Complete Markets - a Wishart based approachMar 22, 2016 · LSE, March 22, 2016 R. Trabalzini Complete Markets - a Wishart based approach LSE, March 22, 2016 1 / 27. Completing markets

Completing markets with options - (5)A good family of factor processes - (9)Completion, hedging & numerics - (7)

Recursive calibration - (4)

Complete Markets - a Wishart based approach

Romano Trabalzini

Department of MathematicsImperial College of Science and Technologyjoint work with Prof. Mark H. A. Davis

LSE, March 22, 2016

R. Trabalzini Complete Markets - a Wishart based approach LSE, March 22, 2016 1 / 27



Index

1 Completing markets with options - (5)The problem under scrutiny & literatureOptions as system of PDEs; solving backward Kolmogorov equationsHedging via predictable representation

2 A good family of factor processes - (9)Affinity, cross-correlation, cardinality of the filtrationWishart processes: characterization and propertiesMinimal parameterization, strong solution & boundary behavior

3 Completion, hedging & numerics - (7)A general theoremEncoding market completenessNumerics: simulation, calibration & FFT pricing

4 Recursive calibration - (4)Bona fide small noise filteringConclusions, addenda & open questions

R. Trabalzini Complete Markets - a Wishart based approach LSE, March 22, 2016 2 / 27



The problem under scrutiny & literatureOptions as system of PDEs; solving backward Kolmogorov equationsHedging via predictable representation

MARKET COMPLETION & SPANNING SET OF OPTIONS

NOVEL RESULTS:

Modelling the non-redundancy of contingent claims

Framing observed options as solutions of Backward Kolmogorov equations

Thm. 2.15: (Replication strategy)

References:[Ross, 1976][Romano and Touzi, 1997][Davis, 2004][Davis and Ob loj, 2008]

R. Trabalzini Completing markets with options - (5) Complete Markets - a Wishart based approach LSE, March 22, 2016 3 / 27




A market is complete if there exists a finite cardinality set of spanning securities

It is well known that the Black-Scholes financial market model, consisting of alog-normal asset-price diffusion and a non-random money market account, iscomplete: sharing same source of randomness with underlying, intuition goes thatevery contingent claim is replicated by a portfolio formed by dynamic trading inthe two assets. Options are therefore redundant.

This result is predicated on the martingale representation property of BM.

Trying to correct the empirical deficiencies of the asset model by including, say,stochastic volatility, completeness is lost if the original two assets are regarded asthe only tradables: there are no longer enough assets to span the market.

However there are traded options markets for many assets such as single stocks orstock indices, so it is a natural question to ask whether the market becomescomplete when these are included.

An early result in this direction was provided by [Romano and Touzi, 1997] whoshowed that a single call option completes the market when there is stochasticvolatility driven by one extra Brownian motion.

Providing a general theory has been difficult, hence fresh start: “MartingaleModel”





Work on probability space (Ω,F ,Ftt∈R+ ,Q) - usual conditions; let D ⊂ Rn, openconnected

the martingale measure Q is the only one taken in consideration in this work

consider the following vector valued diffusive SDE - ”the factor process”:

dξt = µ(t, ξt )dt + σ(t, ξt )dWt , ξ0 = x ∈ D (1)

where µ : R+ × D → Rn and σ : R+ × D → Rn×n are Lipschitz continuous functions and

σσT is unif elliptic, W := Wt ∈ Rn,Ft ; 0 ≤ t <∞ mFt BM

Ftt∈R+ is the natural filtration of Wt

for any v(t, x) ∈ C 1,2 a Cauchy final value problem for (1), with h(x) being time Tboundary data, under technical conditions on µ, σ and h, is( ∂

∂t+ L

)v(t, x) = 0, for all (t, x) ∈ (0,T )× D (2)

v(T , x) = h(x), for all x ∈ D (3)

(Lf )(t, x) :=n∑

i=1

µi (t, x)

∂

∂x if (t, x) +

1

2

n∑i,k=1

aik (t, x)∂2

∂x i∂xkf (t, x) + c(t, x)f (t, x)

with

aik (t, x) :=n∑

j=1

σij (t, x)σkj (t, x) =

(σ(t, x)σT (t, x)

)ik.





