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

Work with Tianhui Michael LiPrinceton: Bendheim Center and ORFE

1. Option hedging (version 1)

time tT

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

Hedge portfolio Xt shares

X t = X 0 +Z T

✓s ds

mark-to-market

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

evaluate on

mean and variance

option expiryor

market close

Option hedging (version 2)

time tClose

Asset price Pt

P t = P 0 + W t + himpacti

Hedge shares Xt

X t = X 0 +Z T

✓s ds

Mark-to-marketvalue

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?

Questions

1. What is a reasonable market model?

2. What do solutions look like?

3. How do they compare to Black-Scholes?

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

Permanent impact

t = instantaneous rate of trading

X t = X 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

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

buy at ask

sell at bid

P t = P t +1

2s sgn(✓t)

2s sgn(✓t) · ✓t t =

2s|✓t|t

“Linear” model: cost to trade sharest t

Solutions with bid-ask spread cost

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

Proportional cost model

H(0) = 0

concave(empirical)

Linear for simplicity

Quadratic cost:

H(✓) =1

H(✓) · ✓t =

P t = P t + H(✓t)

Our solutions with proportional cost

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

X t = X 0 +

✓s ds

P t = P 0 + W t + ⌫(X t X 0)

˜P t = P t +

2 ✓t

Hedge holding:

Public market price:

Private trade price:

F t = filtration of W t

Black-Scholes option value

g(T,p) = g0(p)

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

Final value specified

Intermediate values defined by Black-Scholes PDE

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

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

g(t,p) = option payout to trader

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

Longcall

Longput

< 0 < 0

> 0 > 0

Γ= sign and size of option position

Final portfolio value

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

P t ✓t dt

Option value Portfolio value Cash spent

Mark to marketwithout transaction costs

2Include permanent impact

in liquidation cost:

We neglect: gives manipulation opportunities

dominated by risk aversion

Integrate by parts

RT = R0 +

X t + g0(t,P t)

= R0 + Z T

= R0 +

Y t dW t +

Y t ⌫✓t dt 1

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

infinitevariation

smooth

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

RT = R0 +

Y t dW t + ⌫

Y t ✓t dt 1

ERT = R0 + ⌫ E

0Y t ✓t dt

VarRT = complicated

RT is random: optimize expectation and variance

mishedge

Variance of RT

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

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

P t ✓t dt

= RT + Y T P T 0 P T

Z T 0T

t, P t

T, P T

= RT + Y T P T ⇠

P T ,⇠ have mean zero

Version 2 objective function

Random variables

Terminal

mishedge

⇠ distribution depends only on P T

P T mean 0, independent of F T

✓2⇥

⇣Y T P T ⇠⌘2

Y 2t dt ⌫

Y t ✓t dt +1

✓ 2t dt

3. Solution

J(t,p,y) = inf ✓s :tsT

Y T P T ⇠⌘2

tY 2s ds ⌫

tY s ✓s ds +

t✓ 2

P t = p, Y t = y #

Value function

Dynamic programming

0 = inf ✓

2 2y 2 y ⌫✓ +

+ J t +1+ ⌫

✓J y + ⌫✓J p

2J pp + 2 J py +1

2 2J yy

2 2y 2

h⌫y J p

1+ ⌫ )J y

+ J t +1

2 2J pp + 2 J py +

HJB PDE:

✓ =1

⇣⌫y

1+ ⌫

J y ⌫J p

⌘optimal control

Solvable in 2 special cases

1. Constant gamma

2. No permanent impact

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

⌫ = 0

Γ measures position size and size

Constant Γ

A1 = 0, A0(T

t,p) = A0(T t)

Instantaneousmishedge

ratecoefficient

function of time remaining

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

✓t = hY t

What happens to price process?

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

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 (1+ ⌫ )=

2(1+ ⌫ )s

Combined processes

P t = P 0 + 1

1+ ⌫

⌫Y t + Z t

= P 0 +

1+ ⌫

W t + ⌫ Z

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

t = (T t) Y t + bias term

Restriction on sign of trading

inf ✓2⇥+

(· · ·)solve numerically

Extensions

Restricted sign

Unrestrictedstrategy

Restrictedstrategy

4. Possible applications

Price pinning to strike near expirationif hedgers are net long Γ

Intraday volume patterns

hedge near open and close

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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