Volatility Model Risk Measurement and Strategies Against Worst Case

Volatility Mo del Risk measurement and

strategies against worst case volatilities

Risklab Pro ject in Mo del Risk

1 Mo del Risk: our approach

Equilibrium or (absence of ) arbitrage mo dels, but also p ortfolio management

applications and risk management pro cedures develop ed in nancial institu-

tions, are based on a range of hyp otheses aimed at describing the market

setting, the agents risk app etites and the investment opp ortunityset. When

it comes to develop or implement a mo del, one always has to make a trade-o

between realism and tractability. Thus, practical applications are based on

mathematical mo dels and generally involve simplifying assumptions which

may cause the mo dels to diverge from reality. Financial mo delling thus in-

evitably carries its own risks that are distinct from traditional risk factors

such as interest rate, exchange rate, credit or liquidity risks.

For instance, supp ose that a French trader is interested in hedging a Swiss

franc denominated interest rate book of derivatives. Should he/she rely on

an arbitrage or an equilibrium asset pricing mo del to hedge this b o ok? Let us

assume that he/she cho oses to rely on an arbitrage-free mo del, he/she then

needs to sp ecify the number of factors that drive the Swiss term structure

of interest rates, then cho ose the mo delling sto chastic pro cess, and nally

estimate the parameters required to use the mo del.

The Risklab pro ject gathers: Mireille Bossy, Ra jna Gibson, François-Serge Lhabitant,

Nathalie Pistre, Denis Talay and Ziyu Zheng. It is a joint collab oration between the

Risklab institute (Zürich), INRIA Sophia Antip olis, and University of Lausanne. Author's

current addresses: M. Bossy, D. Talay and Z. Zheng: INRIA Sophia Antip olis, 2004

route des Lucioles, BP 93, 06902, Sophia Antip olis Cedex, France. N. Pistre: ENSAE, 3

avenue Pierre Larousse, 92245 Malako Cedex, Paris, France. R.Gibson: Swiss Banking

Institute, 14 Plattenstrasse, 8032, Zurich, Suisse. F-S. Lhabitant: UBS AG, Global Asset

Management, Klausstrasse 10, 8034 Zurich.

To app ear in the Journal of the French Statistics So ciety. 1

In order to characterize the random evolution of the term structure of

interest rates, mo dels with one-factor, generally chosen as the short term

rate, have been developp ed b ecause they are easy to implement (see, e.g.,

Merton [21] , Vasicek [25], Cox, Ingersoll and Ross [10], Hull and White [17],

etc.), even if most empirical studies using a principal comp onent analysis

have decomp osed the motion of the interest rate term structure into three

indep endent and non-correlated factors, which resp ectively capture the level

shift in the term structure, the twist in opp osite direction of short and long

term rates, and the buttery factor that captures the fact that the inter-

mediate rate moves in the opp osite direction of the short and the long term

rates. The multi-factor mo dels do signicantly b etter than single-factor mo d-

els in explaining the dynamics and the shap e of the entire term structure,

but the latter provide analytical expressions for the prices of simple interest

rates contingent claims, whereas a multi-factor mo del generally leads to nu-

merically or quasi analytically solve partial dierential equations in a higher

dimension to obtain prices and hedge ratios for the interest rate-contingent

claims.

The factor(s) dynamics sp ecication is another source of Mo del Risk. The

dynamics sp ecications cover a large sp ectrum of distributional assumptions

such as pure diusion pro cesses or mixed jump-diusion pro cesses. The

sto chastic pro cesses can have time-varying or constant drift and/or volatility

parameters and they can rely on a linear or a non-linear sp ecication of the

drift (see Ait- Sahalia [3])

Once the mo del is xed, one has to estimate its parameters. This step

do es not really provide help to verify the adequacy of the mo del on past

data, since the theory of parameter estimation generally assumes that the

true mo del b elongs to a parametrized family of mo dels. Moreover, a mis-

sp ecied mo del do es not necessarily provide abad t to the data.

A study by Jacquier and Jarrow [18] prop oses to incorp orate mo del error

and parameter uncertainty into a new metho d of contingent claims mo d-

els' implementation. Using Markov Chains, Monte Carlo estimators, their

conclusion, for a single sto ck option case study, suggests that the pricing

p erformance of the "extended" BlackandScholes mo del dominates the sim-

ple Black and Scholes mo del within but not out-of-sample. Usually, the

mo del parameters must be estimated by tting a given set of market data.

