recent topics in financial mathematics

Recent topics in Financial Mathematics

Martin FordeKing’s College London

Apr 2016

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Martin Forde King’s College London Apr 2016 Recent topics in Financial Mathematics

Contents

I The Robust hedging problem with tradeable barrier options.

I Portfolio optimization - linear and non-linear price impact underexponential utility.

I Portfolio optimization under proportional transaction costs with logutility and a jump-to-default, using shadow prices.

I Fractional stochatic volatility models - basic properties of fBM,large deviations, application to small-time asymptotics for a stochasticvolatility model driven by fBM; extensions using rough paths theory.

I Optimal order execution for an Almgren&Chriss-type model andthe Bayraktar-Ludkovski power-law limit order book model withstochastic liquidity.


The Skorokhod embedding problem and robust hedging

I Compute supτ∈T (µ) E(Φ(Bτ∧., τ)), where Φ is some path-dependentfunctional of B and T (µ) is the set of all stopping times for whichBτ ∼ µ with

∫µ(dx)dx = 0 and

∫|x |µ(dx) <∞ and the stopped

process (Bτ∧t)t≥0 is uniformly integrable.I Azema-Yor/Vallois: Φ = 1Bτ>x(1Lτ>x), τ = inf{t : Bt ≤ K∗µ(Bt)}(τ = inf{t : Bt /∈ (φ−(Lt), φ

+(Lt))}), proved using excursion theory.- Perkins: Φ = 1Bτ<x , τ = inf{t : Bt /∈ (−γ+(Bt), γ

−(−B t))}.- Root/Rost: Φ = ∓(τ − K )+, τ = inf{t : sgn(R±(Bt)− t) = ±1}.- [BHR01]: supτ1∈T (µ1)≤τ2∈T (µ2) P(Bτ2 > y), then τ1 = τ2 = τAYµ1 if

ξ2(Bτ1 ) ≥ K∗µ1(y)(Bτ1 ), else τ1 = τAYµ1 , τ2 = inf{t : Bt ≤ ξ2(Bt)}, where

ξ2(y) = argminζ2≤y [ c2(ζ2)y−ζ2

− 1ζ2>K∗µ1(y)(

c1(ζ2)y−ζ2

− c1(K∗µ1(y))

y−K∗µ1(y) )].

- [HK13]: Minimize E(|XT2 − XT1 |) over all martingales X : XT1 ∼ µ,XT2 ∼ ν, T1 < T2. Optimal joint law is such that XT2 |XT1 has a trinomialdistribution with points p(XT1 ),XT1 , q(XT1 ) for some functions p, q.

I In [FK15], we consider: P(µ) := supτ :(Bτ ,Bτ )∼µ E(Φ) and

supτ :Bτ∼µ,Bτ∼ν E(Φ) ; we adapt existing results in [GTT15] + use theRogers necc+suff. condition on µ; we work under Wasserstein topologyW1 on space of admissible µ’s (instead of peacocks as in [GTT15]).


I Usual problem: compute minimal superhedging cost for an path-dependent option payoff Φ(X .) on an (unspecified) continuousmartingale stock price process X , when we have observed tradeableEuropean call options at all strikes K with a single fixed maturity T att = 0, and we can dynamically trade X . We adapt this problem tocompute minimal superhedging cost D(µ) when we can also tradebarrier options, and we prove that D(µ) = P(µ).I Recall the DDS time-change result: any continuous martingale Xcan be written as time-changed Brownian motion Xt = B〈X〉t for someBrownian motion B and Bt = XAt where At = inf{s : 〈X 〉s > t}.I The duality result shows that P(µ) equals

supP∈Mµ

EP[Φ(X )]

= infλ∈Λ{∫λ(x , y)dµ(x , y)|∃γ :

∫ T

0

γtdXt + λ(XT ,XT ) ≥ Φ(X ) a.s.∀P ∈M}

Mµ is the set of all P: X is a cts martingale and (XT ,XT ) ∼ µ, where µis the measure implied by observed barrier option prices. UseProkhorov’s thm and bi-conjugate theorem applied to P(µ) under W1

to prove P(µ) = infλ∈Λ supτ E[Φ− λ(Bτ ,Bτ ) + µ(λ)]. ThenDoob-Meyer to the associated Snell envelope, MRT to construct γt .