Crucially, the “martingale model” defines options as the expected value of payoffs h(·) under Ft ,which amounts to solve the BK equation for (2)-(3):

v(t, x) := E[h(ξT )|ξt = x], x ∈ D (4)

The following generalization of [Davis, 2004, Thm. 3.2] holds:

Theorem (Replication strategy [Trabalzini, 2015, Thm. 2.15])

W.r.t. EMM Q, let the market correspond to the set of all Ftt∈[0,T ]-adapted contingent claimspayoffs

Ht,i : D 7→ R+, Ht,i ∈ L2(D); t ∈ t0, . . . , tT−1, i ∈ 1, . . . and let Y := Yt ∈ R,Ft ; 0 ≤ t ≤ t1 be the value process of a self-financing portfolio anddefine Ok,i (t, x) =: vk,i (t, x) as in (4). Consider:

Assumption

∃ C ,m ∈ R+ s.t. |σ(t, x)−1| ≤ C(1 + |x|)m, for all (t, x) ∈ [0, t1]× D.

Assumption

For all (t, x) ∈ [0, t1]× D, rank|Jn,n(t, x)| = n in eq. (5), i.e. is full rank.

Under the above assumptions, a trading strategy for Yt can be explicitly provided.





Sketch of proof: expressing with ∇v the gradient of a function v in the x variables, fixing time

index k ∈ 1, ...,T and strike index i ∈ 1, ...,mk, a (vega weighted) delta hedging for theself-financing portfolio Yt := Y (t, ξt ) gives

dY (t, ξt ) = wT,0(t, ξt )dF (t, ξt ) +T∑

k=1

mk∑i=1

wk,i (t, ξt )dOk,i (t, ξt ), Y0 = y ∈ R

=(

wT,0(t, ξt )∇vT,0(t, ξt ) +T∑

k=1

mk∑i=1

wk,i (t, ξt )∇vk,i (t, ξt ))σ(t, ξt )dWt ,

= w(t, ξt )T J(t, ξt )σ(t, ξt )dWt , J(t, x) :=

∇vT,0(t, x)· · ·

∇vk,j (t, x)· · ·

∇vT,MT (t, x)

. (5)

Given Ht,i ∈ mFT , exercise value of an arbitrary contingent claim, Martingale Representation gives

Ht,i (ξt ) = H0,i +

∫ t

0

w T (s, ξs )J(s, ξs )σ(s, ξs )dWs , for all t ≥ 0

=: H0,i +

∫ t

0

WT(s, ξs )dWs ,

implying H0,i ∈ R. The existence of an unique, progressively mble stochastic processW impliesthe trading strategy map w(·, ·) which, identifying the combination of contracts spanning themarket filtration, is constructively given by

w T (t, x) =WT(t, x)σ−1(t, x)J−1(t, x), for all (t, x) ∈ [0, t1]× D.




Affinity, cross-correlation, cardinality of the filtrationWishart processes: characterization and propertiesMinimal parameterization, strong solution & boundary behavior

APPROPRIATE FAMILY OF FACTOR PROCESSES

NOVEL RESULTS:

The ‘right’ choice of factor process

I affine form of the probability model of state → Fourier analysis, FFT;I factors cross correlation → geometry of state space;I filtration cardinality → strong solution implies minimal parameterization.

Prop. 3.30 : (Minimal parameterization in arbitrary dimension)

Thm. 3.43 : (Existence of strong solutions)

References:[Duffie and Kan, 1996][Duffie et al., 2003][Bru, 1991][Filipovic and Mayerhofer, 2009][Cuchiero et al., 2010][Graczyk and Mayerhofer, 2012][da Fonseca et al., 2008]

R. Trabalzini A good family of factor processes - (9) Complete Markets - a Wishart based approach LSE, March 22, 2016 8 / 27




I (1/3): AFFINE FORM OF THE STATE VECTOR PROBABILITY MODEL

A vanilla option payoff can be expressed as a linear combination of components

Ga,b(y ; x , τ) := E[e〈a,ξτ 〉1〈b,ξτ 〉≤y|ξ0 = x]

where a, b ∈ Rn, y ∈ R, x ∈ D.

