almgren, li - closed-form solutions for option hedging with market impact

7/27/2019 Almgren, Li - Closed-Form Solutions for Option Hedging With Market Impact

1/30

Submitted to Operations Researchmanuscript (Please, provide the mansucript number!)

Closed-Form Solutions for Option Hedging withMarket Impact

Tianhui Michael Li*Princeton Bendheim Center for Finance and Operations Research and Financial Engineering

Robert AlmgrenQuantitative Brokers, LLC and NYU Courant Institute

We present a highly-intuitive closed-form delta-hedging result for a large investor whose trades generateadverse market impact. Unlike the complete-market or proportional-transaction-cost cases, the agent nolonger finds it tenable to be perfectly hedged or even within a fixed distance of being hedged. Instead, hemay find himself arbitrarily mishedged and optimally trades towards the classical Black-Scholes delta, withtrading intensity proportional to the degree of mishedge and inversely proportional to illiquidity. Whenviewed in light of a recent result of Garleanu and Pedersen (2009), this suggests that delta-hedging canbe thought of as a Merton problem where the Merton-optimal portfolio is the Black-Scholes delta hedge.

Both the discrete-time and continuous-time problems are solved. We discuss a number of applications of ourresult, including the implications of our model on intraday trading patterns and stock pinning at optionsexpiry. Finally, numerical simulations on TAQ data based on intraday hedging of call options suggest thatthis strategy is able to significantly reduce market impact cost with respect to Black-Scholes delta hedging.

1. Introduction

The solution to hedging an option in a complete market is well known (Black and Scholes 1973,Merton 1973). However, the assumption of frictionless trading that underlies complete markets is farfrom reality. Practioners who trade large positions are familiar with the concept of market impactwhen trading in real markets with limited liquidity. For instance, a buyer who lifts orders from theoffer eats up liquidity and pushes his execution price ever higher. Liquidity of the underlying oftendries up during financial crises such as the 1987 Crash, the LTCMs collapse, the recent subprimedebacle, or the London Whale scandal and this has only further heightened investor concerns aboutliquidity.

We are interested in the optimal hedging of an option by a large risk-averse investor in an illiquidmarket. There is a large literature on trading under transaction costs. Part of the literature isinterested in super-replication (Cetin, Soner, and Touzi 2010, Soner, Shreve, and Cvitanic 1995).Our paper relaxes this requirement by having a finite penalty for being mishedged. Our paper ismore closely linked to these using a utility-based framework (Cvitanic and Wang 2001, Davis andNorman 1990, Hodges and Neuberger 1989, Janecek and Shreve 2004, Leland 1985, Shreve andSoner 1994).

Trading frictions have been modeled via various mechanisms. One strand of the literature is cen-tered around illiquidity (Cetin, Jarrow, and Protter 2004). These authors hypothesize a stochasticsupply curve whose steepness coincides with illiquidity. Similar work was done by Bank and Baum(2004). The major drawback of both their theories is that liquidity costs can be completely avoidedif trades are finite-variation. Hence, the cost of replicating an option is asymptotically zero as thewriter can take finite-variation approximations to the classical Black-Scholes -hedging technique.

* This research is based upon work supported by the National Science Foundation Graduate Research Fellowshipunder Grant No. DGE-0646086 and the Fannie and John Hertz Foundation Graduate Fellowship. The author wouldlike to thank both organizations for their generous support.

1


2/30

Author: Option Hedging with Market Impact

2 Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!)

To get around this problem, Cetin, Jarrow, Protter, and Warachka (2006) impose an artificialconstraint on the minimum time between discrete trades.

Another strand of the literature models trading frictions as a cost proportional to trade size,typically interpreted as arising from the bid-ask spread. Mathematically, these problems exhibita scaling which allows the authors to simplify their formulas. This branch of the literature uses

singular control and the optimal solution is typically in the form of a tracking band. As the portfolioexits this band, the trader makes singular corrections to his holdings to keep it strictly within thelimits of the band (Cvitanic 1999, Cvitanic and Karatzas 1996). This is analogous to the results ofDavis and Norman (1990), Shreve and Soner (1994) for the Merton Problem. Typically, there is noanalytic form for these no-trade regions and much work has been done on their numerics (Clewlowand Hodges 1997, Davis, Panas, and Zariphopoulou 1993) or asymptotics (Soner and Barles 1998,Whalley and Wilmott 1997).