However, in nance it has b een proved that natural estimators suchasmax-

imum likeliho o d and generalized metho d of moments estimators applied to

time-series of interest rates may require a very large observation p erio d to

converge towards the true parameter values, yet it seems highly unrealis-

tic to assume constant parameters over such a long period, see, Fournié &

Talay [13]. Another imp ortant problem arises from the time discretization 2

when we numerically compute the statistical estimates of the parameters of

a sto chastic mo del.

All the sources of Mo del Risk we have just listed, have strong nancial

rep ercussions (losses incurred by a bank or a nancial institution due to

Mo del Risk are fairly common) on the pricing, hedging, risk management

and the denition of regulatory capital adequacy rules.

As far as derivatives pricing/hedging is concerned, a large part of the lit-

erature relies on the assumption of absence of arbitrage opp ortunities since

the seminal articles of Black and Scholes [8] and Merton [21]. The principle

is simple: given a distribution on a primitive asset price (for a sto ck, a b ond,

or a commo dity), if wemake the simplied hyp othesis that there are no fric-

tions and that the risk structure is not to o complex, one can show that the

derivative cash-ows can b e replicated by a dynamic trading strategy involv-

ing the primitive assets. The Absence of Arbitrage hyp othesis stipulates that

the price of the derivative should b e the same as the price of the replicating

p ortfolio or there would be arbitrage prots (free lunches) in the market.

However, in this framework the pricing or hedging p erformance is not

indep endent of the mo del used for the underlying asset price. One imp ortant

problem is that the pricing mo dels are often derived under a p erfect and

complete market paradigm, whereas markets are actually incomplete and

imp erfect. Violations of the p erfect market assumption generally preclude us

from observing uniqueness of the asset's theoretical price, and we are often

left only with bounds to characterize transaction prices. In an interesting

study fo cusing on equity options, Green and Figlewski [15] thus explain why

option writers will charge a volatility mark-up to protect their Prot and

Loss against Mo del Risk.

The presence of Mo del Risk will also aect the p erformance of mo del

based hedging strategies.

In a continuous-time and frictionless market, a market maker, for ex-

ample, the seller of an option, can synthetically create an opp osite p osition

(called delta-hedging strategy) which eliminates his/her risk completely. If

the hedger uses an alternative (wrong) option pricing mo del, his/her price

for the option diers from the true (market) price and provide an incorrect

hedge ratio. In the presence of Mo del Risk, even though we assume fric-

tionless markets, the self-nancing delta-hedging strategy do es not replicate

the nal pay-o of the option p osition. Bossy and al. [9] study the case of

b ond option hedging, the Mo del Risk is dened by the Prot and Loss of the

seller of an option who b elieves that the true mo del is one of the univariate

Markov mo dels nested in the Heath-Jarrow-Morton [16] framework, whereas

the true mo del is actually another mo del b elonging to the same class. In a

subsequent study, Akgun [4] extends the metho dology develop ed by Bossy 3

et al. [9] to include omitted jumps by the trader who uses the wrong pure

diusion univariate term structure mo del to hedge his derivatives exp osure.

In his setting, the jumps are driven by a nite state-space comp ound Poisson

pro cess. This in turn allows him to show that omitting jump risk can be

fairly devastating, as evidenced by simulated forward Mo del Risk Prol and

Loss probability distributions, esp ecially when shorting in and at-the-money

naked or spread option p ositions.

Indeed, it is imp ortant for regulators to measure the trading and the

banking books interest-rate risk exp osures correctly. The Basle Committee

on Banking Sup ervision [1], [2] issued directives to help nancial institutions

evaluate the interest-rate risk exp osures of their exchange traded and over-

the-counter derivative activities, as well as for their on and o-balance sheet

items. Regulators ask the banks and other nancial institutions to set aside

equity in order to cover market risk driven losses.