Portfolio optimization with transaction costs - linear andnon-linear price impact

I Consider a market with a safe asset earning zero interest and a riskyasset whose best quoted price St satisfies GBM:

dSt = St(µdt + σdWt) .

I Unlike a frictionless market, trades in the risky asset are not realized atthe best quote St , but rather at a less favourable price which effectivelypenalizes the trader for making large trades in a short period of time.More precisely, we assume that the average price for trading is

St := St(1 + λSt θt)

where θt is the number of shares held at time t, which we assume isdifferentiable in t. Let Ct denote the cash position, which must evolve as

dCt = −Stdθt .

I We now let ut = θtSt , Xt = θtSt + Ct denote the total wealth, andYt = θtSt the risky wealth.


I We assume the investor has exponential utility and a long-timehorizon, and thus trades to maximize lim infT→∞− 1

αT logE(e−αXT ).

I This is a stochastic control problem - we are looking for the optimalut process. From standard stochastic control arguments, for anyadmissible control ut , V (t,Xt ,Yt) is a supermartingale and is amartingale for the optimal control ut , and applying Ito’s lemma toV (t,Xt ,Yt) +setting drift=0, V (t, x , y) satisfies a HJB eq. We thensubstitute the ansatz V (t, x , y) = −e−αxeαβteα

∫ y0q(ζ)dζ .

I In [FWZ15], we find that the optimal trading policy is

ut = u(Yt) ∼ −ασ2(Yt − Y )

σ√

2α

1√λ

(λ→ 0)

where Y = µ/(ασ2) is the frictionless target, i.e. the optimal Y -value

when λ = 0 (which is constant), and β(λ) = µ2

2ασ2 − c1

√λ+ o(

√λ). We

expect u(y) to blow up as λ→ 0, because in the frictionless case

Yt = θtSt = Y so θt = YSt

, which clearly is not differentiable in t a.s.

I We later extend to the non-linear price impact :St = St(1 + λ|St θt |γsgn(θ)). c1(γ) is now determined (numerically) asthe unique value for which the solution s(w) to the non-linear ODE:

−c + w2 − |s(w)|1+ 1γ + s ′(w) = 0 satisfies s(w) ∼ |w |

2γ1+γ as |w | → ∞.


In the upper graph , we have plotted a Monte Carlo simulation of how thenumber of shares θt = Yt/St evolves using the (asymptotically) optimal

trading strategy u(y) = −√ασ(y−Y )√

21√λ

under linear price impact (dark

blue) against the evolution of θt in the frictionless Merton setting (lightblue) and the optimal θt process under the [GMK15] model (red) withproportional transaction costs but no price impact. We see that theprice impact curve smoothly tracks the frictionless Merton portfolioand the transaction costs curve is a.s. piecewise constant because ofthe no-trade region. Here the parameters are µ = .05, σ = .1,λ = .0.00001, α = 1, the size of the proportional transaction costs isε = .0001 and the time horizon here is t = 1 year. The lower graph showsthe corresponding evolution of the risky wealth under all three models.


Transaction costs with jumps

I First consider a (frictionless) stock price process which evolves as

dSt = St−[µtdt + σtdWt − dJt ] (1)

where J is a standard Poisson process with intensity λJ and µt , σtare progressively measurable process with

∫ t

0µ2sds,

∫ t

0σ2s ds <∞.

I Let φt denote the amount of stock. Then Vt = V0 +∫ t

0φudSu is our

total wealth at time t, and we wish to maximize E[logVT ].I From the change of variable formula we can show that

E(VT ) = V0 + E[

∫ T

0

e−λJ t(πt−µt −1

2π2t−σ

2t + λJ log(1− πt−)) dt] .

I Differentiating with respect to π, the optimal π∗t− satisfies

µt − π∗t−σ2t −

λJ1− π∗t−

= 0 (2)

and for µt ∈ (0, 1) the unique π∗t− = 12 [θt + 1−

√(1− θt)2 + 4λJt ]

where θt = µt/σ2t and (λJ)t = λJ/σ

2t .