Yet another strand of the literature models the full order book. Alfonsi and Schied (2009),Alfonsi, Fruth, and Schied (2007), Predoiu, Shaikhet, and Shreve (2010) model a resilient orderbook where levels gradually replenish after large trades. They generally derive optimal strategiesin which the broker makes large block trades at the beginning and end of the trading period withmany small trades in between. These models often allow for general order book shape and are

mathematically rather challenging to work with.The final strand of related literature focuses on modeling effective market impact. Rather than

dwelling on the the details of microstructure, it models effective microstructure. The literaturebegan with Bertsimas and Lo (1998) and Almgren and Chriss (1999) who decompose price impactin terms of both temporary and permanent price impact. The model was subsequently refined inObizhaeva and Wang (2005) by the addition of a transient price impact term. While earlier worksin the literature concentrated on the liquidation a fixed number of shares in a finite period, theseimpact models were applied to portfolio allocation (Garleanu and Pedersen 2009) and optionshedging (Rogers and Singh 2007), the liquidation of portfolios (Schied and Sch oneborn 2007),trading in both lit and dark venues (Kratz and Schoneborn 2010), and multiple-agent competitiveliquidation (Schoeneborn and Schied 2010).

In contrast to proportional-transaction-cost models, the source of incompleteness for our model

is price impact: the more aggressively one trades, the more impact is incurred proportionally.The cost is no longer linear in the shares purchased and the problem no longer exhibits linearscaling. We readily conceded that an ideal transaction cost model would naturally take into accountboth bid-ask spread and market impact. However, such a model presents a different set of (veryintriguing) mathematical challenges that are beyond the scope of this paper. Nonetheless, we arguethat neglecting an explicit spread term is a more realistic model of the effective market impactthan including only the spread. The following cases serve to illustrate this point.

The spreads-only cost model assumes all orders are active (at the offer for purchases, at thebid for sales). Most executions in US equities occur through electronic brokers who execute bothpassively and aggressively. If the passive/active fraction is 50%, then the average execution is atthe mid price, i.e. the orders incur zero market impact (this assumes that all passive and activeorders execute at the top of the book). Most brokers are able to attain near-mid execution fornon-urgent orders.

For more urgent orders, the fraction of aggressive fills typically increases. For example, fillsthat are 80% aggressive and 20% passive result in an effective market impact of 30% of the bid-askspread (again, assuming top-of-the-book execution). We can observe that the effective market-impact is actually a smooth function of urgency and not realistically modeled by a fixed, discretetransactions cost as in the spread-only model.

For super-urgent orders, the top-of-the-book fills assumption is no longer realistic. A spread-only model limits the market impact to the half-spread and is completely unable to account forexecution deeper into the book.


3/30


Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) 3

In order to realistically account for the last point in a microsturcture sensitive way, we wouldneed to directly model the order book and both active and passive orders. This is addressed inanother strand of the literature (Alfonsi and Schied (2009), Predoiu et al. (2010)) by explicitlymodeling the order book. However, doing so poses a different set of mathematical challenges thatare beyond the scope of this article. Our effective model is chosen to judiciously capture the

effective trading costs while minimizing mathematical fuss. Yet, we are still able to place this modelwithin a tractable framework to obtain a highly intuitive closed-form trading solution.Despite this mathematical parsimony, our solution more realistically handles the case when the

portfolio is far from being hedged, for example due to a sudden movement in underlying price.Under only proportional transaction costs, the optimal solution in this situation is to cross thespread with market orders in order to re-enter the tracking band. In real markets, such an orderwould likely eat several levels into the order-book, incurring significant market-impact, not modeledby the naive, proportional-transaction-costs model. Under our market-impact model, the optimalsolution in this situation is to trade aggressively towards being hedged, in a way that takes intoaccount both available liquidity and the degree of the mishedge. Our trading strategy is muchsmootherindeed, we are approximating the impact-free Black-Scholes Delta, an infinite variationprocess, with trading positions that are differentiable. A comparison is given in Figure 1 to illustrate

the point.

Figure 1 Comparison of the proportional-transaction-cost fxied-tracking-band strategy (top) and our dynamicstrategy (bottom). In the former, our (green) trading position changes only when the (blue) target leavesthe (gray) trading-band. In the latter, our (green) trading position smoothly adjusts to the same (blue)target. Compare the smooth trading flow and position of the latter strategy to the abrupt trading ofthe former.

We separate market impact into temporary and permanent impact terms following the frameworkdeveloped by Almgren and Chriss (2001). The temporary impact affects only the execution price Pt


4/30



but has no effect on the fair value or fundamental price Pt. In contrast, the permanent impactdirectly affects Pt while having no direct effect on the execution price Pt. Thus we can think ofthe temporary impact as connected to the liquidity cost faced by the agent while the permanentimpact as linked to information transmitted to the market by the agents trades (see, for example,Back (1992), Kyle (1985)).

