francesco chiminello

8/12/2019 Francesco Chiminello

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Reducing Monte-Carlo Noise in

Complex Path-Dependent Trades

Francesco Chiminello

Barclays Capital

This presentation has been prepared by Barclays Capital - the investment banking division of Barclays Bank PLC and its affiliatesworldwide (Barclays Capital). This publication is provided to you for information purposes, any pricing in this report is indicative and isnotintended as an offer or solicitation for the purchase or sale of any financial instrument. The information contained herein has beenobtained from sources believed to be reliable but Barclays Capital does not represent or warrant that it is accurate and complete. Theviews reflected herein are those of Barclays Capital and are subject to change without notice. Barclays Capital and its respective officers,directors, partners and employees, including persons involved in the preparation or issuance of this document, may from time to timeactas manager, co-manager or underwriter of a public offering or otherwise deal in, hold or act as market-makers or advisors, brokers orcommercial and/or investment bankers in relation to the securities or related derivatives which are the subject of this report. NeitherBarclays Capital, nor any officer or employee thereof accepts any liability whatsoever for any direct or consequential loss arising fromanyuse of this publication or its contents. Any securities recommendations made herein may not be suitable for all investors. Pastperformance is no guarantee of future returns. Any modelling or back-testing data contained in this document is not intended to be astatement as to future performance. Investors should seek their own advice as to the suitability of any investments described herein fortheir own financial or tax circumstances. This communication is being made available in the UK and Europe to persons who areinvestment professionals as that term is defined in Article 19 of the Financial Services and Markets Act 2000 (Financial Promotion Order)2001. It is directed at persons who have professional experience in matters relating to investments. The investments to which is relatesareavailable only to such persons and will be entered into only with such persons. Barclays Capital - the investment banking division ofBarclays Bank PLC, authorized and regulated by the Financial Services Authority (FSA) and member of the London Stock Exchange.

Copyright in this report is owned by Barclays Capital ( Barclays Bank PLC, 2005) - no part of this report may be reproduced in anymanner without the prior written permission of Barclays Capital. Barclays Bank PLC is registered in England No. 1026167. Registeredoffice 1 Churchill Place, London E14 5HP.



Risks by MC: re-simulation

Re-simulation approach take a mkt parameter q

bump it by an amount dq

re-run the simulation

risks by finite differences: PX(q+dq) – PX(q-dq)

Pros: simple to implement

unbiased by construction

consistent with PL by construction

Cons: computational requirements

MC noise



Risks by MC: re-simulation

MC noise in risks

amplified by taking small differences

larger bumps than in analytic/PDE prices

reduced by the high correlation betweensimulations

reuse of same random numbers

theta can problematic

affected by local vol surface

compute risks by finite differences

bootstrap/term structure noise



Risks by MC: other approaches

Pathwise derivatives derivation on option-by-option basis

no discontinuous payouts

difficult to compute transition probabilities

Likelihood ratio works well with single-step MC

works very badly for daily steps MC

Benchmark instruments similar to weighted MC

need a complete base of benchmarks

a lot of benchmarks to be chosen and computed

incomplete basis will give nonsensical results



Risks by MC

In the following, it is assumed that computation byre-simulation is the only practically availablemethodology for risks.

How can the adverse effects of MC noise bemitigated, under the constraint of finitecomputational power?

Note that by the sqrt(N) law, even modestimprovements can be significant.



Example TARN

An Energy TARN will be used for the numericalexamples.

Rationale of the choice:

complex path-dependent trade justifies use of MC realistic commodities trade payout is defined in terms of Asian averages underlying is a complex portfolio of options

digital component to the payout not Longstaff-Schwartz



Example trade: TARN

Monthly payments, 1 year duration first payment is evaldate+2months

Underlying portfolio of options is a 3-way:

short 2 puts at 65 $/bbl long 1 call at 90 $/bbl short 1 call at 105 $/bbl observing the prompt contract at all times

Early termination if cumulative sum of call spreads reaches 40$, deal

terminates maximum total positive payout is 40$

adjustment to last coupon in case of earlytermination



Example trade: market, model, simulation

simplified forward curve: all forwards 85 $/bbl

realistic implied volatilities from the market ATM ranging from 42% down to 32%

Skews ranging from -2% to 7%

2-factors model with term structure of volatility

all simulations are run with 10000 antithetic paths

interested in MC noise

3 simulations per test (with different seeds)

bumps: delta, vega 0.2$; vega 1%



Basic results

Seed 1 Seed 2 Seed 3 Average St Dev

MC PV -28.7252 -28.5957 -31.1591 -29.4933 1.4440

MC Error 1.0041 1.0026 1.0383 1.0150

Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

May-10 Aug-10 Nov-10 Feb-11 Jun-11 Sep-11

Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45




Basic results

Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45




Basic results

Vega

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05


Gamma

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2




Control variates

Add and subtract to exotic payout a portfolio ofvanillas whose price is known analytically

The weights are computed so to minimise the MCvariance (linear algebra)

The vanillas are defined for the example TARN aseach of the monthly 3-ways



Basic v. control variates results

Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45





Gamma

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2


Gamma

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2





Vega

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05


Vega

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05




Basic v. synthetic asset results

Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Delta

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45





Gamma

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2


Gamma

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2





Vega

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05


Vega

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




Weighted MC

Choose a set of analytic control trades One size fits all or trades-specific?