With v ∈ R, the FT with parameter v of any of these is well-defined:

Ga,b(v ; x , τ) :=

∫R

e ivy Ga,b(dy ; x , τ) =

∫R

e ivy E[e〈a,ξτ 〉1〈b,ξτ 〉≤dy|ξ0 = x], ∀x ∈ D

Define the integral transform map

Ψ : Cn × D × R+ 7→ C via Ψ(γ, x , τ) := E[e〈γ,ξτ 〉|ξ0 = x], γ ∈ Cn

For a probability model of affine form, i.e. where (i) drift vector and (ii)instantaneous covariance matrix have affine dependency on the state vector, thatrelation can be easily simplified via Riccati ODE. Defining maps:

A : Cn × R+ 7→ R, B : Cn × R+ 7→ Rn,

it holdsΨ(γ, x , τ) := exp (A(γ, τ) + 〈B(γ, τ), x〉).





I (2/3): FACTORS CROSS CORRELATION & DEGREES OF INTERACTION

[Filipovic and Mayerhofer, 2009, Thm. 2.2]: X x affine iff a(x), b(x) affine in x

Cross correlated CIR diffusions (#2) on canonical state space X := Rm+ × Rn−m

will not do:

d

(x1

x2

)=

[√x1 0

0√

x2

]d

(W1

W2

)=

[ √x1 0

ρ√

x2 ρ√

x2

]d

(W1

W⊥1

)=: Σ(x1, x2)d

(W1

W⊥1

)where 〈W1,W2〉t = ρt, ρ :=

√1− ρ2 and

a(x1, x2) := Σ(x1, x2)Σ(x1, x2)T =

[x1 ρ

√x1x2

ρ√

x1x2 x2

]is not an affine function - so the characteristic function will not have affine form unless ρ = 0.





I (3/3): FILTRATION CARDINALITY & ADEQUATE NUMBER OF DRIVERS

identification problem: determining the minimal number of securities adapted tothe filtration Ftt∈R+ , i.e. the filtration cardinality, as the length of anyminimal sequence of non-zero orthogonal martingales generating the space

Both (i) an empirical and (ii) a theoretical question:

The (i) empirical question is best solved with PCA: series of spot, RR, BFLY and ATMpoints for various currency pairsEmerging markets and more stable currencies; series examined stretching in strike up to25 delta and in time from a month up to 5 years; observation span is 4 yearsPrincipal Components to unscaled data, i.e. Covariance matricesFor each surface, the total variance explained by different components in the eigenspaceis analyzed in detail, testing for persistence in time of latent factors.Currencies pairs with marked skew (e.g. USDJPY) have performed distinctly better inthe analysis than other pairsOverall, percentages of explained variances are reasonably stable in timeThe standard account of a level, a curvature and a skew effect can be justified presentacross most of the pairs (think to SABR σ0, α, ρ or Heston v0, ξ, ρ).





Principal components on cumulative variation (%): plots show time-series(2008-2014) of superimposed first three eigenvalues of volatility surfacecovariance matrices

Figure: USDJPY - 25 days windows Figure: USDJPY - 50 days windows

On a (ii) theoretical level, the replication argument is entirely based on the strongmartingale representation, hence existence of strong solutions for state process isrequired

The completeness idea makes sense iff the market filtration is the naturalfiltration of the factor process, hence same cardinality is required





Let S+d be the cone of positive semidefinite symmetric matrices, d ∈ N

Define a Wishart processWISd (x , α,M,Q)

to mean the process Σ := Σt ∈ S+d ,Ft ; 0 ≤ t <∞ given by matrix valued SDE

dΣt =(αQT Q+MΣt +Σt MT

)dt+

(√Σt dWt Q+QT dW T

t

√Σt

), Σ0 = Σ ∈ S

+d

(6)where Q,M ∈ M(d , d) , and α ≥ d − 1 and

W := Wt := (W(1)t , ...,W

(d2)T ),Ft ; 0 ≤ t <∞.

