Risk

Some Practical Issues in FX and EquityDerivatives

Phenomenology of the Volatility SurfaceThe volatility matrix is the map of the implied volatilities quoted by themarket for options of different strikes and different maturities. Impliedvolatility is the parameter σ needed to calculate the B&S formula. Inpractice the matrix can be built according to two main rules:

• Sticky Delta: the matrix of implied volatilities is mapped, for eachexpiry, with respect to the ∆ of the option; this rule is usually adoptedin the over the counter markets (e.g.: fx options). Options are priceddepending on their ∆. That has subtle implications for the running of abook of options.

• Sticky Strike: the matrix of implied volatilities is mapped, for eachexpiry, with respect to the strike prices; this the rule is usually adoptedin official markets (e.g.: equity options and futures options). Impliedvolatilities remain constant for each strike, even if the underlying assetprice changes.

Phenomenology of the Volatility SurfaceSticky Delta Rule: For each strike K and expiry T the implied volatilityof any option is dependent on the level of its ∆ and consequently on themovements of the underlying asset (and on the elapsing of time).

Phenomenology of the Volatility Surface

Sticky Strike Rule:For each strike K and expiry T the implied volatility ofany option is independent of the movements of the underlying asset.

Sticky Strike Arbitrage

• Consider the P&L of a perfectly hedged portfolio in a short period dt:

P&L = dΠ + f = dC − ∆dS + f

where f is the financing cost of the position.

• Let’s assume the underlying asset’s evolution is commanded by aBrownian motion:

dS = µSdt + σtSdZ

and apply Ito’s lemma (note that σt is the actual and not the impliedvolatility)

dC = (Θ +1

2σ2

t S2Γ + µ∆)dt + ∆σtSdZ


• Since f is:

f = (−rd(t)C + rd(t)∆S − rf(t)∆S)dt

= (−rd(t)∆S + Dfdrd(t)KΦ(d2) + rd(t)∆S − rf(t)∆S)dt

we get

Θdt + f = −1

2σ2

KS2Γdt

and hence (recalling Θ definition):

P&L =1

2S2Γ[σ2

t − σ2

K]dt


• Now, let’s assume we have two call option C1(K1, σ1) and C2(K2, σ2)with σ1 > σ2. We build a portfolio such as: long Γ1 options C2 andshort Γ2 options C1

• The P&L in the interval dt is.

P&L =Γ1Γ2

2S2[σ1 − σ2]dt

which is a positive amount.

• Thus Sticky Strike rule imply an arbitrage if it is realized in the market

Sticky Delta Arbitrage

• The Sticky Delta rule imply arbitrage as well, if it is realized in themarket, though in less evident way.

• One should consider the shape of the surface in terms of slope andconvexity and then build complex portfolios

• Generally speaking, with relatively symmetric matrices (such as in theEur/Usd FX market), a position short butterfly grants a positive Γ andpositive Θ portfolio.

• A short butterfly = long an ATM straddle and short a ∆ symmetricstrangle (e.g.: short a 25∆ Call and a 25∆ Put), in an amount such thatthe total Vega of the portfolio is nil

Conclusions on the Sticky Strike and Sticky Delta Rules

• Matrices with different implied volatility for different levels of strike showthat the constant volatility assumption of the B&S model is not realistic.

• The two Sticky Strike and Sticky Delta rules imply arbitrage should theyactually be realized: so they both cannot be considered as a feasiblemodel of the evolution of the volatility surface.

• They can be considered just as two convention for quoting volatilitysurfaces and they are respectively chosen according to their suitability todifferent markets.


The evolution of the volatility surface can be decomposed in three mainmovements, for each expiry:

• Parallel Shift

• Convexity Increase/Decrease

• Slope Increase/Decrease.

To represent these movements in terms of market instruments, one canconsider:

• The ATM straddle volatility as an indicator of the level

• The Vega Weighted Butterfly as an indicator of the convexity

• The Risk Reversal as an indicator of the slope.


Parallel


Convexity


Slope


Composite

Model Risk in Hedging Derivatives Exposures

• Every time we choose a model to price and to hedge a derivative, wemake more or less realistic assumptions about the risk factors we wantto take into consideration.