Prop osition 6 of the Basle Committee Prop osal [2] states that banks can

calculate their market risk capital requirements as a function of their fore-

casted ten-days-ahead value-at-risk. The value-at-risk (VaR) is dened as

the quantile at a 99% condence interval of the distribution of the future

value of the considered activity. The aim is to estimate the p otential loss

that would not b e exceeded with a 99% probabilityover the next ten trading

days. An imp ortant source of Mo del Risk arises from the approximation tech-

niques that a bank adopts to incorp orate non-linear payo securities in the

VaR mo del. Pritzker [22] investigates the trade-o b etween accuracy and the

computational time for six alternative VaR computation metho ds. Among

them, the delta-gamma Monte Carlo metho d provides the b est trade-o but

still leads to signicant errors in the VaR gures, esp ecially for deeply out of

the money options. Under these directives, banks are allowed to apply their

own internal risk measurement mo dels to calculate the VaR which will in

turn determine their regulatory capital charge. No particular typ e of mo del

is prescrib ed, as long as the internal mo del captures the relevant market risks

run by a nancial institution.

Even if internal mo dels are allowed, regulators have intro duced a back-

testing pro cedure to assess the accuracy of a given VaR mo del and p enalties

in the form of multipliers: the market risk capital charge is computed using

the bank's own estimate of the value-at-risk, times a multiplier whose value

dep ends on the numb er of exceptions over the last 250 days detected with the

help of the back-testing pro cedure. The regulator has xed the value of the

multiplier b etween 3 and 4 in order to keep a security margin against p ossible

mo del errors made in the computation of the VaR. Lop ez [20] compares three

commonly used back-testing pro cedures and shows that they all have very

low power against alternative VaR mo dels. Thus, even at the nal stages of 4

mo del assessment and compliance, the accuracy detection metho d can induce

regulatory `Mo del Risk and thus lead to over or under-capitalized nancial

institutions'.

Sp ecifying a prop er loss function to assess a mo del's accuracy is thus

a rst step to mitigate Mo del Risk. This loss function should dep end on

the sp ecic applications asso ciated with the mo del and be adapted to the

time-horizon and the contractual features of the p ositions b eing valued or

hedged, to the division and/or the resp onsibility levels involved (trading desk

versus senior management), without leading to excessive risk taking b ehavior

esp ecially below the critical downside risk thresholds. The metho dology of

Bossy and al. [9] can be used to measure the risk implied by the choice of

an erroneous univariate term structure mo del. It measures the distribution

of the losses due to this error, including (but not reduced to) estimation

errors. It also takes into account hedging errors in addition to pricing errors

(that can be avoided by the calibration of a wrong mo del on true market

data). This metho dology is briey discussed in Section 2.1 b elow. Quantile

approximation from Talay and Zheng [24] is presented at the end of Section

This metho dology is however restricted to the comparison of one (p oten-

tially incorrect) mo del against one or several (p ossible true) mo dels among a

class of univariate Markov term structure mo dels. This class do es not contain

all p ossible term structures mo dels. We present a more general metho dology

and the results develop ed by Talay and Zheng [23] which aim at selecting a

hedging strategy which minimizes the exp ected utility of the Loss due his/her

mo del risk under the worst p ossible movements of nature (as characterized

by forward rates' volatility tra jectories). This metho dology is discussed in

Section 3.

This article is a synthesis of the following articles: Gibson, Lhabitant,

Pistre and Talay [14], Bossy, Gibson, Lhabitant, Pistre and Talay [9], Talay

and Zheng [23], [24]. Of course, our nancial and numerical metho dologies

can easily be extended to a wide family of cases: Europ ean options with

Black and Scholes typ e mo dels, markets with transaction costs, etc.

2 Mo del Risk measurement

We briey recall the expression of the Prot and Loss obtained in Bossy et

al. [9] for a trader who b elieves in a term structure mo del of the univariate

Markov Heath-Jarrow-Morton family whereas the true term structure fol-

lows another mo del mo del in the same family. In reality the `true' mo del is

unknown. Thus one must consider this Mo del Risk analysis as b eing p er- 5

formed with resp ect to `b enchmark' mo dels selected bytheinvestor, the risk

controller or the regulator.

2.1 Expression of the Prot and Loss in the Heath-

Jarrow-Morton mo del

We are given a probability space ( ; F ; P ) equipp ed with the augmented

ltration (F ); t 2 [0;T] generated by a real valued Brownian Motion (W ;t 2

t t

[0;T]). We supp ose that the yield curve of the nancial market follows the

Heath-Jarrow-Morton mo del and, as here our analysis is fo cused on volatility

Mo del Risk, we also supp ose that the premium risk pro cess ( ) is null, thus

we have: for all time T , the instantaneous forward rate f (t; T ) solves the

sto chastic dierential equation

Z Z

t t

f (t; T )= f (0;T )+ (s; T ) (s; T )ds + (s; T )dW ; (1)

0 0

with

(s; T ):= (s; u)du:

In Bossy et al. [9], the Mo del Risk analysis is based on Monte Carlo

simulations of the Prot and Losses of the self-nancing strategies of a trader

who aims at hedging a Europ ean option written on a bond of maturity T .