Symmetrization in (6) guarantees evolution in space of symmetric d × d matrices

Boundary: a WISd (x , α,M,Q) on S+d with α ≥ d + 1,QT Q 6= 0 does not hit

bdry





The Markov state vector (”factor process”)

On (Ω,F ,Ftt∈R+ ,Q), d ∈ N define two stochastic processes

X := Xt ∈ R,Ft ; t ∈ [0,∞) Σ := Σt ∈ S+d ,Ft ; t ∈ [0,∞), by

dXt = −1

2Tr[Σt ]dt + Tr

[√Σt dZt

], X0 = x0 ∈ R (7)

dΣt = (2pQT Q + MΣt + Σt MT )dt + (√

Σt dWt Q + QT dW Tt

√Σt ), (8)

Σ0 = Σ ∈ S+d

with

W := Wt := (W (1)t , ...,W (d2)

t ),Ft≥0 , Z := Zt := (Z (1)t , ..., Z (d2)

t ),Ft≥0

mFt -matrix valued BM. are mFt -matrix valued BM, and (8) of the type in (6).

Define, for some R ∈ M(d, d),

Zt := Wt RT + Bt

√Id − RRT , B := Bt := (B(1)

t , ...,B(d2)t ),Ft ; 0 ≤ t <∞

where d2 components of W and d2 components of B are independent. Rewrite (7)-(12) as

Xt = X0 +

∫ t

0

−1

2Tr[Σs ]ds +

∫ t

0

Tr[√

Σs

(dWs RT + dBs

√Id − RRT

)](9)





Requirements are met:

I Affine form of the probability model of the state vector (here: Xτ ):

Ψ(γ, x ,Σ, τ) := E[e〈γ,Xτ 〉

∣∣∣X0 = x ,Σ0 = Σ], x ∈ R,Σ0 = Σ, γ ∈ R

= exp(Tr[A(τ)Σ0] + b(τ)X0 + c(τ)

)∣∣∣∣∣X0=x,Σ0=Σ

(10)

A ∈ M(d , d), b ∈ R, c ∈ R are obtained by solving Riccati ODEs;I Factors cross correlation: see [da Fonseca et al., 2008]I Filtration cardinality & avoiding redundancy of Brownian drivers

Proposition (Minimal parameterization [Trabalzini, 2015, Prop. 3.30])

Let d ∈ N s.t. the volatility process Σt ∈ S+d , for all t > 0 in (12), i.e.

dΣt = (αQT Q + MΣt + Σt MT )dt + (√


√Σt ), Σ0 = Σ ∈ S

+d .

It has 2d2 independent Brownian drivers. Given symmetry, only d(d+1)2

are indeednecessary. Moreover, let K be the probability filtration multiplicity, as defined above

where K = 1 + d(d+1)2

. Then the number of Wiener drivers in (9) can be reduced toone plus the number of drivers entering Σt , thus matching K , where the vector BMhas all components orthogonal.





Theorem (Uniqueness of PDE classical solution [Trabalzini, 2015, Thm. 3.43])

Let D be a open connected subset of R and T ∈ (0,∞). For given mble func.

h : D 7→ [0,∞), g : [0,T ]× D 7→ (−∞, 0] c : [0,T ]× D 7→ R

take v : [0,T ]× D 7→ [0,∞] as identified by the stochastic representationv(t, x) := E[h(XT )|Xt = x], x ∈ R and consider the initial value problem (2)-(3), where theprocess entering the stochastic representation is given by

dXt = −1

2Tr[Σt ]dt + Tr

[√Σt (dWt RT + dW T

t

√Id − RRT )

], X0 = x0 ∈ R

and the infinitesimal generator LX,Σ is given by

LX,Σ := −1

2Tr[Σ]

∂

∂x+

1

2Tr[Σ]

∂2

∂x2+ Tr[(αQT Q + MΣ + ΣMT )D + 2ΣDQT QD] + 2Tr[ΣRQD]

∂

∂x

If α ≥ d + 1 and (αQT Q + MΣt + Σt MT ) in (12) is positive semidefinite, v(t, x) satisfies the

IVP above, v(t, x) ∈ C 1,2([0,T ]× D) and the classical solution to the PDE is unique.