Previous work involving the Almgren-Chriss model has involved trading in the presence of DarkPools (Kratz and Schoneborn 2010), portfolio liquidation (Schied and Schoneborn 2007), and com-petitive liquidation (Schoeneborn and Schied 2010). Hernandez-Del-Valle and Pacheco-Gonzalez(2009) extends the work to Geometric Brownian Motion and Horst and Naujokat (2008) extendsit to derivative valuation. Rogers and Singh (2007) also examine the delta hedging problem but foragents with different risk preferences and were only able to obtain analytic expressions asymptot-ically. Our work differs from theirs in that we derive closed-form expressions by using simpler riskpreferences. The paper most closely related to ours is Garleanu and Pedersen (2009). They solvethe infinite-horizon Merton Problem under only temporary market-impact assumptions. As inour setup, they use a linear-quadratic objective rather than the traditional expected utility setupof the classical Merton Problem. They find that trading intensity at time t (Proposition 5) is givenby

t = h Xt targett 1/where Xt is the number of shares, is an urgency parameter with units of inverse time, h > 0 isa dimensionless constant of proportionality related to the convexity of the continuation value, andthe target portfolio targett is related to the frictionless Merton-optimal portfolio. That is, withmarket-impact costs, it is no longer optimal to hold the instantaneous Merton-optimal portfolio butinstead, the agent trades towards a new target portfolio (which is a new Merton-optimal portfolio).The intensity of trading t is proportional to the distance between the current holdings and target(Xt targett) and is inversely proportional to the square-root of the illiquidity parameter .

We use an analogous finite-horizon setup on [0, T] with both temporary and permanent impactand obtain that the trading intensity is

t = h((T t))

Xt targett

where the target portfolio targett is now related to the frictionless Black-Scholes delta-hedge andh() is a positive dimensionless function, which comes from the finite-horizon nature of the setup(compare to (15) with = 0 so K= 1). Hence, as in Garleanu and Pedersen (2009), an agent facingmarket illiquidity no longer maintains the zero-liquidity-cost optimal portfolio but instead tradestowards a new target to correct this mishedging. Furthermore, also as in Garleanu and Pedersen(2009), the agents trading intensity t is proportional to the difference between his current holdingsand the optimal no market-impact portfolio (Xt targett) and inversely related to the market-impact cost . If the option as well-approximated by having a constant gamma, we find that target tis actually the Black-Scholes hedge (Theorem 1). This similarity to Garleanu and Pedersen (2009)

suggests that we can think of delta-hedging in an illiquid market as a Merton optimal investmentproblem where the Merton portfolio is the Black-Scholes hedge portfolio. For more general options,the target targett has an extra term accounting for the non-zero third derivative with respect tospot of the Black-Scholes option price (Theorem 2).

The subsequent sections are as follows. We motivate our assumptions and formally set up theproblem in Section 2. Closed-form solutions for two general cases are presented in Section 3. InSection 4, we give a number of applications and implications of our result. In Section 5, we givea discrete-time formulation of the problem. This is necessary for performing discretized numericalsimulations on TAQ data. The discrete-time solution is used in the hedging simulation in Section 6.


5/30



2. Problem Setup

In this section, we first give a heuristic justification for our objective (6). The resulting optimalpolicy for this objective is then rigorously proved in Section 3. We give the market impact model,our model for a European option, and our formal objective in the next three subsections.

2.1. Market Impact Model

Let (,F,P) be a complete probability space with Ft being the filtration generated by Wt. Considerthe trading model

Xt = X0 +

t0

u du

where Xt is the number of shares held by the agent and t is the intensity of trading. The funda-mental price is given by

Pt = P0 +

Xt X0

+ Wt (1)

where > 0 is the coefficient of permanent impact, > 0 is the absolute volatility of the fair value,and Wt is a standard one-dimensional Brownian Motion. Using a linear Brownian Motion rather

than a geometric one is appropriate over the short time horizons considered in the paper and leadsto dramatic simplifications in our result. Throughout this paper we neglect interests rates, whichis appropriate for the short time-periods under consideration.

To model temporary impact, we assume that the tth share (at time instant t) will cost a linearpremium t over the fair value

Pt(t) Pt = twhere > 0 is the market impact parameter. Therefore, the total temporary impact (rate) ofbuying t shares is t

0

[Pt() Pt]d= t

0

d= 2t2

. (2)

We can think of temporary impact as coming from a limit-order book with constant depth 1/

and instant resilience (see, for example, Alfonsi and Schied (2009), Predoiu et al. (2010)). In sucha model, purchasing t shares would consume all the shares priced from Pt to Pt + t on the book,thus pushing the execution cost of the last share up by t. The limit orders that were eaten upare replaced immediately after execution. Thus our temporary impact coefficient is related tothe corresponding term in Almgren and Chriss (2001) by = 2.

2.2. European Option

We think of hedging a European contingent claim over a finite time horizon [0 , T]. For a hypotheticalsmall trader whose execution has no price-impact and trades in a complete market, the value ofthe options price g

t, Pt

is a function of the time t and the price Pt of the underlying asset,

g : [0, T]R

R .