In this subsection we give an outline of the metho d for the case where the

function is deterministic.

Supp ose that the trader do es not know the map (s; T ). Instead, he or

(s; T ) and tries to hedge the she cho oses a deterministic mo del structure

contingent claim according to this mo del.

V be the value of the trader's p ortfolio at time t and V be the Let

t t

value of the p erfectly hedging p ortfolio. The option seller's Prot and Loss

is dened by:

P &L := V V :

t t t

Given aprice P , dene the forward price P by

P := ;

B (t; T )

O O

where B (t; T ) is the price of the bond of maturity T in the mo del driven

by (t; T ).

It can be shown that the self nancing constraint implies that

F F F

= dV (t; B (t; T ))dB (t; T ); (2)

@x 6

where B (t; T ) is the forward price of the discount b ond, and that the forward

Prot and Loss P &L satises

F F

V (0;B (0;T)) P &L =

0 t

F F

+ (t; B (t; T )) (t; B (t; T ))

(3)

1 @

F F 2

(s; B (s; T ))B (s; T ) +

2 @x

O 2 O 2

f( (t; T ) (t; T )) ( (t; T ) (t; T )) gds;

where is the solution to the following parab olic PDE parametered by the

function :

@ (t; x) (t; x) @ 1

2 2

(t; T ) (t; T )) + x ( =0;

(4)

@t 2 @x

(T; x)=(x):

Thus the gamma of the p osition is shown to b e essentially in the quantityof

Mo del Risk induced by the p osition of the trader. This means that limiting

the Mo del Risk of a trader implies limiting the gamma of the p osition, and

that the Mo del Risk exp osure of an option p osition is not similar to its

interest rate exp osure and thus has to b e managed sep erately.

As justied by Artzner and al.[5], quantiles of the negative part of P &L O ,

E [U (P &L )] where U is a utility function, are go o d candidates of Mo del

Risk measurements. Bossy et al. [9] discuss numerical results obtained by

Monte Carlo metho ds for simple and agregate strategies. In our next subsec-

tion we analyse the accuracy of such Monte Carlo metho ds. As the pro cess

F F

(B (t; T );P&L ) is the solution of a sto chastic dierential equation (see

Equation (7) b elow), we fo cus our attention to the accuracy of the Monte

Carlo metho d to compute the quantiles of diusion pro cesses.

2.2 Quantile approximation of a diusion pro cess by

Monte Carlo metho ds

Given a random variable X and 0 < < 1, the quantile of level is the

smallest ( ) such that

P [X ( )] = :

The case where X = X , X solution to a sto chastic dierential equation, is

of sp ecial interest for us.

Let (X ) be a real valued pro cess, solution to

(

dX = b(X )dt + (X )dW ;

t t t t

(5)

X = x;

0 7

where (W ) is a r -dimensional Brownian motion.

The Euler scheme for (5) is

n n n n

+ (X )(W W ): X = X + b(X )

(p+1)T=n pT =n

pT =n (p+1)T=n pT =n pT =n

Dene ( ) by

n n

P [X ( )] = :

We supp ose that (5) satises the same uniform hyp o ellipticity condition as in

Bally &Talay [7]. We do not rewrite here this technical hyp othesis because

it requires to o much material to be stated. We only emphasize that it can

be shown that this hyp othesis is generally satised for the equation (7).

Theorem 2.1. Under the above uniform hypoel lipticity condition, there exist

strictly positive constants C (T ) and p (( )) such that

C (T )

; 8n: j( ) ( )j

q ( )n

For the pro of of Theorem 2.1, see Talay & Zheng [24]. The pro of shows

that

q ( ):= inf p (y );

T X

y 2(( )1;( )+1)

where p is the density of the distribution of X .

X T

As classical estimates show that the standard deviation of the statistical

error of the Monte Carlo metho d, that is, the error due to the approxima-

tion of the exp ectation by the average over the simulations, is governed by

C (T )=(p (( )) N ), N b eing the number of simulations. To get estimates

on the discretization step and the number of simulations which are necessary

to obtain a desired accuracy with a given condence interval one needs an

accurate lower bound of the density of X . For the strictly uniform elliptic

generators, see, e.g., Azencott [6]. In the degenerate case, under restrictive

assumption on b, see Kusuoka & Stro o ck [19]. Such assumptions are not

satised in our Mo del Risk study, and therefore we need the supplementary

results of our next subsection.