Prop. 3.30 proves that the number of drivers is not redundant .Multiple stoc. integral can bewritten as a single in the case under investigation (→ for cardinality)

Thm. 3.43 proves that the stoc. representation v(t, x) := E[h(XT )|Xt = x] yields a classicalPDE solution (→ for strong Martingale representation).




A general theoremEncoding market completenessNumerics: simulation, calibration & FFT pricing

COMPLETION, DYNAMIC HEDGING & NUMERICS

NOVEL RESULTS

Thm. 4.12-13: (Market completion thms.)

Prop. 4.16: (Spanning set of Laplace transforms)

References:[Davis and Ob loj, 2008][Kramkov and Predoiu, 2014][Rouah, 2013][Benabid et al., 2008][Benabid et al., 2010][Gauthier and Possamaı, 2009]

R. Trabalzini Completion, hedging & numerics - (7) Complete Markets - a Wishart based approach LSE, March 22, 2016 17 / 27




Necessary condition for the invertibility of the market matrix is

Assumption (ii) - Full rank of market matrix (Jacobian)

For all (t, x) ∈ [0, t1]× D, rank|Jn,n(t, x)| = n in eq. (5), i.e. is full rank.

Assumption is quite difficult to verify. By imposing additional structure on the maps thatappear in the final value problem in (2)-(3) a sufficient condition holds:

Theorem

([Davis and Ob loj, 2008, Thm. 4.2]) Verification of Assumption (ii) over the full time-spacedomain, i.e. for all (t, x) ∈ [0, t1]× D, can be reduced to its verification in a single point(t0, x0) ∈ [0, t1]× D under the following conditions:

1 (Conditions on real analyticity): µ, σ, hi are real analytic functions;

2 (Conditions on factor process): Eq. (1) has unique strong solution, i.e. for some ε > 0

σ(t, x)σ(t, x)T > εI , for all (t, x) ∈ (0, t1)× D (strong ellipticity) and µ, σ are such thatξ := ξt ∈ D,Ft ; 0 ≤ t ≤ T admits a positive density on D, t ≤ T ;

3 (Condition on payoff maps): hi : D 7→ [0,∞) have at most polynomial growth, 1 ≤ i ≤ n;

4 (Conditions on PDE classical solutions): vi : [0,T ]× D 7→ [0,∞], 1 ≤ i ≤ n are the uniquesolutions of eqs. (2)-(3) with at most polynomial growth.

The theorem exploits the non-trivial fact that real analyticity of µ, σ, hi maps implies realanalyticity of maps vi





The target is to explicitly compute the trading strategy map

wT (t, x) =WT(t, x)σ−1(t, x)J−1(t, x), for all (t, x) ∈ [0, t1]× D (11)

as a linear combination of existing contracts spanning the market filtrationBy Theorem (3), condition on Jacobian needs being verified at a single point

Theorem (Market completion - simpler case [Trabalzini, 2015, Thm. 4.13])

For a factor process of the form

dΣt = (αQT Q + MΣt + Σt MT )dt + (√


√Σt ),

with Σ0 = Σ ∈ S+d where matrices are diagonal, the trading strategy in (11) can be

explicitly computed and hence a market completion exists for all time t ≤ t1. It can beencoded in the coefficients of the corresponding solution.

A full extension of the theorem above, relaxing assumption on diagonal structureand introducing the log-spot process (7) as well, needs further investigation, andit is conjectured:

Theorem (Market completion - full form )

Assumption (ii) is verified for the model whose factor process is given by eqs. (7)-(8)hence a market completion exists for all time t ≤ t1, the shortest expiration claim(s)in the spanning set.





