options hedging with market impact [r. almgren] presentation. 2012

7/28/2019 Options Hedging With Market Impact [R. Almgren] Presentation. 2012

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Robert Almgren

Option Hedgingwith Market Impact

New York UniversityCourant Institute of Mathematical Sciences

Market Microstructure, Paris, Dec 2012



Outline

1. The problem

2. Formulation

3. Solution

4. Examples

2

Work with Tianhui Michael LiPrinceton: Bendheim Center and ORFE



1. Option hedging (version 1)

3

time tT

Asset price PtP t = P 0 + W t + himpacti

Hedge portfolio Xt shares

X t = X 0 +Z T

0

✓s ds

Final

mark-to-market

valueg0(P T ) + X T P T + cash

evaluate on

mean and variance

option expiryor

market close



Option hedging (version 2)

4

time tClose

T

Asset price Pt

P t = P 0 + W t + himpacti

Hedge shares Xt

X t = X 0 +Z T

0

✓s ds

Mark-to-marketvalue

Open

T’

X T 0 = X T

Overnight

g0(P T 0) + X T P T 0 + cash



Applications

1. Broker execution algorithm:Client specifies ∆ and Γ (possibly varying)

Execute to achieve optimal hedge at closeone direction trading (buy or sell)

2. What happens after you buy an option?Seller must hedgeWhat does his hedging do to price process?

5



Questions

1. What is a reasonable market model?

2. What do solutions look like?

3. How do they compare to Black-Scholes?

6



Market impact models

Two types of market impact(usually both are active):

• Permanentdue to information transmissionaffects public market price

• Temporary

due to finite instantaneous liquidity“private” execution price not reflected in market

Many richer structures are possible

7



Permanent impact

8

t = instantaneous rate of trading

X t = X 0 +

Z T 0

✓s ds

Linear to avoid arbitrage (Huberman & Stanzl, Gatheral)

P t = P 0 + W t + ⌫(X t X 0)

G(✓)=

⌫ ✓

dP t = dW t + G(✓t)dt



Temporary impact

9

We trade at P t 6= P t

P t depends on instantaneous trade rate ✓t

Require finite instantaneous trade rate

imperfect hedging

P t = P t + H(✓t)



Example: bid-ask spread

10

✓t

buy at ask

sell at bid

P t

P t s

P t = P t +1

2s sgn(✓t)

1

2s sgn(✓t) · ✓t t =

1

2s|✓t|t

“Linear” model: cost to trade sharest t



Solutions with bid-ask spread cost

11

Ideal Black-Scholes hedge

Target band(no-trade region) Actual hedge holding

Davis & Norman, Shreve & Soner,Cvitanic, Cvitanic & Karatzas



Critique of linear cost model

independent of trade size

not suitable for large traders

in practice, effective execution near midpointspread cost not consistent with modern cost models

liquidity takers act as liquidity providers

12



Proportional cost model

13

t

H(0) = 0

concave(empirical)

Linear for simplicity

Quadratic cost:

H( t)

H(✓) =1

2✓

H(✓) · ✓t =

1

2✓

2t

P t = P t + H(✓t)



Our solutions with proportional cost

14

Ideal Black-Scholes hedge

Actual hedge holding

pursuit

Gârleanu & Pedersen:investment with proportional cost

✓t = h (T t)

·

X t target



2. Formulation

Market model

15

X t = X 0 +

Z T

0

✓s ds

P t = P 0 + W t + ⌫(X t X 0)

˜P t = P t +

1

2 ✓t

Hedge holding:

Public market price:

Private trade price:

F t = filtration of W t



Black-Scholes option value

16

g(T,p) = g0(p)

g(t, p), t < T , p 2 R

g +1

2

2g

00= 0

Final value specified

Intermediate values defined by Black-Scholes PDE

Def:(t,p) = g0(t,p)

(t,p) = 0(t,p) = g00(t,p)



17

Short

call

g(t,p) = option payout to trader

(t,p) = g0(t,p) (t,p) = 0(t,p) = g00(t,p)

Longcall

Short

put

Longput

> 0

< 0 < 0

< 0

< 0

> 0

> 0 > 0

p p

pp

g

gg

g

Γ= sign and size of option position



Final portfolio value

18

RT = g(T,P T ) + X T P T

Z T 0

P t ✓t dt

Option value Portfolio value Cash spent

Mark to marketwithout transaction costs

1

2⌫X

2Include permanent impact

in liquidation cost:

We neglect: gives manipulation opportunities

dominated by risk aversion



Integrate by parts

19

RT = R0 +

Z T 0

X t + g0(t,P t)

dP t

1

2

Z T 0

✓2

t dt

= R0 + Z T

0

Y tdP

t

1

2

Z T

0

✓2

tdt

= R0 +

Z T 0

Y t dW t +

Z T 0

Y t ⌫✓t dt 1

2

Z T 0

✓2

t dt

R0 = g(0, P 0) + X 0 P 0Initial value:

Mishedge:

(constant)

Y t = X t (t,P t) = X t + g0(t,P t)



Mean-variance evaluation

20

infinitevariation

smooth

Y t = X t + g0(t,P t) 6= 0

RT = R0 +

Z T

0

Y t dW t + ⌫

Z T

0

Y t ✓t dt 1

2

Z T

0

✓2

t dt

ERT = R0 + ⌫ E

Z T

0Y t ✓t dt

1

2E

Z T

0✓

2t dt

VarRT = complicated

RT is random: optimize expectation and variance

mishedge



Variance of RT

21

Small portfolio size, or

“market power”(Almgren/Lorenz 2007)

µ = X/T ⇤ T

price impact of trading whole position

price changefrom volatility

Neglect uncertainty of market impact term

in comparison with price uncertainty

(“Mean-quadratic-variation” Forsyth et al 2012)

VarRT ⇡ VarZ T 0

Y t dW t = 2Z T 0

Y 2t dt



Version 2: Overnight risk

T = market close todayT’ = market open tomorrow

23

RT 0 = g(T 0, P T 0) + X T P T 0 Z T

0

P t ✓t dt

= RT + Y T P T 0 P T

+

Z T 0T

hg0

t, P t

g0

T, P T

idP t

= RT + Y T P T ⇠

P T ,⇠ have mean zero



Version 2 objective function

24

Random variables

Terminal

mishedge

⇠ distribution depends only on P T

P T mean 0, independent of F T

inf

✓2⇥

E"1

2

⇣Y T P T ⇠⌘2

+1

2 2

Z T

0

Y 2t dt ⌫

Z T

0

Y t ✓t dt +1

2

Z T

0

✓ 2t dt

#



3. Solution

25

J(t,p,y) = inf ✓s :tsT

E

"1

2⇣

Y T P T ⇠⌘2

+1

2 2

Z T

tY 2s ds ⌫

Z T

tY s ✓s ds +

1

2

Z T

t✓ 2

s ds

P t = p, Y t = y #

Value function



Dynamic programming

26

0 = inf ✓

⇢1

2 2y 2 y ⌫✓ +

1

2✓2

+ J t +1+ ⌫

✓J y + ⌫✓J p

+1

2

2J pp + 2 J py +1

2

2 2J yy

=1

2 2y 2

1

2

h⌫y J p

1+ ⌫ )J y

i2

+ J t +1

2 2J pp + 2 J py +

1

2 2

2J yy

HJB PDE:

✓ =1

⇣⌫y

1+ ⌫

J y ⌫J p

⌘optimal control



Solvable in 2 special cases

1. Constant gamma

2. No permanent impact

28

g0(t,P t) = g0(t,P 0) + P t P 0

⌫ = 0

Γ measures position size and size



Constant Γ

29

A1 = 0, A0(T

t,p) = A0(T t)

Instantaneousmishedge

ratecoefficient

function of time remaining

=

s 2

timeconstant

risk / temporary impact

✓t = h⇣ (1+ ⌫ )(T t)

⌘Y t



Summary of hedge strategy

Far from expiration, h=1

Near expirationh increases if overnight risk large

h decreases if overnight risk small

h becomes negative (!) if no overnight risk

✓t = Y t

31

✓t = hY t



What happens to price process?

32

constant Γ

(t,P t) = 0 (P t P 0)

Y t = X t

= X t X 0 + (P t P 0)

dX t = ✓t dt

dP t = dW t + ⌫ ✓t dt

✓t = Y t (h = 1)dynamic hedge



Combined processes

33

dY t = dX t + dP t= (1+ ⌫ )Y t dt + dW t

Z t = ⌫(X t X 0) + (P t P 0)

dZ t = dW t

def: same dWt

hY 2t i = 2

2

2 (1+ ⌫ )=

2 2

2(1+ ⌫ )s

2



Combined processes

34

P t = P 0 + 1

1+ ⌫

⌫Y t + Z t

= P 0 +

1+ ⌫

W t + ⌫ Z

t

0

e⌫(1+⌫ )(ts)dW s

!

stationarymodifiedvolatility

Γ<0: hedger is short the option1+νΓ<1: overreaction, increased volatility

Γ>0: hedger is long the option

1+νΓ>1: underrreaction, reduced volatility



General Γ

35

t = (T t) Y t + bias term

Restriction on sign of trading

inf ✓2⇥+

(· · ·)solve numerically

Extensions



Restricted sign

36

Unrestrictedstrategy

Restrictedstrategy



4. Possible applications

Price pinning to strike near expirationif hedgers are net long Γ

Intraday volume patterns

hedge near open and close

37



Conclusions

Simple market impact modeltemporary/permanent

linear model

Explicit solution (at least for constant Γ)hedge position tracks toward Black-Scholes

Large hedger can change volatilitymarket impact on implied and realised vol

options hedging with market impact [r. almgren] presentation. 2012

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