2.3 Application to Mo del Risk

Dene as the solution of the same parab olic problem as (4) with instead

. Set of

u (t):= (t; T );

u (t):=( (t; T ) (t; T ));

(6)

@ @

(t; x) (t; x): '(t; x):=

@x @x 8

Thus the forward price of the bond and the forward Prot and Loss P &L

satisfy

(

F F F

dB (t; T )=B (t; T )u (t)u (t)dt + B (t; T )u (t)dW ;

1 2 2 t

(7)

F F F

dP &L = '(t; B (t; T ))dB (t; T ):

We are interested in the quantile of P &L O = P &L , that is

P [P &L O ( )] = :

We must rst prove that P &L O has a density, that is, the following Theo-

rem:

Theorem 2.2. Suppose that ju (t)j a>0 for al l t in [0;T ], u (t), u (t)

2 1 2

F F

are bounded, and B (0;T)'(0;B (0;T)) 6=0. Then the law of P &L O has

a smooth density p O . Moreover, p O is strictly positive on its support.

T T

For the pro of, see Talay &Zheng [24].

In view of our comments for Theorem 2.1, it would be useful to obtain

. This is done in Talay an accurate pointwise lower b ound estimate for p

& Zheng [24].

3 Mo del Risk management against worst case

volatility

The previous metho dology is restricted to the comparison of the p otentially

incorrect mo dels against the p otentially true (but unknown) or benchmark

mo dels among a class of univariate Markov term structure mo dels. The

following metho dology is far more general, since it allows to optimize the

choice of the trader's strategy against `all' p ossible actual volatility pro cesses.

3.1 Motivation

The ob jective is to prop ose a new strategy for the trader which, in a sense,

guarantees go o d p erformances whatever is the unknown pro cess (; ). The

construction corresp onds to a `worst case' worry and, in this sense, can be

viewed as a continuous time and rigorous extension of discrete time strategies

based up on prescriptions issued from VaR analyses at the b eginning of each

period.

Roughly sp eaking, the idea can b e expressed by the following graph: 9

Trader = Minimizer of Risk.

Market = Maximizer of Risk.

Trader vs Market.

Thus the Mo del Risk control problem is set up as a two players (Trader

versus Market) zero-sum sto chastic dierential game problem. We notice

that our approach is related to, but dierent from, Cvitanic and Karatzas

dynamic measure of risk(see [11]). Moreover, the solution at time 0 of our

sto chastic game problem can b e viewed as a `reserve p osition' for a nancial

institution.

3.2 The sto chastic dierential game and the Hamilton

Jacobi Bellman Isaacs equation

Let ( ) be the delta pro cess chosen by the trader. The self nancing con-

straint implies that one has

(

F F F

dB (t; T )= B (t; T )u (t)u (t)dt + B (t; T )u (t)dW ;

1 2 2 t

(8)

F F

d V (t)= (t)B (t; T )u (t)u (t)dt + (t)B (t; T )u (t)dW ;

1 2 2 t

where u (t) and u (t) are dened as in (6).

1 2

We adopt the denition of admissible controls and strategies of Fleming

& Souganidis [12]. The set of all admissible controls for the market on [t; T ]

is denoted by Ad (t) and the set of all admissible strategies for the investor

on [t; T ] is denoted by Ad (t). These admissible controls and strategies take

value in compact sets K and K resp ectively.

For given 2 Ad (r ) and u 2 Ad (r ), we dene the ob jective function

F O

J (r;x;y;;u ):= E [F (f (B (T ;T)) V )]

r;x;y

where F is a utility function and f is the prole of the payo function . The

function F should be chosen according to the denition of measures of risk

intro duced in Artzner et al. [5].

Denition 3.1. The value function of the Model Risk control problem with

initial data (r;x;y) is dened by

V (r;x;y):= inf sup J (r;x;y;;u ): (9)

2Ad (r )

u 2Ad (r )

: u

For the sake of simplicity our notation do es not emphasize that the pro cess

1 2 F

V ) is parametered by (u (t);u (t); ). (B (t; T );

t 10

We have the following result:

Theorem 3.2. Suppose that

x) y )j P (jxj; jy j; jxj; jy j)(jx x j + jy y j); jF (f (x) y ) F (f (

where P (jxj; jy j; j xj; jy j) is a polynomial function.