Sketch of proof: analytic representation of the market matrix & approximation argument

Consider again the Market Matrix J : [0,T ]× D 7→ Rn×n

J(n,n)(t, x) =

∇vT,0(t, x)· · ·

∇v1,n−1(t, x)

:=

∇vT,0(t, (x1, . . . , xn)T )· · ·

∇v1,n−1(t, (x1, . . . , xn)T )

, for all (t, x) ∈ [0, t1]× D.

Proposition (Reduction to analytic components [Trabalzini, 2015, Prop. 4.14])

Let i ∈ 0, . . . , n − 1 be index for the market observables, k index for the tenors observed in thespanning set (not necessarily distinct) τk := T − tk and with ai ∈ Rn, n ∈ N, n-vectors of real

numbers. Let Σ := Σt := (Σ1t , . . . ,Σn

t )T : Σt ∈ D,Ft ; 0 ≤ t ≤ T be the factor process of theform in (12) (no spot component). Assume that the market trades products of the form

vτk ,i(0,Σ) = vτk ,i

(0, (Σ1, . . . ,Σn)T )

:= E[e〈ai ,Στk〉|Σ0 = Σ], for all Σ := (Σ1

, . . . ,Σn)T ∈ D. (12)

Then J(n,n) can be expressed in terms of conditional MGF and has real analytic components.

Proposition (Spanning Laplace Transforms [Trabalzini, 2015, Prop. 4.16])

For any payoff H : D 7→ R+, H ∈ L2(D) there exists a sequence(γn, τn,wn

)n∈N

with wn ∈ C:

H(x) =∑n≥0

wnΨ(γn, x,Σ, τn), for all x ∈ D

where Ψ(γ, x,Σ, τ) is the Laplace Transform in (10).





Sketch of Euler-Maruyama scheme for a matrix valued process

[ V 0 , D 0]= e i g ( s 0 ) ; e i g S=s o r t ( d i a g ( D 0 ) ’ ) ;. . .v o l a=V 0∗ s q r t ( D 0 )∗V 0 ’∗ ( s q r t ( h )∗ normrnd ( 0 , 1 , d , d ))∗Q;. . .w h i l e min ( d i a g ( e i g S ))< 0 , h=0.5∗h , r e p e a t s tep , end ;

Figure shows log-asset paths with WIS variance (params from [Gauthier and Possamaı, 2010])

S0 1.T 1.h 0.01d 2α 4.8X0 log S0

Table:Evolutionparams

Σ0 :=

[0.5 00 0.5

]

Q :=

[0.35 0

0 0.4

]M :=

[−0.6 0

0 −0.4

]R :=

[−0.5 0

0 −0.5

]

Figure: Log-asset sample paths (Wishart variance)





Table (2) gives the input market parameters and Table (3) the calibrated parametersfor the above model, as in [Gauthier and Possamaı, 2010].

S0 61.90rd 0.03rf 0.00T 1.00

Table: Input data

θ1 0.10 θ2 0.15σ1 0.10 σ2 0.20ρ1 -0.50 ρ2 -0.50v0

1 0.36 v02 0.49

Table: Calibrated parameters

α(CM) 1.75υ(UB) 400

N 1024

Table: Parameters

Strike Price Maturity Analytic Fourier Inv Error %61.90 1 yr 19.4538 19.4538 0.00%61.90 10 yrs 41.3942 41.3940 -0.06%43.33 1 yr 27.6047 27.6047 0.00%43.33 10 yrs 45.2796 45.2793 -0.07%80.47 1 yr 13.9276 13.9276 0.00%80.47 10 yrs 38.2718 38.2719 0.00%

Table: Diagonal Wishart call prices via Fourier inversion





Same (FFT):