• In practice, markets never behave in the way predicted by the model, sothat the risk one incurs in by using a misspecified model is very high.

• In order to minimize the model risk, one can analyze which is the hedgingerror arising from the non-realistic modeling of the factors included inthe model; subsequently, one tries to minimize this error.

• In what follows we will analyze which is the hedging error produced bythe false assumption of a constant volatility (as in the B&S model).

• We also analyze which is the error produced by the more realisticassumption of a changing implied volatility, in case we are not able tocorrectly describe its evolution.

Model Risk: Hedging by B&S (constant volatility)

• We have shown above that the P&L in short interval dt of a perfectlyhedged (under B&S’s assumptions) portfolio is:

P&L =1

2S2Γ[σ2

t − σ2

K]dt

• We make a profit if the realized volatility σt higher than the impliedvolatility σK, and the magnitude of this profit is directly linked to thelevel of the Γ (which is always positive in the case of a plain vanilla calloption).

• If we integrate over the entire option’s life, we obtain the total P&Lresulting from running a ∆-hedged book at constant implied volatility:

P&L =

∫ T

0

1

2S2

t Γ(St, σK, t)[σ2

t − σ2

K]dt

Model Risk: Hedging by B&S (constant volatility)From the formula above we can infer that:

• Continuous ∆-hedging of a single option revalued at a constant volatilitygenerates a P&L directly proportional to the Γ of the option;

• In general the P&L of a long position in the option, continuously re-hedged, is positive if the realized volatility is, on average during theoption’s life, higher than the constant implied volatility; it is negative inthe opposite case;

• The previous statement is not always true since the total P&L isdependent on the path followed by the underlying: if periods of lowrealized volatility are experienced when the Γ is high, whereas periods ofhigh realized volatility are experienced when the Γ is negligible, then thetotal P&L is negative, though the realized volatility can be higher thanimplied volatility for periods longer then those when it is lower.

Model Risk: Hedging by a Floating Implied Volatility

• In practice, every day the trader’s book is marked to the market, so tohave a revaluation as near as possible to the true current value of theassets and other derivatives.

• That means that the book is revaluated at current market conditionsregarding the price of the underlying asset and the implied volatility (wedrop for the moment the fact that also the interest rates are updated tothe current level).

• We would like to explore now the impact on the hedging performancewhen the implied volatility is floating and continuously updated to themarket levels.


• Under the real probability measure P, the underlying asset’s price evolvesaccording to the following SDE:

dS = µSdt + σtdZ1

• The implied volatility σK is now considered as a new stochastic factorand model its evolution, under the real probability measure P, as:

dσK = αdt + νtdZ2

where dZ1 and dZ2 are two correlated Brownian motion.• Under the equivalent martingale measure Q:

dS = (rd− rf)Sdt + σtdW1

dσK = αdt + νtdW2

where dW1 and dW2 are again two correlated Brownian motion withcorrelation parameter ρ.


• Under the real probability measure P, the option’s price evolves as:

dC =

(

∂C

∂t+

1

2

∂2C

∂S2σ2

t S2 +

∂C

∂SµS +

∂C

∂σK

α

+1

2

∂2C

∂σ2

K

ν2

t +∂2C

∂σK∂SρσtSνt

)

dt +∂C

∂SσtSdZ1 +

∂C

∂σK

νtdZ2

• or, writing the partial derivatives as Greeks:

dC =(

Θ +1

2Γσ2

t S2 + ∆µS + Vα +

1

2Wν2

t + XρσtSνt

)

dt + ∆σtSdZ1 + VνtdZ2

• where V is the Vega, X is the vanna and the W is the volga.


