2012 Tokyo Ebig Yau

7/27/2019 2012 Tokyo Ebig Yau

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Equity-Based Insurance Guarantees Conference

June 18, 2012

Tokyo, Japan

Market Risk Modeling

Eric Yau


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Eric YauConsultant, Barrie & Hibbert Asia

[email protected]

Eric YauConsultant, Barrie & Hibbert Asia

[email protected]

18 June 2012 (1150 1230 hours)18 June 2012 (1150 1230 hours)


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Agenda

+ VA market risk modeling: motivation and building blocks

+ Calibrating to the Japan market

+ Risk factor modeling

Interest rate

Equity

Credit

+ Hedging and hedge projection

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mot i vat i on and bui l d i ng blocks


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Motivation of VA modeling

VA modeling aims to answer a few fundamental questions:

ProductDesign /Pricing

What is the cost of guarantees

embedded in VA?

How do product features impact this

cost?

HedgingValuation

management strategies

affect cashflow profile?

How should we

implement a hedging

What is the right level

of reserve?

How hedging

strategies affect

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strategy and test its

effectiveness?

reserves?


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Building blocks of VA modeling

Economic

Assumption

Model

Assumption

Model

Choice

Market

Data

calibration

Risk Monitoring Reports:

* Asset and liability valuation*

Economic scenarios:

Economic Scenario Generator

* Risk limits and utilization

Base and Sensitivities

Liability

Portfolio ALM SystemAsset

Portfolio

Trading System

Asset Portfolio Optimizer

Generate, for both asset and

liability portfolios:

* Valuation

* Hedge Strategy

* RebalancingRules

* Risk Limits

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Trading

Engine

* Greeks

* Mismatch position

va a e

InstrumentsCalculation

Engine


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VA risk management and modeling

Product design and pricing

- Cost of guarantees

Valuation

- Reserve/capital calculation- ens t v ty ana ys s - eserve pro ect on

Ensure fair and adequate pricing / valuation / capital

Internal hedging

-

Hedge projection

-- Automated calibration

- Variance reduction

Capture key risk exposure of hedging strategy

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Desirable ESG features for VA

+ Integrated modeling

Interest rates, multiple equity indices, credit risks and alternative assets ons stent mar et-cons stent an rea -wor mo e ng

+ Multiple equity modeling choices E.g., Local volatility, Heston with jumps, to support analysis of model risk

Exact fit to starting yield curve (for interest rate models)

Accurate fit to option-implied volatility

And for hedging:

+ Ability to be run on efficient hardware configurations -

+ Fast calibration tools to facilitate re-calibration / sensitivities

+ Automation of scenario production7


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The market

Bond / Interest rate market+ Longest available swap / JGB up to 40 years

+ Swap liquid tenors up to ~10 years

Derivative implied volatilities+ E.g. Nikkei 225 options up to ~10 years

What happens if we need to discount long term liabilities?

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Linkage to liability valuation

+ This generally apply to a number of areas

Interest rate volatility

Equity volatility10


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Simple extrapolation: interest rate

12%USD government forward

8%

9%

10%

11%

strate

rates assuming constant

rate beyond 30 years

for1985-2007

5%

6%

7%

Forwardin

ter

Very conservative and willgenerate very high

volatility in the MTM value

of ultra long-term cash

2%

3%

0 10 20 30 40 50 60 70 80 90 100

Maturity(years)

.

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Macroeconomic extrapolation

Three questions:

latility

Limiting, unconditionalforwardrate/IVassumption1) What is the longest

market interest rate that

we can observe?

ate/Option

V

Marketforwards

2) What is an appropriateassumption for the very

long-term

unconditional or

InterestRorwar ra e

3) What path should beset between the longest

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0 10 20 30 40 50 60 70 80 90 100 110 120

Term(years)mar e ra e an e

unconditional forward

rate?


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Extrapolation of interest rate

A common approach

+ Fitting to available market data+ Setting a target for the ultra long-term forward rate

+ Developing an economically sensible functional form

11%

12% 11%

6%

7%

8%

9%

10%

ardinterestrate

5%

6%

7%

8%

9%

ardinterestrate

2%

3%

4%

5%

0 10 20 30 40 50 60 70 80 90 100

For

Maturity(years)1%

2%

3%

4%

0 10 20 30 40 50 60 70 80 90 100

For

Maturity (years)

+ Unconditional anchor: stability in mark-to-model valuations

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Extrapolation of derivative implied vol

+ A perfect fit to market data? + Economically robust, stable

+ How should we extrapolate?

extrapolation lead to stable,

sensible valuation

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Example: equity implied vol

+ One possible extrapolation approach:

Real-world volatility estimate as the limit of extrapolation... ... Adjusted for empirical option implied volatility / real-world volatility bias

How fast do we revert to this long-term position?

N225Im liedVolatilitiesandExtra olation

30%

35%

40%

45%

latilities

10%

15%

20%

25%

EquityImpliedV

Q42007

Q42008

Q42009

Q42010

15

0%

5%

0 5 10 15 20 25 30 35 40 45 50

Term(years)

Q42011


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What makes VA modeling difficult?