S0 61.90rd 0.03rf 0.00T 1.00

Table: Input data

θ1 0.10 θ2 0.15σ1 0.10 σ2 0.20ρ1 -0.50 ρ2 -0.50v0

1 0.36 v02 0.49

Table: Calibrated parameters

η 0.3910λ 0.0157b 8.0346

Table: Parameters

Strike Price Maturity Analytic FFT Inv Error %61.90 1 yr 19.4538 19.4538 0.00%61.90 10 yrs 41.3942 41.3940 0.06%43.33 1 yr 27.6047 27.6052 -0.18%43.33 10 yrs 45.2796 45.2794 0.03%80.47 1 yr 13.9276 13.9284 -0.57%80.47 10 yrs 38.2718 38.2721 -0.07%

Table: Diagonal Wishart call prices via FFT




Bona fide small noise filteringConclusions, addenda & open questions

RECURSIVE CALIBRATION & THE NON-LINEAR FILTERING

NOVEL RESULTS:

Interpreting the factor tracking as a non-linear filtering problem

Thm. 6.2: (Recursive calibration via Gauss Newton)

Main references:[Picard, 1991][Schwartz, 1969][Bell and Cathey, 1993],

R. Trabalzini Recursive calibration - (4) Complete Markets - a Wishart based approach LSE, March 22, 2016 24 / 27




Once performed, the calibration needs not to be repeated in time, from t tot + ∆t;

Observing a vector of option prices at each time ti i ∈ 1, . . . , gives access to asurface of option prices across the strike and maturity ladder;

Modelling the deterministic set of market prices at time kδ by the DE

Yk = h(y ; Xkδ,T − kδ), k = 1, 2, ...

where h(·) is the observation function, the connection with non-linear filteringcan be performed via introducing some noise in the observation process, as

Yk = h(y ; Xkδ,T − kδ) + εBk , Y0 = y0 ∈ Rn, ε > 0.

The small noise filtering comes from a bona fide small noise, in the sense thatwhat is really observed is an DE, but phenomena like quantization, transactioncost, bid-ask spread lead to a stochastic treatment;

The computational advantage of a well-understood computational strategy asEKF will be revealed in the following, where the two stages above (i) predictionand (ii) correction will be interpreted as (i) convolution and (ii) Gauss-Newtonoptimization for inversion of the Jacobian.





Theorem (Recursive calibration: Non-linear filter formulation [Trabalzini, 2015, Thm 6.2])

On a complete filtered probability space (Ω,F, Ft≥0,Q), denote by

Yk := FYk = σY1, ...,Yk the σ-field generated by the observation process and let

X := Xt ∈ D,Ft ; 0 ≤ t ≤ t1

Y := Yk ∈ D,FYk ; k = 1, 2...

be given by equations

dXt = µ(t,Xt )dt + σ(t,Xt )dWt , X0 = x ∈ D, (13)

Yk = h(y ; Xkδ,T − kδ) + εBk , Y0 = y0 ∈ Rn, ε > 0. (14)

where Bk , k = 1, 2... i.i.d.N(0, 1), and W := Wt ∈ Rn,Ft ; 0 ≤ t ≤ t1 is a Rn BM,Ftt∈[0,T ] is the natural filtration of W , Bk ⊥ W , k = 1, 2, ...,

y := yuT, y ∈ Rn

, u ∈ 1, . . . , n

is the vector of strikes and the map h : Rn × D × [0,T ] 7→ Rn is given by

h(y ; x, τ) = h(yu ; x, τ)T, u ∈ 1, . . . , n

:=

E[(

yue〈0,Xτ 〉 − e〈a,Xτ 〉)1 〈a,Xτ 〉≤ ln yu

∣∣∣X0 = x]T

,

where a ∈ Rn, τ ∈ [0,T ] and 〈·, ·〉 the usual inner product. Then the factor process in equation(13) can be recovered from the observation process (the option prices) in equation (14) via thecombination of a time update step and a convergent measurement update step.