I (See Appendix for primer on shadow prices). With transactioncosts, we now make the following ansatz for the shadow price:assume that S0 = S0 = 1, and if S increases from 1 to s withoutsetting a new minimum, then we guess that St = g(St) for0 ≤ t ≤ τs , for some g ∈ C 2 and target value s, to be determined.

I Make an initial trade at t = 0, + postulate that the optimal tradingstrategy involves no further trading until τs (i.e. a no-trade region).

I Applying Ito’s formula to St , we obtain

dSt = dg(St−) = g ′(St−)dSt +1

2g ′′(St−)σ2S2

t−dt − g(St−)dJt

⇒ dSt = St [µ(St)dt +1

2σ(St)dWt − dJt ]

where µ(s) = [g ′(s)sµ+ 12g′′(s)σ2s2]/g(s), σ(s) = g ′(s)σs/g(s).

Hence we see that that St follows a process of the form in (1).

Setting c = ϕ0t /ϕt we see that πt− = ϕg(St)

ϕ0+ϕg(St)= 1

1+ cg(St )

I Combining this with (2) we obtain the non-linear ODE:

µ(s) − 1

1 + cg(s)

σ(s)2 − λJ

1− 11+ c

g(s)

= 0 .


I Setting π∗(s) = g(s)g(s)+c = 1

1+c/g(s) , then

1

2σ2s2(π∗)′′(s) + sµ (π∗)′(s)− λJπ∗(s) = 0.

which is an Euler ODE, so π∗(s) = Asα+

+ Bsα−

for some α±.I The true shadow price process for all t is then given by

St = mtg(Stmt

)

with mt defined as in [GMS13].I (ϕ0, ϕ) is self-financing strategy, and satisfies

dϕ0t = −Stdϕt = −mtdϕt

an such thatϕ0

t

ϕt= cmt , so we see that we buy when mt decreases, as

required for the definition of a shadow price process. Thus for t < τs

dϕ0t = cmtdϕt + cϕtdmt = −cdϕ0

t +ϕ0t

mtdmt

sodϕ0

t

ϕ0t

= 11+c

dmt

mt(we can perform similar analysis on the sell

boundary S = (1− λ)S).


Asymptotics and the implied welfare

I Using the implicit function theorem we can show that

s(λ) = 1 +1

A13

λ13 + o(λ

13 ) ,

c(λ) = c +φ′(1)

A13

λ13 + o(λ

13 )

for some A which is easily computed in terms of the parameters.

I Welfare: Expected long term log utility of wealth is

δ := limT→∞

E(logVT ) = V0 + limT→∞

E[

∫ T

0

e−λJ t Υ(Stmt

) dt]

where

Υ(y) := π∗(y)µ(y)− 1

2π∗(y)2σ(y)2 + λJ log[1− π∗(y))] .

I log St

mtis identical in law to a geometric Brownian motion

dYt = YtσdBt with two reflecting barriers at 1 and s = s(λ). Thus

δ = V0 + limT→∞

E[

∫ T

0

e−λJ tΥ(Yt) dt] = V0 +1

λJE[Υ(Yτ )] .


Numerics

1.005 1.010 1.015 1.020 1.025 1.030

-1. ´ 10-6

-8. ´ 10-7

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Figure: We set µ = .01, σ = .2, λ = .000001, λJ = .0001, we find (numerically)that the correct c-value is c = 3.102951 and s = 1.03231. Here we haveplotted g(s) − s and −λs, and we see that the two curves are tangential at s.


Fractional Brownian motion and stoc vol models

Recall that a zero-mean real-valued Gaussian process (Zt)t≥0 is astochastic process such that on any finite subset {t1, .., tn} ⊂ R ,(Zt1 , ...,Ztn) has a multivariate normal distribution with mean zero. Thelaw of a Gaussian process is entirely determined by its covariancefunction R(s, t) = E(ZsZt). A zero-mean Gaussian process BH

t is calledstandard fractional Brownian motion (fBM) with Hurst parameterH ∈ (0, 1) if

R(s, t) = E(BHt BH

s ) =1

2(|t|2H + |s|2H − |t − s|2H) .