Embedded derivative nature

+ Path-dependent payoff+ Multiple assets, multiple time periods

Uncertainty

+ Insurance risks

+ Management actions

+ Policyholder options

ut are po cy o ers pr ce-sens t ve or u y rat ona

Complex exposure to various financial risk factors

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- , - , -

for a realistic estimation of risk exposure and valuation

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Sample ESG model structure

Property returnsAlternative asset returns

(e.g. commodities)

Credit risk model

Corporate bond returnsEquity returns

Initial swap and

government nominal

bonds

Nominal short rate

Real-economy; GDP

and real wages

Exchange rate

(PPP or IRP)

Inflation

expectations

Real short rateIndex linked

government bonds

Realised Inflation and

alternative inflation

Foreign nominal

short rate and

inflation

o n s r u on

Correlation relationships between the shocks to different models

Economically rational structure 18

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Real-world vs Market-consistent

A clarification of terminology

- -

Question to answer: What is the probability

distribution of future asset

prices?

What is the current

market-consistent value of

future cashflows?

Usage: Financial projections for

ALM, cashflow testing,probability of ruin analysis

Fair valuation of liabilities

(and Greeks)

Calibration: Calibrated to best-estimate Calibrated to market

arge s op on- mp e vo a es

Risk premium: Y N

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The section below focuses on market-consistent modeling

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n eres ra e mo e s

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Key consideration

+ Provide good fit to market option-implied volatility surface

+ Take yield curve as input+ Flexibility in volatility factor specification and modeling

From a modeling perspective it generally means

+ A number of popular yield curve choices for MC modelling

Hull-White

ox- ngerso - oss

Heath-Jarrow-Morton

+

In particular LIBOR Market Model is a market standard for rate derivatives trading+ Fast robust calibration tools should be available for fre uent re-

calibrations

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Low interest environment

+ Negative interest rate issues with

Gaussian models like HW Lognormal models with displacement

+ Theoretically acceptable, but in practice

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Model choice matters

Hull-White /

Vasicek

Black-

Karasinski

LMM DDLMM + SV

Fit to initial

yield curve

Depends on

implementation

Depends on

implementation

Fit to swaption

pr ces

Calibrationefficiency

Negative

interest rate

Yes No No Yes

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qu y mo e s

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Black-Scholes model as a starting point

+ Model assumption affects fair valuation / pricing of liability

+ Among other assumption: Returns are normally distributed

Volatility is constant

Constant volatilit assum tion

0.7%

0.8%

0.9%

1.0%

y

Historic (20th Century)

Stochastic Vol Model

Normal Distribution

35.00%

40.00%

0.2%

0.3%

0.4%

0.5%

0.6%

Frequenc

10.00%

15.00%

20.00%

25.00%

30.00%

25

0.0%

0.1%

-30% -25% -20% -15% -10%

Equity returns in excess of ri sk-free rates

0.

25 0

.5

0.

75

1

2

34

57 1

0

0.00%

5.00%

Maturity

Strike

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Key consideration

+ Provide good fit to market option-implied volatility surface

+ Support fast and frequent re-calibration+ Provide simultaneous fit to multiple equity indices vol surfaces

From a modeling perspective it generally means

+ Going beyond Black-Scholes (constant volatility), e.g.

Stochastic volatility

orre a on e ween re urn an vo a y

Mean reverting volatility and volatility clustering

+ But still provide (semi-) analytical calculation

Technology infrastructure for daily recalibration

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Market-consistent valuation

+ Model should first fit to market prices of derivativesMarket Volality Sur face Model Volality Surface

35.00%

40.00%

35.00%-40.00%

30.00%-35.00%

25.00%-30.00%

20.00%-25.00% 30.00%

35.00%

40.00%

35.00%-40.00%

30.00%-35.00%

25.00%-30.00%

20.00%-25.00%

5.00%

10.00%

15.00%

20.00%

25.00%

.

15.00%-20.00%

10.00%-15.00%

5.00%-10.00%

0.00%-5.00% 5.00%

10.00%

15.00%

20.00%

25.00% 15.00%-20.00%

10.00%-15.00%

5.00%-10.00%

0.00%-5.00%

0.2

5 0.

50.

75

1

2

3

45

7 10

0.00%

Maturity

Strike

0.

25 0

.5

0.

75

1

2

3

4

5

7 10

.

Maturity

Strike

s mp e ac - c o es mo e canno a vo a y sur ace - u

market implies one

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What is SVJD?

