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PRICING BASKET OPTIONS WITH SKEW

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TRANSCRIPT OF PRICING BASKET OPTIONS WITH SKEW

Pricing Basket Options With SkewDong Qu Quantitative Products GroupDerivatives &
Structured ProductsAbbey 1. Introductionvolatility surface and analyse a few
examples. We will develop further the copula implied basket volatility
construction technique (ref. 5) and examine 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.

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 single 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 certain 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 basket can be priced using various techniques (ref. 14), 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 technique 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

2. Basket RisksA typical basket consists of several underlyings with associated
weightings. The basket spot is:n

SB =i=1

w i Si

Where wi is the weightings and Si the individual underlyings. The values of
European call and put options on the basket are calculated from:EQ Max(SB K, 0)
; EQ Max(K SB , 0) ; Call 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.Wilmott magazine

58

TECHNICAL ARTICLE 2

3. Historic Basket Volatility SurfacesIt is possible to construct a historic
basket volatility surface from the historic time series of the individual basket
components. One such

1.5%

1.0%

Basket Specific Risks Cross Gamma Risks

Comments The cross Gamma is defined as2P , i = j, Si Sj where P is the basket
option price, Si and Sj0.5%

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. 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. In
a Quanto or Compo basket, the FX correlation with each individual basket
component will impact the price. If merge and acquisition activity happens to
the basket member, the basket composition will change and so will its volatility
structure. 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.

0.0% 70%

90%

110%

130%

150%

Figure 1: Historic PDF (Mean Drift = 9%). where N defines the time interval. The
bucketed log-returns RT 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 ) = EQ (ln Q ln P)

Basket Volatility Skew Risks

FX Correlation Risks

By minimising relative entropy S(P, Q ), one can solve for Q , subject to the
following constraints,Q (ST ) ST dST = Forward Q (ST ) dST = 1

Contractual Risks

Jump Risks

Additional optimisation constraint can also be applied, such as imposing the ATM
volatility. The optimisation solution for the problem posed above is:Q (ST ) =
P(ST ) exp(ST ) P(S) exp(S) dS

approach is to minimise the relative entropy (ref. 6) in which a risk neutral
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 Si the following steps are
taken to build a historic basket volatility surface: Step 1: Recover the
historic PDF for the basket at time interval T by calculating the log-return at
appropriate time interval:RT = ln St+N St

Where is a constant that can be obtained by iterating constraints during the
numerical optimisation. Note that given P(ST ) is always positive, Q (ST ) 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 Ki
:Call (Ki , T) = DF Put (Ki , T) = DF 0

(ST Ki ) Q (ST ) dST

K K

^

(Ki ST ) Q (ST ) dST

Wilmott magazine

59

1.5% 50% 1.0% 40% 0.5% 30% 20% 0.0% 70% 90% 110% 130% 150% 10% 0% 50% 80% 2.0
110% 140% 170% 0.08 200% 7.0

Figure 2: Risk Neutral PDF (Mean Drift = 6.75%). Where DF is the discount
factor. The option prices can then be used to recover the implied volatility
points at Ki . 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 volatility 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 constitutes a relatively small proportion of the
time series. For day-to-day trading 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.

Figure 4: FT-Banks Historic Vol Surface.

25% 20% 15% 10% 5%

3.0 0.75 70% 90% 0.08 110% 130%

50% 40% 30% 20% 10% 0% 50% 80% 2.0 110% 140% 170% 0.08 200% 7.0

0%

Figure 5: Index Basket Historic Vol Surface.

4. Implied Basket Volatility SurfacesIt is conceivable that one can construct an
implied basket volatility surface from the implied volatility surfaces of
individual basket components. 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
efficientWilmott magazine

Figure 3: FTSE Historic Vol Surface.

60

TECHNICAL ARTICLE 2

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.

1.0

4.1.

The Copula Technique to Construct Basket Volatility Surfaces

0.5

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 distribution. The copula technique allows us
to specify a joint distribution function for known marginal distributions with a
dependence (correlation) structure. Specifically, for given univariat

ZERO-SKEW TREES

LEFTIST SKEW HEAPS

SKEW DESIGN

THE CBOE SKEW INDEX - SKEW

ACCURATE PRICING OF BASKET OPTIONS A PRAGMATIC APPROACH .A COMMON FINANCIAL
PRODUCT THAT ALLOWS TO

MULTI-FACTOR APPROACH FOR PRICING BASKET INTERNATIONAL JOURNAL OF BUSINESS AND
FINANCE RESEARCH ♦ VOLUME 5 ♦ NUMBER 4 ♦ 2011 115 MULTI-FACTOR APPROACH FOR
PRICING BASKET CREDIT

PRICING BASKET OPTIONS WITH AN EYE ON SWAPTIONS ASPREMON/PDF/ آ  PRICING BASKET
OPTIONS WITH AN EYE

.BASKET CODE BASKET #1 BASKET #2 BASKET #3 BASKET #4 BASKET #5 BASKET #6 BASKET
#7 BASKET #8 BASKET

A NON-GAUSSIAN OPTION PRICING MODEL WITH SKEW - ARXIV 2008. 2. 2.آ  LISA BORLAND
ˆ— AND JEAN-PHILIPPE

N CLOCK SKEW

PRICING OF PATH-DEPENDENT BASKET OPTIONS USING A COPULA ... PRICING OF
PATH-DEPENDENT BASKET OPTIONS

RQUANTLIB: INTERFACING QUANTLIB FROM R PRICING ENGINES (ASIAN, BARRIER, BASKET,
CAP/FLOOR, CLIQUET,

HOT OR NOT: FINGERPRINTING HOSTS THROUGH CLOCK SKEW EFFECT ON CLOCK SKEW €¢
SKEW OF TYPICAL CLOCK CRYSTAL

آ« HYPOTHESIS OF CURRENCY BASKET PRICING OF CRUDE OIL: AN IRANIAN PERSPECTIVE
2011-06-14آ  PERSPECTIVE

CSE 248 SKEW 1KAHNG, UCSD 2011 CSE248 SPRING 2011 SKEW.

PRICING FXOPTION WITH SKEW PETER CARR

SPREAD AND BASKET OPTION PRICING: AN APPLICATION TO ... ... 2. SPREAD OPTION
PRICING: A LITERATURE REVIEW

CERTIFICATE IN FINANCIAL ENGINEERING CFE EXAM (CURRICULUM).PDFآ  NUMERICAL
SCHEMES FOR PRICING BASKET

BREAKEVEN SKEW III

BRAIDED SKEW MONOIDAL SKEW MONOIDAL CATEGORIES ARE ALSO CALLED LEFT SKEW.) IF
CIS SKEW MONOIDAL THEN



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