ConclusionsIn this talk, market completion - i.e. using information contained in a spanningset of traded assets for pricing and hedging purposes, has been investigated in anentirely novel way. A rich but tractable class of factor processes based on theWishart family has been singled out and computable conditions have beenfurnished under which any contingent claim in the market can be replicated by acombination of options and the forward. The dynamic hedging problem underscrutiny has further been connected in a novel way to non-linear filteringtechniques and Gauss-Newton optimization ideas.

Addenda & Open questionsNumerics for stability of dynamic hedging strategiesApplication in automarket: the opaque nature of Wishart based processes parameters isno issue for automatic markets (FX space)Applications in Risk Management and Portfolio OptimizationBasket case: where is the correlation coming from in single asset case?Wing behaviour: asymptoticsADJ: adding jumps to log-asset and/or variance process





Bell, B. M. and Cathey, F. W. (1993).

The Iterated Kalman Filter Update as a Gauss-Newton Method.IEEE Transactions on automatic control, 38:294–297.

Benabid, A., Bensusan, H., and Karoui, N. E. (2008).

Wishart Stochastic Volatility: Asymptotic Smile and Numerical Framework.https://hal.archives-ouvertes.fr/hal-00458014v2.

Benabid, A., Bensusan, H., and Karoui, N. E. (2010).

Wishart Stochastic Volatility: Asymptotic Smile and Numerical Framework.

Bru, M. (1991).

Wishart Processes.Journal of Theoretical Probability, 4(4).

Cuchiero, C., Filipovic, D., Mayerhofer, E., and Teichman, J. (2010).

Affine processes on positive semidefinite matrices.http://papers.ssrn.com/sol3/Papers.cfm?abstract_id=1481151.

da Fonseca, J., Grasselli, M., and Tebaldi, C. (2008).

A multifactor volatility Heston model.Quantitative Finance, 8:591–604.

Davis, M. H. A. (2004).

Complete-market models of stochastic volatility.Proceedings of the Royal Society: Mathematical, Physical and Engineering Sciences, 460:11–26.

Davis, M. H. A. and Ob loj, J. (2008).

Market completion using options.In Stettner, L., editor, Advances in Mathematics of Finance, volume 43, pages 49–60. Banach Center Publications, PolishAcademy of Sciences, Warsaw.

Duffie, D., Filipovic, D., and Schachermayer, W. (2003).

Affine Processes and Application in Finance.


http://papers.ssrn.com/sol3/Papers.cfm?abstract_id=1481151




Annals of applied probability, pages 984–1053.

Duffie, D. and Kan, R. (1996).

A Yield-Factor Model of Interest Rates.Mathematical finance, 6:379–406.

Filipovic, D. and Mayerhofer, E. (2009).

Affine diffusion processes: Theory and applications.http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1333155.

Gauthier, P. and Possamaı, D. (2009).

Efficient Simulation of the Wishart Model.http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1474728.

Gauthier, P. and Possamaı, D. (2010).

Efficient Simulation of the Double Heston Model.http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1434853.

Graczyk, P. and Mayerhofer, E. (2012).

Stochastic Analysis Methods in Wishart Theory.http://arxiv.org/abs/1201.6634v2.

Kramkov, D. and Predoiu, S. (2014).

Integral representation of martingales motivated by the problem of endogenous completeness in financial economics.Stochastic Processes and their Applications, 124:81–100.

Picard, J. (1991).

Efficiency of the extended Kalman filter for non-linear systems with small noise.SIMA J. Appl. Math., 51:843–885.

Romano, M. and Touzi, N. (1997).

Contingent claims and market completeness in a stochastic volatility model.Mathematical Finance, 7:399–410.

Ross, S. A. (1976).


http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1333155






Options and efficiency.The Quarterly Journal of Economics, 90:75–89.

Rouah, F. D. (2013).

The Heston Model and Its Extensions in Matlab and C].John Wiley & Sons, Inc., Hoboken, New Jersey.

Schwartz, S. C. (1969).

On the Estimation of a Gaussian Convolution Probability Density.SIAM Journal on Applied Mathematics, 17:447–453.

Trabalzini, R. (2015).

Complete markets and Wishart stochastic variance.working paper, Imperial College.


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