

^

Wilmott magazine 63

different correlation coefficients (0

%

and 80

%

). Overall the basket skew

tends to be reduced compared with those of the basket underlyings. This

can be qualitatively understood from the central limit theorem. Given

the skew indicates a non-log-normality, and when there are more basket

underlyings contributing towards the joint distribution, the log-normal-

ity tends to be preserved and this reduces the skew.

4.3. Basket Skew Under A Skewed Correlation

Structure

In contrast to a constant correlation coefficient, a skewed correlation

structure illustrates a more complex but realistic relationship among the

basket components. Fig. 14 contains 10 years worth of historic spot time

series for FTSE-100, ESTX (EUROSTOCK 50) and S&P. As can be seen in the

figure, typically, when markets fall dramatically, they tend to fall

together, indicating a much more correlated market move. A skewed cor-

relation structure incorporates the fact that the correlation becomes

larger when basket component spots are falling.

The skewed correlation structures can be built into the basket

implied volatility surface during the construction process outlined in

section 4.1. One such approach is to impose a linear correlation structure

depending on where the average basket underlying spot is in a simulated

path. Fig. 15 plots three basket volatility curves versus strike for a basket

of three underlyings. All curves are subtracted by the ATM volatility to

highlight the basket skew. The first curve is generated using a flat corre-

lation coefficient of 44

%

, which is chosen such that the basket ATM

volatility equals to that in the second and third curve illustrated below.

The second curve is constructed with a skewed linear correlation struc-

ture in the local volatility space in small steps. For each small step, the

initial correlation of 40

%

is scaled up linearly to a maximum of 95

%

when the average of the basket underlying spots falls below the prevail-

ing average at the start of that small time step. It is apparent from the fig-

ure that the skewed correlation structure increases the slope of the bas-

ket volatility curve and the basket skew. The third curve is created also

with a skewed linear correlation structure but using one big step, in

which the linear correlation coefficient (between 40

%

and 95

%

) is

dependent on half of the average of the terminal basket spot drop. The

large step approach is effectively an approximation to the small step local

volatility approach. This can be clearly seen in the figure as the third

curve is very close to the second curve.

In practice, different practitioners may prefer to impose different cor-

relation structures depending on different market views and hedging

needs. The copula large step approach is flexible enough to allow various

correlation structures to be incorporated into the basket. The correlation

structure impact on the basket skew and basket pricing and risk expo-

sures could be significant.

TECHNICAL ARTICLE 2

15%

25%

35%

45%

55%

60% 80% 100% 120% 140

%

Strike

Cor = −20%

Cor = 0%

Cor = 50%

Cor = 90%

Figure 11: Basket Implied Volatility (1Y).

−25%

−20%

−15%

−10%

−5%

0%

−20% 0% 20% 40% 60% 80% 100%

Correlation

Basket

Und 1

Und 2

Und 3

Figure 12: Skew (1Y) vs Correlation.

−40%

−30%

−20%

−10%

0%

0123 5 746

Underlying 1

Underlying 2

Basket (Cor = 0%)

Basket (Cor = 80%)

Figure 13: Skew Term Structure.

W

64 Wilmott magazine

TECHNICAL ARTICLE 2

5. Conclusions

A basket volatility surface construction technique is developed using the

copula methodology. It captures the information in the implied volatili-

ty surfaces of the individual basket components. The copula basket

volatility surface construction technique is computational efficient as it

is in large steps at finite time slices. Once the basket volatility surface is

built, the basket can be treated as a new single underlying and the sur-

faces could be interpolated or extrapolated for pricing and risk manag-

ing basket products in a consistent manner. The risk parameters relevant

to individual volatility skew as well as the correlation effects can be gen-

erated within the same framework.

Both volatility and skew are positive functions of the corre-

lation. Skewed correlation structures can be imposed onto the

basket during the construction process. The correlation struc-

ture impacts the basket skew and basket option prices. A real-

istic and consistent basket skew framework built by incorpo-

rating a skewed correlation structure should be able to cap-

ture some risk exposures of dramatic market falls.

It is possible to extend the basket volatility surface con-

struction technique to other multi-asset basket structures. For

example, in the following multi-asset option structures:

Call On Worst Of : max(min( S

1

, S

2

, ···, S

n

) − K, 0)

Put On Worst Of : max(K − min(S

1

, S

2

, ···, S

n

), 0)

Call On Best Of : max(max(S

1

, S

2

, ···,,S

n

) − K, 0)

Put On Best Of : max(K − max(S

1

, S

2

, ···, S

n

), 0)

the worst-of

S

min

= min(S

1

, S

2

, ···, S

n

)

and the best of

S

max

= max(S

1

,

S

2,

···, S

n

)

can be treated as the new single

underlying similar to the basket underlying

S

B

. The new sin-

gle underlying’s terminal spots can be simulated using the copula tech-

nique which links the individual CDFs with an appropriate dependence

(correlation) structure. The simulated spots can then be used to calculate

the option prices at different strikes. The implied volatility surfaces for

the new single underlying can be backed out from option prices, and the

skew risk measurements of these multi-asset option structures can be

conducted similarly.

Finally, although this paper deals with the copula and correlation

skew in the context of equity basket option, the techniques and the

thought process can potentially benefit credit derivatives professionals

too. It will be very interesting to see cross fertilization between equity

and credit derivatives professionals.

0

2500

5000

7500

0 530 1060 1590 2120 2650

FTSE

ESTX

S&P

−2.0%

−1.0%

0.0%

1.0%

2.0%

80% 90% 100% 110% 120

%

Strike

Flat Cor (44%)

Skewed Cor (Local Vol)

Skewed Cor (Large Step)

Figure 14: Historic Spot (FTSE, ESTX, S&P).

Figure 15: Basket Implied Volatility (1Y)

■ D. Gentle, “Basket Weaving”, Risk, Vol. 6, No. 6, p. 51 (1993)

■ M. A. Milevsky and S. E. Posner, “A Closed-Form Approximation for Valuing Basket

Options”, The Journal of Derivatives, Summer 1998, p. 54

■ C. B. Huynh, “Back to Baskets”, Risk, Vol. 7, No. 5, p. 59 (1994)

■ M. Curran, “Valuing Asian and Portfolio Options by Conditioning on the Geometric

Mean Price”, Management Science (1994), No. 12, p. 1705–1711.

■ D. Qu, “Basket Implied Volatility Surface”, Derivatives Week, 4 June 2001.

■ J. Zou and E. Derman, “Strike-Adjusted Spread: A New Metric For Estimating The

Value Of Equity Options”, Goldman Sachs Quantitative Strategies Research Notes,

July 1999.

REFERENCES