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Robert Almgren
Option Hedgingwith Market Impact
New York UniversityCourant Institute of Mathematical Sciences
Market Microstructure, Paris, Dec 2012
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Outline
1. The problem
2. Formulation
3. Solution
4. Examples
2
Work with Tianhui Michael LiPrinceton: Bendheim Center and ORFE
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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
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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
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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
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Questions
1. What is a reasonable market model?
2. What do solutions look like?
3. How do they compare to Black-Scholes?
6
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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
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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
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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)
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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
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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
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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
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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)
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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
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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
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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)
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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
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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
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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)
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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
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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
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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
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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
#
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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
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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
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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
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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
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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
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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
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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
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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
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General Γ
35
t = (T t) Y t + bias term
Restriction on sign of trading
inf ✓2⇥+
(· · ·)solve numerically
Extensions
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Restricted sign
36
Unrestrictedstrategy
Restrictedstrategy
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4. Possible applications
Price pinning to strike near expirationif hedgers are net long Γ
Intraday volume patterns
hedge near open and close
37
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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