I fBM is continuous a.s. and H-self-similar, i.e. (Bat)t≥0(d)= aH(Bt)t≥0

(i.e. they have the same finite dimensional distributions). For H 6= 12 , BH

does not have independent increments;I BH

t − BHs ∼ N(0, |t − s|2H); thus BH has stationary increments.

I Bk − Bk−1 and Bk+n − Bk+n−1 are positively correlated if H ∈ ( 12 , 1)

and negatively correlated if H ∈ (0, 12 ). Thus BH is persistent when

H > 12 and anti-persistent when H < 1

2 .

I Sample paths of BH are α-Holder-continuous, for all α ∈ (0,H).

I BHt is neither a Markov process nor a semimartingale (see [Nual96]).


Large deviations for fBM

I There is a Volterra-type representation of fBM on the interval [0, t]:

BHt =

∫ t

0

KH(s, t)dBs .

I√εBH satisfies the LDP on C0[0, 1] as ε→ 0 with speed 1

ε and rate

function Λ(f ) = 12

∫ 1

0h(s)2ds if f (t) =

∫ t

0KH(s, t)h(s)ds ∀t ∈ [0, 1]

and some h ∈ L2[0, 1], and Λ(f ) =∞ otherwise.I The space of functions with Λ(f ) <∞ is known as the reproducingkernel Hilbert space H of fBM, with 〈f1, f2〉H = 〈h1, h2〉L2[0,1].I We have used this to compute small-t asymptotics for the model{

dSt = Stσ(Yt)(√

1− ρ2dWt + ρdBt) ,dYt = dBH

t

(3)

I In [FZ15] we show that tH−12 log St satisfies the LDP as t → 0 with

speed 1t2H and rate I (x) = inf f∈H1

[ (x−ρG(f ))2

2ρ2F (KH f ′)+ 1

2‖f ‖2H1

], where

F (f ) =∫ 1

0σ(f (s))2ds, G (f ) =

∫ 1

0σ((KH f

′)(s))f ′(s)ds,

(KH f )(t) =∫ t

0KH(s, t)f (s)ds , σ(.) can be unbounded but must satisfy

a linear growth condition which still allows for moment explosions for St .Martin Forde King’s College London Apr 2016 Recent topics in Financial Mathematics

Generalizing the model using rough paths theory

I Rough paths theory is concerned with differential equations of type

dYt =d∑

i=1

V i (Yt)dXit

where the driving signals X it may be rougher than standard Brownian

motion (rough in the sense of Holder continuity) e.g. fBM.I Eqs of this type can be solved pathwise using rough paths theory.Lyons proved that the Ito map which takes the signal X to the solutionY is continuous in an appropriate rough path topology, which forH > 1

3 requires keeping track of the higher order iterated integrals of X :

X is,t = X i

t − X is , Xij

s,t =

∫ t

s

∫ r

s

dX iudX

jr .

I We let xs,t = (1,Xs,t ,Xs,t). We can then define a norm and metric onthe space in which x lives, and Lyon’s Universal limit theorem says thatsthe Ito map is continuous under the topology associated with this norm,which makes proving LDPs much simpler, using the contraction principle.


I We prove a similar small-time LDP for a general model where thestock price St satisfies

dSt = St [σ(Yt)dWt + η1dB1t ],

dYt = V 1(Yt)dBH1t + V 2(Yt)dB

H2t (4)

where BH1t =

∫ t

0KH1 (s, t)dB1

s , BH2t =

∫ t

0KH2 (s, t)dB2

s and W ,B1,B2 are

3 independent standard Brownian motions and 14 < H1 <

12 < H2 < 1,

under suitable regularity conditions on the coefficients.