• Under equivalent martingale measure Q the dynamics of the call optionis:

dC = Θ +1

2Γσ2

t S2 + ∆(rd

− rf)S + Vα +1

2Wν2

t + XρσtSνt = rdC

• Let’s build a portfolio of: long one call option and short quantity ∆ ofthe underlying; it’s P&L over a small period dt is:

dΠ = dC − ∆dS + f

where f is the cost born to finance the position:

f = (−rd(t)C + rd(t)∆S − rf(t)∆S)dt


• After a few substitutions we get

dΠ =(

Θ+1

2Γσ2

t S2+∆(rd

−rf)S+Vα+1

2Wν2

t +XρσtSνt

)

dt−rdCdt+VνtdZ2

• Adding and subtracting Vα and by means of previous equations we have

dΠ = VνtdZ2 + (Vα − Vα)dt = V(dσk − αdt)

• By integrating over the option’s life:

P&L =

∫ T

0

dΠ =

∫ T

0

V(dσk − αdt)

Model Risk: Hedging by a Floating Implied VolatilityFrom the formula above we can infer that:

• Continuous ∆-hedging of a single option revalued at a running impliedvolatility generates a P&L proportional to the Vega of the option.

• In general the P&L of a long position in the option, continuously re-hedged, is positive if the realized variation in implied volatility is, onaverage during the option’s life, higher than its expected (risk-neutral)variation, it is negative in the opposite case.

• The previous statement is not always true since the total P&L isdependent on the path followed by the underlying: if periods of lowrealized variations of the actual implied volatility are experienced whenthe Vega is high, whereas periods of high realized variations of the actualimplied volatility are experienced when the Vega is negligible, then thetotal P&L is negative, though the realized variations could be on averagegrater then expected implied volatility’s variation.

Hedging Volatility Risk in a B&S World

• In practice, the a trader’s book is frequently updated in terms of theunderlying asset price and implied volatility. If the book is re-valued andhedged as in a B&S world, then we know from the previous analysis thatwe have to minimize the model risk by minimizing the Vega exposure.

• Then the book will be ∆-hedged against the movements of the underlyingasset; it will be Vega-hedged against the change in the implied volatility.

• Vega-hedging must be considered in a very extended meaning: theportfolio must remain Vega-hedged even after movements in the impliedvolatility and/or the underlying asset.


So, hedging a book in a B&S world implies setting to zero the followingGreeks:

• ∆

• V

• X (Vanna or DvegaDspot)

• W (Volga or DvegaDvol)

The ∆ exposure is (usually) easily set to zero by trading in the underlyingasset’s cash market. The volatility-related Greeks are set to zero bytrading (combinations of) other options.


Tools to cancel Vega exposures are:

• ATM straddle: this structure has a strong Vega exposure, low Volgaexposure and nil Vanna.

• Risk Reversal 25∆: no Vega and Volga exposures, strong Vanna exposure.

• Vega Weighted Butterfly 25∆: no Vega and Vanna exposures, strongVolga exposures.

By combining the three structures above, traders make their bookVega-hedged, and the keep this hedging stable to implied volatility adunderlying asset movements.


Exotic option (e.g.: barriers and One Touch) often show more sensitivityto the Volga and to the Vanna than to the Vega.

As an example we consider an Up&Out Eur Call Usd Put option:

• Spot Ref.: 1.2183

• Expiry: 6M

• Strike: 1.2250

• Barrier Up&Out: 1.3100


Value

-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

1.05

1.07

1.10

1.12

1.14

1.16

1.19

1.21

1.23

1.26

1.28

1.30

1.32

1.35

1.37

1.39

1.41

spot 1.0

4

1.0

7

1.1

0

1.1

3

1.1

6

1.2

0

1.2

3

1.2

6

1.2

9

1.3

2

1.3

5

1.3

8

1.4

1

183

139

96

52

8

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

0.0600

time to(


Vega

-0.1000

-0.0800

-0.0600

-0.0400

-0.0200

0.0000

0.0200

0.0400

1.05

1.07

1.10

1.12

1.14

1.16

1.19

1.21

1.23

1.26

1.28

1.30

1.32

1.35

1.37

1.39

1.41

spot 1.0

4

1.0

7

1.1

0

1.1

3

1.1

6

1.2

0

1.2

3

1.2

6

1.2

9

1.3

2

1.3

5

1.3

8

1.4

1

183

139

96

52

8

-0.4000

-0.3500

-0.3000

-0.2500

-0.2000

-0.1500

-0.1000

-0.0500

0.0000

0.0500

0.1000

time to


Volga

-1.5000

-1.0000

-0.5000

0.0000

0.5000

1.0000

1.5000

2.0000

2.5000

3.0000

1.05

1.07

1.10

1.12

1.14

1.16

1.19

1.21

1.23

1.26

1.28

1.30

1.32

1.35

1.37

1.39

1.41

spot 1.0

4

1.0

7

1.1

0

1.1

3

1.1

6

1.2

0

1.2

3

1.2

6

1.2

9

1.3

2

1.3

5

1.3

8

1.4

1

183

139

96

52

8

-4.0000

-3.0000

-2.0000

-1.0000

0.0000

1.0000

2.0000

3.0000

4.0000

5.0000

6.0000

time to


Vanna

-1.5000

-1.0000

-0.5000

0.0000

0.5000

1.0000

1.5000

2.0000

2.5000

1.05

1.07

1.10

1.12

1.14

1.16

1.19

1.21

1.23

1.26

1.28

1.30

1.32

1.35

1.37

1.39

1.41

spot 1.0

4

1.0

7

1.1

0

1.1

3

1.1

6

1.2

0

1.2

3

1.2

6

1.2

9

1.3

2

1.3

5

1.3

8

1.4

1

183

139

96

52

8

-20.0000

-10.0000

0.0000

10.0000

20.0000

30.0000

40.0000

time t


Another example

We consider a Down&Out Eur Put Usd Call option:

• Spot Ref.: 1.2183

• Expiry: 3M

• Strike: 1.2000

• Barrier Down&Out: 1.0700


Value

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

0.0350

0.0400

1.10

1.11

1.13

1.15

1.16

1.18

1.19

1.21

1.23

1.24

1.26

1.27

1.29

1.30

1.32

1.34

1.35

spot 1.0

9

1.1

1

1.1

4

1.1

6

1.1

8

1.2

0

1.2

2

1.2

4

1.2

6

1.2

8

1.3

0

1.3

3

1.3

5

91

69

48

26

5

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

time to(d


Vega

-0.6000

-0.5000

-0.4000

-0.3000

-0.2000

-0.1000

0.0000

0.1000

0.2000

1.10

1.11

1.13

1.15

1.16

1.18

1.19

1.21

1.23

1.24

1.26

1.27

1.29

1.30

1.32

1.34

1.35

spot 1.0

9

1.1

1

1.1

4

1.1

6

1.1

8

1.2

0

1.2

2

1.2

4

1.2

6

1.2

8

1.3

0

1.3

3

1.3

5

91

69

48

26

5

-0.6000

-0.5000

-0.4000

-0.3000

-0.2000

-0.1000

0.0000

0.1000

0.2000

time to(


Volga

-6.0000

-4.0000

-2.0000

0.0000

2.0000

4.0000

6.0000

8.0000

10.0000

1.10

1.11

1.13

1.15

1.16

1.18

1.19

1.21

1.23

1.24

1.26

1.27

1.29

1.30

1.32

1.34

1.35

spot 1.0

9

1.1

1

1.1

4

1.1

6

1.1

8

1.2

0

1.2

2

1.2

4

1.2

6

1.2

8

1.3

0

1.3

3

1.3

5

91

69

48

26

5

-6.0000

-4.0000

-2.0000

0.0000

2.0000

4.0000

6.0000

8.0000

10.0000

time to(


Vanna

-12.0000

-10.0000

-8.0000

-6.0000

-4.0000

-2.0000

0.0000

2.0000

4.0000

6.0000

8.0000

10.0000

1.10

1.11

1.13

1.15

1.16

1.18

1.19

1.21

1.23

1.24

1.26

1.27

1.29

1.30

1.32

1.34

1.35

spot 1.0

9

1.1

1

1.1

4

1.1

6

1.1

8

1.2

0

1.2

2

1.2

4

1.2

6

1.2

8

1.3

0

1.3

3

1.3

5

91

69

48

26

5

-15.0000

-10.0000

-5.0000

0.0000

5.0000

10.0000

15.0000

20.0000

25.0000

30.0000

35.0000

time to


• Given the Vega, Vanna and Volga of an option, we calculate theequivalent position in terms of three basic options (ATM, 25∆ Call and25∆ Put): these quantities can be easily converted in amounts of thehedging instruments we have shown above.