+ The SVJD model is a combination of two well known models of

quantitative finance

The Heston Stochastic Volatility Model

er on s ump us on o e

+ Benefits:

Fairly realistic but parsimonious model

Generally provides a good fit to volatility surface at both long and short maturities

Semi-analytic (i.e. fast) valuation formulae for vanilla option prices

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Technical specification

+ Two parts:

Stochastic volatility part, Heston model (red) Jump diffusion part, Merton model (blue)

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Example simulationSimulation of the SVJD Model

Example Simulation

45%200.00

10 Year Total Return Index

Index No Jumps Index Jumps Only Index Stochastic Volatility

30%

35%

40%

120.00

140.00

160.00

.

ility

ex

15%

20%

25%

60.00

80.00

100.00

Stochastic

Volati

TotalReturn

In

0%

5%

10%

0.00

20.00

40.00

30

Month

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Calibrating to implied vol surface

Market Feature Model Component Key Parameters

Implied volatility term

structure

Stochastic variance Initial variance, mean

reversion level, andspeed of mean reversion

Lon term skew/smile Stochastic Variance Return-variance

correlation and volatility

of variance

Short term skew/smile Jump Diffusion Jump parameters

+ Realistic description of underlying dynamics is key

volatility surface

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re mo e s

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Impact of credit models

+ The impact of credit risky bonds can be highly significant

On both guarantee costs / pricing

And required hedge portfolio

2.60%

2.80%

3.00%

)

2.00%

2.20%

2.40%

'tees(%

InitialFund

premium GMAB & GMDB for a 45year-old male for 20 years 50%

invested in equities and 50% in a

1.40%

1.60%

.

Costof-yr on un ...

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1.00%

.

Govt AA BBB

AssumedBondStrategy

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Credit Modelling: Key Features

A good credit risk model should be

+ Arbitrage-free

+ Stochastic credit rating migrations and defaults+ Stochastic variations in credit spreads

+ Integrated with other market risks (e.g. equities and interest rates)

o e ynam c ers or eren app ca ons:

+ Real world modelling Pricing matrix more severe than underlying transition matrix

=>

+ Risk neutral modelling

Pricing matrix as severe as underlying transition matrix

=> risk neutral

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hedge projection

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Projection of hedging strategies

+ Real-world model capturing hedging strategys key risk exposures

Vega: Increases in option-implied volatility levels

- -

+ Integrated modelling of equity total returns and changes in the level

of equity option-implied volatilities

25%

30%

35%

40%

45%

50%

gVolatility

0.5

0.6

0.7

0.8

0.9

1.0

Correlation

UK Volatility (LHS)

US Volatility (LHS)

UK vs US Correlation (RHS)

0%

5%

10%

15%

20%

-63

-68

-73

-78

-83

-88

-93

-98

-03

5Y

Rolli

0.0

0.1

0.2

0.3

0.4

5Y

Rollin

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De

De

De

De

De

De

De

De

De

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Long-term Hedging P/L Analysis

+ Long-term profitability of delta hedging strategy driven by how

realized volatility behaves relative to implied volatility levels

Expected real-world volatility vs option-implied volatility

Variability of real-world volatility

15.0%1.50%

-2.5%

0.0%

2.5%

5.0%

7.5%

10.0%

12.5%

-0.25%

0.00%

0.25%

0.50%

0.75%

1.00%

1.25%

PriceChane(%

)

edgingLosses

rlyinhgfund)

CumulativeDeltaHedgingP/L

-17.5%-15.0%

-12.5%

-10.0%

-7.5%

-5.0%

-1.75%-1.50%

-1.25%

-1.00%

-0.75%

-0.50%

DailyS&P500

Cumulative

(%ofunde

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- .- .

1 3 5 7 9 1113151719212325272931333537

TradingDay(October1st November20th'08)

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Variance reduction for risk assessment

+ Hedge projection requires Greeks projection

Estimation of Greeks at each point in each real-world simulation requires

stochastic-on-stochastic (theoretically)

+ Variance reduction for Greeks calculation

E.g. Least Squares Monte Carlo to efficiently capture future liability valuation

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Example LSMC for projected Greeks

+ Simple Black-Scholes example

projection of 10-year vanilla put option

+ 10,000 outer scenarios and 2 inners per outer

Higher numbers of inner sims can be used to increase accuracy

+ Fit cubic function and differentiate to estimate delta

0.4

0.5

True value (Black-Scholes)

I-0.2

-0.1

0

0.5 1 1.5 2 2.5

Value (after 1 year) Delta (after 1 year)

y = -0.1367x3 + 0.74x2 - 1.4881x + 1.1539

0.1

0.2

0.3

OptionValue@Year1 In a -scenaro es mae

Regression estimate

Differentiate

fitted polynomial

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

ption

Delta@Year1

True delta (Black-Scholes)

Regression estimate

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0

0.5 1 1.5 2 2.5

Put

Equity Price @ Year 1

-1

-0.9

Put

Equity Price @ Year 1

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Concluding remarks

+ Market risk modeling impacts

Fair and adequate pricing / valuation / capital

Design / risk exposure of hedging strategy

+ Model choice and calibration, among others, have significant impact

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Copyright 2012 Barrie & Hibbert Limited. All rights reserved. Reproduction in whole or in part is

prohibited except by prior written permission of Barrie & Hibbert Limited (SC157210) registered in

Scotland at 7 Exchange Crescent, Conference Square, Edinburgh EH3 8RD.

.

estimates included in this document constitute our judgment as of the date indicated and are subject

to change without notice. Any opinions expressed do not constitute any form of advice (includinglegal, tax and/or investment advice). This document is intended for information purposes only and is

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2012 Tokyo Ebig Yau