I To construct the solution to (4), we let Dn be a sequence of partitionsof [0,T ] with mesh size → 0. Let πV (0, y0; x) denote the solution to thecontrolled ODE dYt = V 1(Yt)dX

1t +V 2(Yt)dX

2t when X 1

t ,X2t have finite

variation. Then the random sequence of ODE solutions πV (0, y0; xDn)(where xDn is the piecewise linear approximation to x) is Cauchy inprobability under the uniform topology and its unique limit point is aC ([0,T ],R)-valued random-variable which does not depend on the choiceof sequence Dn, and is identified as the random RDE solution (Y )t≥0.


Numerical results: the small-maturity implied volatilitysmile

+

+

+

+

+ +

+

+

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0.1005

0.1010

0.1015

0.1020

0.1025

0.1030

Figure: Here we have plotted the small-maturity implied volatility smile for themodel in (3) for ρ = −0.1, σ(y) = 1 + .05 tanh(y), H = 0.25 and t = .002.verses the values obtained by Monte Carlo using the Willard conditioningmethod with 500,000 simulations and 100 time steps.


Optimal liqidation for the Gatheral-Schied problem withstochastic liquidity

I We consider an extension of an Almgren&Chriss-type model oftransient and permanent price impact, where the price that we pay for astock is

St = St + ηt xt + γ(xt − x0)

where dSt = StσdWt , dηt = ηtαdBt and dWtdBt = ρdt, and xt is thenumber of shares held at time t which we assume is differentiable in t.

I The η term corresponds to the temporary (transient) price impact,which effectively penalizes a trader for making a large change in his stockposition in a small time period, but goes away as soon as he stopstrading. The γ term captures the effect of permanent price impact,whereby a large number of buy (sell) orders permanently moves theunderlying stock price up (down), and S is known as the unaffectedstock price.


I Assume x0 = X > 0, and we implement a liquidation strategy xtover [0,T ] so xT = 0. Then it’s well known that the cost of doing this is:

RT =

∫ T

0

St xtdt =

∫ T

0

St xtdt +

∫ T

0

(ηt xt + γ(xt − x0))xtdt

= ST xT − S0x0 −∫ T

0

xtdSt +

∫ T

0

(ηt x2t + γ(xt − x0)xt)dt

= −S0x0 −∫ T

0

xtdSt +

∫ T

0

ηt x2t +

1

2γ(x2

T − x20 ) + γx2

0

= −S0X −∫ T

0

xtdSt +1

2γX 2 +

∫ T

0

ηt x2t dt .

Thus E(RT ) = −S0X + 12γX

2 + E(∫ T

0ηt x

2t dt) and if ηt = 0 for all t,

we see that E(RT ) is independent of the trading strategy x .I Now recall the Gatheral-Schied[GS11] optimal liquidation problem:

V (T ,S , η,X ) = infv∈V

E( ∫ T

0

(λxtSt + ηtv2t )dt |S0 = S , η0 = η, x0 = X

)(1)

where vt = xt and V is the space of all progressively measurable processes

v for which xT = X +∫ T

0vsds = 0 with certain integrability conditions.


I The expectation in (1) (aside from the constants in (5)) is theexpected cost of liquidation plus an additional λ-term to penalize thetrader for holding large positions. The HJB equation for this problem is

VT =1

2σ2S2VSS + ρσαSηVSη +

1

2α2η2Vηη + λSX + inf

v∈R[ηv2 + vVX ] (5)

with limT→0 V (T ,S , η,X ) = 0 if X = 0 and +∞ otherwise (i.e. infinitepenalty if xT 6= 0) and we see that v∗ = − 1

2V∗X/η.

I Solving the HJB eq, and using a verification argument to verifyoptimality, we find that the unique optimal trade execution strategyattaining the infimum is

x∗t =T − t

T[X − λT

4

∫ t

0

Ssηs

ds]

which implies that x∗t = − x∗tT−t −

λSt(T−t)4ηt

and the value function is

V ∗(T ,S , η,X ) =ηX 2

T+λSTX

2+λ2S2[2− 2eTθ + Tθ(2 + Tθ)]

16ηθ3

where θ = σ2 − 2αρσ + α2.Martin Forde King’s College London Apr 2016 Recent topics in Financial Mathematics

Asymptotics

I We have the asymptotic formulae:

V (T ,S , η,X ) =X 2η

T+

1

2SXλT − S2λ2

48ηT 3 − S2λ2θ

192ηT 4 + O(T 5)

V (T ,S , η,X ) = V (T ,S , η,X )|α=0

+S2λ2ρ (6 + 4Tσ2 + T 2σ4 + 2eTσ

2

(Tσ2 − 3))

8ησ7α + O(α2) .

