calculating provisions for heuristic fx skew...

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Calculating Provisions for Heuristic Calculating Provisions for Heuristic FX Skew ModelsFX Skew Models

Quant Congress Europe 2006Quant Congress Europe 2006London, October 11thLondon, October 11th

Alberto Elices & Eduard GiménezAlberto Elices & Eduard Giménez

2

Outline

Introduction.Objectives.Method description.Results: comparison between heuristic and ATM

Double No Touch.Considerations for provision calculation.Case study 1: Digital option portfolio.Case study 2: Soft barrier double no touch.Conclusions and Future developments.

3

Introduction

There might be different pricing and hedging alternatives for a given product.Traders prefer models that fit exotic market prices even

with some lack of theoretical grounding.Theoretical models may not fit exotic market prices.

Is it possible to have an objective method for comparing models?.How to cope with model skepticism?: provision

calculation.

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Objectives

Develop an objective methodology to compare different models under same conditions.

Calculate provisions consistent with hedging model.

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Method description

The criterion of comparison is the quality of the hedging strategy up to expiry.

P&L distribution at expiry.

Working hypothesis: market behaves as a Heston model with certain parameters.

dtdYdW

dYdtvdv

dWvdtSdS

tt

y

t

ρ

σθκ

µ

,

)(

=><

+−=

+=

6

Method description: details

At each point in time:Heston´s spot and variance are simulated according to certain parameters.Volatility surface is calculated with same Heston parameters.Heston´s two risk factors (underlying and variance) are hedged with underlying and a 6 month vanilla ATM call option.

Pricing models are analytical.This calculation is carried out with the aid of a grid of

computers.

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Method description: market calibration

Heston parameters: calibrated to 1y EUR/USD on August 2006:Kappa (κ) : 1.9006 Theta (θ) : 0.0088Sigma (σ) : 0.1807 Rho (ρ) : 0.1289Var0 (v0 ) : 0.0076

0

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

0.084

0.086

0.088

0.09

0.092

0.094

0.096

0.098

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.086

0.087

0.088

0.089

0.09

0.091

0.092

0.093

0.094

0.095Calibrated implied volatility for maturity 1y

Vol

atili

ty in

per

uni

t

Delta Delta

Maturity in years

8

Method description: EUR/USD, 1y EUR call. S=1.2812, K = 1.2812.

Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb07 Apr070.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8Van1y: Spot price paths; HedgeFreq: 0.25 days

Sp

ot

price

Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb07 Apr070

0.05

0.1

0.15

0.2

0.25

0.3

0.35Van1y: Volatility paths; HedgeFreq: 0.25 days

Vo

latility

9



0.2

0.4

0.6

0.8

1

1.2

1.4Van1y: Hedged delta paths; HedgeFreq: 0.25 days

De

lta

Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb07 Apr07−0.5

0

0.5

1

1.5

2

2.5Van1y: Hedged vega paths; HedgeFreq: 0.25 days

Ve

ga

10


Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb07 Apr07−8

−6

−4

−2

0

2

4

6

8

10

12x 10

−3 Van1y: PnL Paths; HedgeFreq: 0.25 days

Pe

r u

nit o

f n

om

ina

l


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−3 Van1y: StdDev of PnL; HedgeFreq: 0.25 days

Pe

r u

nit o

f n

om

ina

l

11



0.5

1

1.5

2

2.5

3x 10


Pe

r u

nit o

f n

om

ina

l


0.5

1

1.5

2

2.5

3x 10


Pe

r u

nit o

f n

om

ina

l

Comparison between hedging four times and once a day.

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Results: Comparison between heuristic and ATM double no touch

Double no Touch exotic options are usually priced with heuristic models.Once the trade is in the book, they are usually hedged

with an ATM model.

Which model is better for hedging?The heuristic model.The ATM model.

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A 1 year and 2 months double no touch is considered.Hedging is performed once a day.Pricing:

Heuristic model: 0.0839.ATM model: 0.0466.Heston model: 0.1333.

Price

Spot

2130.1 3622.1

2812.1=Spot

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Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−100

−50

0

50SndCDNTch−1y17usd: Hedged delta paths; HedgeFreq: 1.00 days

De

lta


0

50

100

150

200

250SndCDNTch−1y17usd: Hedged vega paths; HedgeFreq: 1.00 days

Ve

ga

Heuristic double no touch:

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There is a consistent bias towards P&L losses.

Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15SndCDNTch−1y17usd: PnL Paths; HedgeFreq: 1.00 days

Pe

r u

nit o

f n

om

ina

l


−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0SndCDNTchATM−1y17usd: Portfolio value; HedgeFreq: 1.00 days

Po

rtfo

lio

price

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Delta hedging error provides consistent revenues ignoring time decay.

( )44 844 764444 84444 76

)()( )()( PnL 11 −− −−−=∆ iOptiOptiHedgeiHedge tPtPtPtP

Price Spot0SDOS UPS

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Vega hedging error provides consistent losses ignoring time decay.