Then the semi-value function V (r;x;y) dened in (9) is the unique vis-

cosity solution in the space,

S := f'(t; x; y )is continuous; 9 A>0;

2 2 2 O

lim '(t; x; y ) exp(A j log(x + y )j )= 0; 8t 2 [0;T ]g

2 2

x +y !1

to the Hamilton-Jacobi-Bel lman-Isaacs Equation,

2 O 2

(t; x; y )+ H (D v (t; x; y );Dv(t; x; y );t;x)= 0 in [0;T ) R ;

v (T ;x;y)=F (f (x) y );

(10)

where

H (A; p; t; x)

1 1

2 2 2 2 2 2 2

u x A + u x A + u x A + p u u x + p u u x ; := max min

11 12 22 1 1 2 2 1 2

2 2 2

u2K 2K

2 2

(11)

for al l 2 2 symmetric matrix A and al l vector p in R .

Moreover, V (r;x;y) satises the Dynamic Programming Principle, that

is,

V (r;x;y)= inf sup E [V (t; x ;y )]: (12)

r;x;y t t

2Ad (r )

u 2Ad (r )

: u

For the pro of, see Talay &Zheng [23].

The value function V (r;x;y) and the optimal strategy can be solved nu-

merically by the nite dierence metho d. In our numerical tests we consider

the Prot and Loss of the seller of a Europ ean call option. The utility func-

tion is

F (x):=(f (x) y ) : (13)

The maturity T of the option is 6 monthes and the option is written on a

discount b ond of maturity T equal to 5 years. The trader uses twobondsto 11

hedge this option: the b ond of maturity 6 monthes and the b ond of maturity

5 years. The strike of the option is K =0:509156.

We set K := [c ;c ]= [1; 1]. The constant c has been chosen after

1 1 1

having computed p erfectlly replicating strategies in the cases of HoLee and

Vasicek mo dels. In addition, we x u 2 [0;c ] and u 2 [c ;c ] with

2 2 1 3 3

c =0:6 and c =0:07.

2 3

We comment Fig. 2. The graph of optimal strategy H (t; x; y ) consists

in three parts:

When x (the forward price of the bond) is comparatively small with

resp ect to y (the forward price of the p ortfolio), then H (t; x; y )= 0. It

is a conservative strategy.

When x is comparatively small with resp ect to y , H (t; x; y ) = c , the

maximum value. It is a risky strategy.

In the other cases, H (t; x; y ) takes values b etween 0 and c . It is similar

to the classical hedging strategy.

Moreover, we observe from Fig. 3 that the optimal volatility u is a bang-bang

control.

Acknowledgement

This study has b een done within the Risklab pro ject `Mo del risk', which is

joint collab oration between the Risklab Institute in Zürich and the Omega

research group at Inria Sophia Antip olis. The authors thank Risklab for the

nancial supp ort.

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Journal of Financial Economics, 5:177188, 1977. 14 ) ;x;y (0 V Graph of 15 unction: F alue V The Model Risk Problem: The value function corresponding to a call option 1:

"modelrisk" Figure

6 5 4 3 2 1 0

1 0.8

0 0.6 0.2 0.4 X-axis 0.4 0.6 0.2 Y-axis 0.8 1 0 ) ;x;y (0 H Graph of 16 Strategy: Optimal The

2: Model Risk Problem: the trader’s optimal strategy for a call option

"optimal" Figure

0.8

0.6

0.4

0.2

1 0.8

0 0.6 0.2 0.4 X-axis 0.4 0.6 0.2 Y-axis 0.8 1 0 ) ;x;y (0 2 u of Graph trol: 17 Optimal Con The Model Risk Problem: the market’s optimal control u2

"u2" Figure 3:

0.6 0.5 0.4 0.3 0.2 0.1 0

1 0.8

0 0.6 0.2 0.4 X-axis 0.4 0.6 0.2 Y-axis 0.8 1 0 ) ;x;y (0 1 u of Graph trol: 18 Optimal Con The Model Risk Problem: the market’s optimal control u1

"u1" Figure 4:

0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

0 0.2

1 0.4 0.8 0.6 X-axis 0.6 0.4 0.8 Y-axis 0.2 0 1