The correction term in the last equation is the leading order correctionto the expected liquidation cost E(RT ) due to stochastic price impact i.e.α, which it turns out has the same sign as ρ. From the penultimateequation, we see that the effect of α is not felt until O(T 4), i.e. veryclose to the terminal time. See left plot on title page for Monte Carlosimulation of x∗t .I xt satisfies a linear coupled FBSDE, using a similar convex analysisargument to the constrained problem in [BSV15] by setting the Gateauxderivative of the functional to be minimized to zero.I Can extend to include stochastic permanent price impact or stochasticinterest rates, and x∗t remains unchanged if S , η are martingale diffusions(i.e. the optimal liquidation strategy is robust in some sense).


A power-law limit order book model

I Following BL14, assume that the unaffected price process St is an(unspecified) martingale and a trader places Nt ∈ N ask orders inthe LOB at price St + δt at time t, where he is free to choose δ > 0as the stochastic control.

I BL14 assume that the number of shares held (Nt)t≥0 is equal to N0

minus a Poisson counting process with controlled intensityΛ(δt) = λδ−αt for α > 1, so trades arrive randomly one at a timeand the trader keeps selling until all the stock is sold, so his cashposition evolves as dCt = −(St + δt)dNt until Nt = 0.

I Let T denote time to maturity and n represent the current stockholding. Then BL14 seek to maximized expected revenue, for whichthe value function V (n,T ) satisfies the (discrete space) HJB eq

VT = supδ > 0

λ

δα[V (n − 1,T )− V (n,T ) + δ]

with boundary conditions:

1. V (n, 0) = 0 for all n (number of shares) - no more revenue isgenerated when time is up.

2. V (0,T ) = 0 for all T - no more revenue after all shares sold.Martin Forde King’s College London Apr 2016 Recent topics in Financial Mathematics

The fluid limit

I We can consider the fluid limit where the share increment is nowequal to ∆� 1 and the Poisson rate is re-scaled by 1/∆. The valuefunction is now for x ∈ {0,∆, 2∆, ...}, for which the HJB equation is

V∆T = sup

δ > 0

λ

δα∆[V∆(x −∆,T )− V∆(x ,T ) + δ∆] .

I As ∆→ 0, we have

VT = supδ > 0{λδ−α(δ − VX )}

which now implies that we now have a continuous stream of tradesso that dxt = −λδ−αt dt, where xt is the number of shares at time t.

I We extend this setup by assuming that λ is stochastic:dλt = βλtdZt where Z is a standard BM independent of S and N,and this is the model that we work with, for which the HJB eq is now

VT =1

2β2λ2Vλλ + sup

δ > 0{λδ−α(δ − VX )} .


Solution

I Using a separable ansatz of the form

V (X ,T , λ) = Xα−1α (λg(T ))

1α

yields that g(T ) = 1κ (1 − e−κT ), where κ = (α−1)β2

2α .I Thus we obtain the optimal spread δ∗t in closed form as

δ∗t =( λtκx∗t

(1− e−κ(T−t)))1/α

.

I The optimal number of shares evolves as:

dx∗t = −λtδ−αt dt = − κx∗t1− e−κ(T−t)

dt

i.e. we trade in a way that ensures that x∗t is deterministic, but δ∗tis not and varies stochastically with λt .

I Note the special case where β = 0 the above reduces to

dx∗t = − x∗tT − t

dt

with solution x∗t = T−tT X .


References

Almgren, R. and N.Chriss, “Optimal execution of portfoliotransactions”, J. Risk, 3:5,39, 2001.

Bank, P., H.M.Soner and M.Voß, “Hedging with Transient PriceImpact”, preprint.

Bayraktar, E. and M.Ludkovski, M., “Liquidation in limit orderbooks with controlled intensity”, Mathematical Finance, 24(4), pp.627-650, 2014.