1

PriceVolatility0σDOσ UPσ

There might be a compensation effect between delta and vega hedging error.

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−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0SndCDNTch−1y17usd: ExpVal of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov070

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05SndCDNTch−1y17usd: StdDev of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

P&L distribution at expiry for heristic model.ExpVal: -0.031.StdDev: 0.045.

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−0.08

−0.06

−0.04

−0.02

0

0.02SndCDNTchATM−1y17: ExpVal of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov070

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1SndCDNTchATM−1y17: StdDev of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

P&L distribution at expiry for ATM model.ExpVal: -0.09.StdDev: 0.093.

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The heuristic model is better than the ATM model:Pricing: it provides a closer price to Heston´s model (the fair value under the assumptions taken).Hedging: it provides a lower P&L dispersion.

[ ]LPEPP &modelHeston +=

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Considerations for provision calculation

Two types of provision schemes are considered:Provision based on model price difference (e.g. Heston vs heuristic model).Provision based on the P&L dispersion of the hedging strategy.

Provisions based on model price difference assume implicitly that the trading portfolio may be sold at mid-price at any time.Provisions based on P&L dispersion take into account:

The uncertainty of incorrect hedge (the model hedged is not the model which gives the fair price).Discrete hedging effect.Discountinuities in payoff thoughout the life of the option.

According to our study, provisions based only on price difference may not be sufficient.

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Considerations for provision calculation

Three referencies for the P&L StdDev:Perfect netting: 0 (long and short positions of same product).Independent P&L diversification when netting is not relevant:

Perfect correlacion:

prodNprodNprodprodprodprod nomnomnom σσσ +++ L2211

2222

22

21

21 prodNprodNprodprodprodprod nomnomnom σσσ +++ L

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Case study 1: Digital option portfolio

A portfolio of three 1y digital options is considered:Underlying spot: 1.2097.Strikes: 1.1797 (ITM), 1.2097(ATM), 1.2397(OTM).

The equivalent portfolio of call spread with 0.0050 strike lag is also considered.

0050.0

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Hedging the digital option vs the call spread.K = 1.2397 (OTM).


0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1DigCS50−1y−1: StdDev of PnL; HedgeFreq: 1.00 days

Pe

r u

nit o

f n

om

ina

l


0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Dig1y−1: StdDev of PnL; HedgeFreq: 1.00 days

Pe

r u

nit o

f n

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l

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Diversification analysis.The use of call spreads seems unnecessary at portfolio level.Portfolio provision is less than considering independent P&L variables.

Option ITM ATM OTM Portfolio Independent P&LK 1.1797 1.2097 1.2397PriceDig 0.5942 0.4701 0.3514 0.4719PriceCS 0.6044 0.4786 0.3604 0.4811StdDevDig 0.0740 0.1040 0.0850 0.0420 0.0511StdDevCS 0.0665 0.0850 0.0660 0.0410 0.0422

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For a given interval of strikes K1, K2 (e.g. 1.1797 to 1.2397), the maximum P&L StdDev reduction would correspond to an evenly distributed (strike and notional) set of digital options between both strikes.This set converges in the limit to a call spread K1, K2. Thus the minimum P&L StdDev of a porfolio of digital

options is the P&L StdDev of the associated call spread.

The maximum diversification given by the call spread is almost achieved by the three digital portfolio.

3DigPortFol CallSpreadStdDev 0.0420 0.0400

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Case study 2: Soft barrier double no touch

It is considered the case in which a double no touch with wider barriers is used for hedging: a soft barrier.

The barriers to compute the price are wider.The option expires when the spot hits the hard barrier (the original one).For calculation of StdDev of P&L, it is assumed that the remaining premium after hitting the hard barrier is returned to the client for accounting purposes only.

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SoftHHardH

HardLSoftL

Start Maturity

[ ] DNThardHitAtHardBarr PPEP +=DNTsoft

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How will provision behave under this new strategy?

Hard Soft1 Soft2High 1.3622 1.3672 1.3822Low 1.2130 1.2080 1.1930P Heuristic 0.0839 0.1118 0.2084P Heston 0.1361 0.1653 0.2641StdDev 1.00d 0.0880 0.0700 0.0250StdDev 0.25d 0.0550 0.0375 0.0270OneStdDevProvision 0.25d 0.1911 0.2028 0.2911

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Conclusions and Future developments

A transparent and objective method for comparing pricing and hedging models is presented.This method gives two meassures. The lower they are

the better the model behaves.Expected value of P&L.Dispersion of P&L.

This meassures allow for provision calculation.Provisions at portfolio level have been studied:

The P&L dispersion of a portfolio of digital options might be lower than assuming independent P&L for each component.A lower bound for the P&L dispersion of a portfolio of digital options has been calculated.

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Conclusions and Future developments

Superhedging:Digital option superhedging through call spread might be unnecessary.Heging DNT with soft barriers does not seem to reduce provisions.

Further developmets should allow the provision calculation for Monte Carlo methods

calculating provisions for heuristic fx skew...

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