Run MC normally, price exotic trade and controls

Alter each path’s probability to match all controls undetermined problem: add a penalty to deviation

form uniform (entropy, quadratic)

Price exotic trade with weighted paths

In this example: controls are fwds, ATM straddles



Basic v. weighted MC results

Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45





Gamma

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2


Gamma

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2





Vega

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05


Vega

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05




Scattered TARN

How to handle the noise from TARN’s cumulativecap?

Obvious approach: soft cap

similar to digital options as call spreads choose a range around cap level fractionally exercise within the range need to guess correlation of consecutive hits:

multiplication (not realistic) minimum (not smooth)

Does not seem to work well



Scattered TARN

How to handle the noise from TARN’s cumulativecap?

Other approach: scattered TARN

choose a range around cap level this example: ±8.92 $

choose a number of evenly-spread levels price a portfolio of TARNs, one for each level average out the result

No need to guess correlation Works quite well



Scattered TARN

Refinements:

Use the above approach to decide if any of thescattered TARNs knocks out, but compute the KO

payout according to the original cap level (smallerbias)

No need to actually price all TARNs literally, justneed to keep a counter of the highest breachedcap level

Use actual trade definition during lifecycling

Can use a range below the original cap level toachieve path-wise conservative pricing



Basic v. scattered results

Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45




Basic v. scattered results

Gamma

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2


Gamma

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03




Combining approaches: control variates and

scattered TARN

Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45





scattered TARN

Gamma

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2


Gamma

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03





scattered TARN

Vega

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05


Vega

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05




Bias comparison and risks recap

Seed 1 Seed 2 Seed 3 Average Stdev

Basic -28.73 -28.60 -31.16 -29.49 1.02

Control variates -29.47 -29.56 -30.57 -29.87 0.43

Synthetic asset -28.08 -28.02 -30.27 -28.79 0.91

Weighted MC -29.77 -29.41 -29.73 -29.64 0.14

Scattered -28.68 -28.49 -31.06 -29.41 1.02

Scattered+CV -29.42 -29.45 -30.48 -29.78 0.42

Delta Avg Delta stdev Gamma Avg Gamma stdev Vega Avg Vega stdev

Basic 0.2445 0.0089 -0.0402 0.0687 -0.1821 0.0358

Control variates 0.2452 0.0084 -0.0416 0.0692 -0.1832 0.0197

Synthetic asset 0.2446 0.0054 -0.0259 0.0571 -0.1815 0.0271

Weighted MC 0.2456 0.0067 -0.0340 0.0845 -0.1822 0.0387

Scattered 0.2446 0.0025 -0.0100 0.0130 -0.1828 0.0353

Scattered+CV 0.2452 0.0036 -0.0106 0.0125 -0.1839 0.0188



Conservative pricing

Why: Bid-ask spread Risk margins Overhedge

as a way to embed risk margins simpler to manage than exact hedge

reduce leverage of pin risk

How: Inequalities on individual paths

Example: scattered TARN with cap levels movedbelow the original trade’s highest scattered TARN = 40$ same spread as unbiased example



Control variates and scattered TARN:

centered v. below

Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Delta

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45





centered v. below

Gamma

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03


Gamma

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08





centered v. below

Vega

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2-0.15

-0.1

-0.05

0

0.05


Vega

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05





centered v. below


Basic -29.47 -29.56 -30.57 -29.87 0.61

Centered -29.42 -29.45 -30.48 -29.78 0.42

Below -29.01 -29.01 -30.08 -29.37 0.44

Delta Avg Delta stdev Gamma Avg Gamma stdev Vega Avg Vega stdev

Basic 0.2452 0.0084 -0.0416 0.0692 -0.1832 0.0197

Centered 0.2452 0.0036 -0.0106 0.0125 -0.1839 0.0188

Below 0.2471 0.0034 -0.0020 0.0208 -0.1808 0.0154



Conservative pricing: approaching autocall

Modest differences at inception

What happens when nearing the TARN cap

market puts TARN in the money?

All past fixings now 100$ Forward curve now 105$




centered v. below: approaching autocall

Delta

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14


Delta

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05






Gamma

-0.05

0

0.05

0.1

0.15

0.2

0.25

Ma y-1 0 Au g-1 0 N ov-1 0 Fe b-1 1 Jun -1 1 Se p-11

Gamma

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1






Vega

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0


Vega

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002







Centered 13.10 13.20 13.10 13.14 0.06

Below 14.60 14.65 14.59 14.61 0.03

Delta Avg Delta stdev Gamma Avg Gamma stdev Vega Avg Vega stdev Theta Avg Theta stdev

Centered 0.0184 0.0013 0.0154 0.0108 -0.0132 0.0015 0.0989 0.0322

Below 0.0048 0.0005 0.0023 0.0066 -0.0029 0.0005 0.0291 0.0132



Acknowledgements

Thanks to Barclays Capital for providing the data and theopportunity.

Thanks to Mircea Marinescu and Yuri Zhestkov for usefuldiscussions.

francesco chiminello

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