• The U&O Eur Call Usd Put and the D&O Eur Put Usd Call have avolatility exposure as presented in the following table:

25∆ put 25∆ call ATM putUp&Out call 79,008,643 54,195,790 -127,556,533

Down&Out put -400,852,806 -197,348,566 496,163,095

Table 1: Quantities of plain vanilla options to hedge the barrier optionsaccording to B&S model.

Hedging Volatility Risk in a Stoch Vol World

• Managing the volatility risk on the B&S’s assumption is inconsistent andincomplete.

• All the volatility related Greeks are zeroed, but the model assumes thatthe impled volatility is constant, so they should not be hedged.

• The book is revalued with one implied volatility (typically the ATM),whereas on the market a whole volatility surface is quoted and it changesover time (the three movements for any expiry have been analyzedbefore).

• The pricing of exotic options is not consistent with a volatility surface.


Need for a model to capture smile effects

• Non-lognormal models (e.g.: CEV)

• Local-volatility models (e.g.: Dupire)

• Stochastc Volatility models (e.g.: Heston, SABR)

• Lognormal Mixture models (e.g.: Brigo & Mercurio; Brigo Mercurio &Rapisarda)

Hedging Volatility Risk in a Stoch Vol WorldExample: we try to hedge the volatility risk by the Brigo, Mercurio &Rapisarda (2004) model.

• The hedging procedure is based on the concept of sensitivity bucketingand reflects what a trader is willing to do in practice. This is possiblethanks to the model capability of exactly reproducing the fundamentalvolatility quotes (at least for the three basic instruments).

• One shifts such a volatility by a fixed amount ∆σ, say ten basis points.One then fit the model to the tilted surface and calculate the price ofthe exotic, πNEW , corresponding to the newly calibrated parameters.

• Denoting by πINI the initial price of the exotic, its sensitivity to thegiven implied volatility is thus calculated as:

πNEW − πINI

∆σ


• In practice, it can be more meaningful to hedge the typical movementsof the market implied volatility curves. To this end, we start from thethree basic data for each maturity (the ATM and the two 25∆ call andput volatilities), and calculate the exotic’s sensitivities to:

• i) a parallel shift of the three volatilities;

• ii) a change in the difference between the two 25∆ wings;

• iii) an increase of the two wings with fixed ATM volatility.

• This is actually equivalent to calculating the sensitivities with respect tothe basic market quotes. In this way we capture the effect of a parallel,a twist and a convexity movement of the volatility surface.

• Once these sensitivities are calculated, it is straightforward to hedge therelated exposure via plain vanilla options, namely the ATM calls, 25∆calls and 25∆ puts for each expiry.


• We use the following volatility surface and interest rate data

σATM σRR σV WB P d(0, T ) P f(0, T )1W 13.50% 0.00% 0.19% 0.9997974 0.99960362W 11.80% 0.00% 0.19% 0.9995851 0.99922021M 11.95% 0.05% 0.19% 0.9991322 0.99838832M 11.55% 0.15% 0.21% 0.9981532 0.99666653M 11.50% 0.15% 0.21% 0.9972208 0.99510186M 11.30% 0.20% 0.23% 0.9941807 0.99025989M 11.23% 0.23% 0.23% 0.9906808 0.98552111Y 11.20% 0.25% 0.24% 0.9866905 0.98078082Y 11.10% 0.20% 0.25% 0.9626877 0.9550092

Table 2: Market data for EUR/USD as of 31st March 2004.


• The hedging quantities calculated according to UVUR model with thescenario approach are shown below.

• The expiry of the hedging plain vanilla options is once again the same asthe corresponding barrier options.

• It is noteworthy that both the sign and order of magnitude of the hedgingoptions is the similar to those of the BS model we calculated before.

25∆ put 25∆ call ATM putUp&Out call 76,409,972 42,089,000 -117,796,515

Down&Out put -338,476,135 -137,078,427 413,195,436

Table 3: Quantities of plain vanilla options to hedge the barrier optionsaccording to UVUR model with the scenario approach.

Analogies between B&S and Stoch Vol Hedging

The sign and magnitude of the hedging quantities show that some analogiesexist between the exposures of an option priced by a B&S model and amodel which consider the smile effects.