H.Brown, D.G.Hobson, and L.C.G. Rogers, “The maximummaximum of a martingale constrained by an intermediate law”,Probab. Theory Relat. Fields, 119(4):558-578, 2001.

Forde, M. and R.Kumar, “Large-time asymptotics for a generalstochastic volatility model with a stochastic interest rate, using theDonsker-Varadhan LDP”, submitted.

Forde, M. and R.Kumar, “Duality for three robust hedging problems- finite strikes and maturities with tradeable exotics, or givenjoint/marginal laws for the minimum and the terminal price, withR.Kumar, 2015, submitted.

Forde, M., M.Weber and H.Zhang,“Portfolio optimization with linearand non-linear price impact under exponential utility”, preprint.

Forde, M. and H.Zhang, “Asymptotics for rough stochastic volatilityand Levy models”, with H.Zhang (2015), submitted.


Gatheral, J. and A. Schied, “Optimal trade execution undergeometric Brownian motion in the Almgren and Chriss framework”,International Journal on Theoretical and Applied Finance, 14 (3),353-368, 2011.

Guasoni, P. and J.Muhle-Karbe, “Horizons, High Risk Aversion, andEndogeneous Spreads”, Mathematical Finance, 25, no.4, p.724-753,2015 .

[GMS13] Gerhold, S., J.Muhle-Karbe and W.Schachermayer, “TheDual Optimizer for the Growth-Optimal Portfolio under TransactionCosts”, Finance and Stochastics, Vol. 17 (2013), No. 2, pp. 325-354.

Guo, G., X.Tan and N.Touzi, “Optimal Skorokhod embedding underfinitely-many “marginal constraints”, 2015.

D.Hobson&M.Klimmek, “Robust price bounds for the forwardstarting straddle”, Finance and Stochastics Vol 19, Issue 1, 189-214,2015

Nualart, D., “The Malliavin Calculus and Related Topics”, 2nd edn.,Probability and its Applications, New York, Springer, Berlin, 2006.

Rogers, L.C.G., “The joint law of the maximum and the terminalvalue of a martingale”, Prob. Th. Rel. Fields 95, pp. 451-466, 1993.Martin Forde King’s College London Apr 2016 Recent topics in Financial Mathematics

Definition of a shadow price

I Definition. A shadow price is a semimartingaleSt ∈ [(1− λ)St ,St ], such that the optimal trading strategy (ϕ0

t , ϕt)for a fictitious market with price process St and zero transactioncosts exists, has finite variation and the number of stocks ϕt onlyincreases when St = St and decreases when St = (1− λ)St .

I Clearly any price process St with zero transaction costs which lies in[(1− λ)St ,St ] leads to more favourable terms of trade than theoriginal market with transaction costs. But a shadow price process isa particularly unfavourable model, for which it’s optimal to only buywhen St = St , sell when St = (1− λ)St + do nothing in between.


Why do we use shadow prices?

I Proposition (Corollary 1.9 in Schachermayer et al.[GMS13]).Let St be a shadow price process whose optimal trading strategy (forzero transaction costs) is given by (ϕ0

t , ϕt), with ϕ0t , ϕt ≥ 0. Then

under non-zero transaction costs, we have

sup(ψ0,ψ)

E[logVT ((ψ0, ψ))] ≥ E[logVT ((ϕ0, ϕ))]

≥ E[logVT ((ψ0, ψ))] + log(1− λ)

for any admissible (ψ0, ψ). Thus if we choose λ suff small so that| log(1− λ)| < ε and take the sup over all (ψ0, ψ), we see that(ϕ0, ϕ) is an ε-optimal trading strategy for the original problem.

I Or take liminf as T →∞ + sup over all admissible strategies, weobtain

lim infT→∞

1

TE[logVT ((ϕ0, ϕ))] = sup

(ψ0,ψ)

lim infT→∞

1

TE[logVT ((ψ0, ψ))] .

Thus the optimal portfolio for the shadow price process isasymptotically optimal for the original problem under transactioncosts, as λ→ 0 and/or as T →∞.


recent topics in financial mathematics

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