• The B&S Vega can be though of as the equivalent of the sensitivity ofthe option price to a parallel shift of the volatility surface.

• The B&S Volga is the equivalent of the sensitivity to a change in theconvexity of the volatility surface, i.e.: an upward or downward movementof the wings with respect to the ATM level.

• The B&S Vanna is the equivalent of the sensitivity to a change in theslope of the volatility surface, i.e.: a twist of the wings with respect tothe ATM level, considered as a pivot point.

More Risks

Other risks, related to plain vanilla and exotic options, have to be managed

• P (Rho) and Φ exposure, i.e.: the sensitivity of the option price to thedomestic interest rate and the foreign interest rate, or dividend yield incase of equity option.

• Risks related to some exotics, e.g.: ∆ gap at the breach of the barrier.

• Correlation risk: many exotic options (especially in the equity market)have as underlying basket of stocks, or the pay-off is contingent on thefuture evolution of a given number of stocks. In these cases, correlationbetween the single assets become a main risk.

Correlation Risk

As an example of correlation risk, we discuss three different option typeswith the following payout structures

• An at-the-money (ATM) call option on an equally weighted basket of n

stocks.

• An option on the maximum performance of n assets

• An option on the minimum performance of n assets.

Payouts are defined relative to Si, i = 1, .., n i.e.: the asset price at theexpiry T = 0.

Correlation Risk

The risk management of multi-asset options implies the canceling of thefirst and second order spot and volatility sensitivities, though in this casewe have to deal with matrices of sensitivities :

• The ∆ vector ∂C∂Si

• The Γ matrix ∂C∂Si∂Sj

• The Vega vector ∂C∂σi

• The Volga matrix ∂C∂σi∂σj

• The Vanna matrix ∂C∂σi∂Sj

• First-order correlation risk (correlation Vega) can be calculated as atriangular matrix ∂C

∂ρijwith i < j.

Correlation Risk

Single stocks and plain vanilla options on single stocks hedge only the ∆the Vega and the diagonal elements of the Γ, Volga and Vanna matrix.The remaining risks, i.e. the nondiagonal elements (cross Γ, cross Volgaand cross Vanna) and the correlation Vega, can be hedged only by othermulti-asset options.

It can be shown that in a B&S world the following relationship holds:

∂C

∂ρij

= SiSjσiσjT∂C

∂Si∂Sj

So that by hedging all the cross Γ exposure one hedges also the correlationVega exposures.

Correlation Risk

The correlation risk affects the price of an options in two ways, dependingalso on the kind of pay-off of the structure:

• It impacts on the volatility of the entire basket of underlying stocks.

• It impacts on the dispersion of the single stocks within the basket.

We make some intuitive considerations on these two effects with respect tothe three kind of exotic options we listed above.

Correlation Risk

Basket options:

• The value of the option is affected only by the basket volatility.

• The dispersion of individual assets does not influence the option price,because the payout only depends on the sum of the asset prices atmaturity.

• Higher correlations increase basket volatility and thus the option price.

• Hence basket options are long in correlation.

Correlation RiskOptions on the Maximum:

• Increasing correlations imply higher volatility of the basket

• Increasing dispersion of the single stocks increases the probability of anystock to reach a very high value at maturity. This effect grows withdeclining correlations.

• So for max options, the two effects operate in opposite directions.

• From moderate to high correlation, option prices decrease with increasingcorrelation: hence, the dispersion effect is stronger than the basketvolatility effect and the max option is short in correlation.

• It should be stressed that this is the initial exposure when the Si(0) arefixed. Situations can occur where the max option is both short and longin correlation depending on the specific levels of correlation and spotprices.

Correlation Risk

Options on the Minimum:

• The min option is affected by both effects, but both take the samedirection in this case.

• The dispersion effect increases option prices as correlations becomehigher, since this minimizes the probability that any asset reaches a verylow level at maturity and maximizes the value of the min option

• Higher correlation implies also a higher volatility of the basket and thisincrease the option value.

• Since both effects operate in the same direction, the correlation sensitivityis positive and especially high for this option type.

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Risk