Hence, the agents total portfolio value at time t is given by XtPt + g(t, Pt). We assume the priceat t = T is given by g0 :RR and the value for t [0, T) is prescribed by Feynman-Kac to be thesolution of the PDE

g(t, p) +2

2g(t, p) = 0 for t, p [0, T) R and g(T, ) = g0 . (3)

Here g represents derivatives with respect to t and g and g represent derivatives with respect top. We can view this as the Black-Scholes option-pricing PDE for the arithmetic Brownian Motion


6/30



Pt and zero interest rates so that g has the interpretation of the option price in the corresponding

(no impact-cost) complete-market. For notational simplicity, we define

(t, p) = g(t, p)

to be the gamma of the option.For a large trader who does face market-impact, the terminal wealth (or revenue) RT is the sum

of the options value, the stocks value, and the cost of acquiring the position,

RT = g(T, PT) T option value

+ XTPT T stock value

T

0

t0

Pt()d acquisition cost

dt . (4)

Using integration by parts, the stock dynamics (1), the total temporary impact (2), and Feynman-

Kac (3), we may rewrite the terminal wealth as

RT = R0 +T

0

Xt + g

(t, Pt)

dPt 2

T0

2t dt ,

where

R0 = g(0, P0) + X0P0 .

It is now convenient to introduce the variable

Yt = Xt + g(t, Pt)

with the interpretation as the portolios net delta exposure. This variable replaces net shares

held Xt as a more relavent variable. Yt also has the interpretation ofdistance from being delta-hedged as the ideal net delta-exposure is 0. Hence, we may decompose our wealth process into

three intuitive parts

RT = R0 +

T0

YudWu IntraperiodFluctuation

+ Yuu du Permanent

Impact

2

2u du Temporary

Impact

. (5)

If we interpret the interval [0, T] as the trading day, we have the wealth RT as the sum of the

fluctuation during the trading day and the liquidity cost from permanent and temporary impacts.

Finally, we make the assumption

Assumption 1. Let g(t, p) have a gamma (t, p) that is bounded everywhere and Lipschitz in p

with a Lipschitz constant that is independent of t.

The above assumption holds, e.g., for the Delta and Gamma of calls and puts except at expiry and

so this formula will be valid for hedging except on the expiration day of the option. The case of

hedging at expiry is well-known for being difficult because of the divergence in Gamma and these

complications are beyond the scope of this paper.


7/30



2.3. Objective

Assumption 2. Assume and PT are L2 random variables satisfying

1. the conditional distributionP [| FT] depends only on PT.2. the conditional distributionP [PT | FT] is independent of FT and has mean zero.

We define our linear-quadratic objective to be

inf

E

2(YTPT )2

TerminalMishedging

+

T0

2

2Y2u

RunningMishedging

Yuu Permanent

Impact

+

22u

TemporaryImpact

du

, (6)

where and P are to be defined later and the control set is given by

:=

predictable : E

T0

2s ds < and t C(1 + |Yt|) a.s. for all t

.

This objective is standard in the literature and can be thought of as a finite-horizon adaptation

of Garleanu and Pedersen (2009). The relationship to utility optimization in the portfolio theoryliterature is well-known and given in Fabozzi (2007, Chapter 2). The first term in (6) gives aterminal (time t = T) penalty for being mishedged. We allow this terminal penalty to be quiteflexible so that it can accommodate multiple interpretations (through multiple intepretations of and P: see Examples 1, 2, and 3). The running terms penalize for intraperiod mishedgingfluctuations, and permanent and temporary impact. Equation (6) has a natural interpretationof balancing the temporary and permanent liquidity costs with the penalty for variance of theintraperiod and terminal mishedging errors.

The state variables have dynamics

dPt = t dt + dWt (7)

dYt = ( 1 + (t, Pt))t dt + (t, Pt)dWt . (8)

(Recall that is the gamma of the option). It is easy to see that Pt is a continuous semimartingaleand hence predictable. We see from (1) that since is bounded that Yt is well-defined for all . We see from (7) that every share purchased t pushes Pt up by because of the permanentimpact. We also see from (8) that every share purchased t increases his net delta position Yt by1 + (t, Pt): 1 for the increase in the stock position and (t, Pt) for the permanent impacts effecton the options delta.

Example 1. We could choose to interpret T as the maturity of the option. In this case, themishedging penalty is proportional to the time to maturity (i.e. the time remaining to potentiallyaccumulate mishedging). Hence, one might choose to not include a terminal term ( PT = 0 and= 0) and no additional penalty is incurred for being mishedged at maturity

E

2(YTPT )2

= 0 .

However, the individual trader may choose to incorporate a penalty because he finds it too riskyto be too mishedged even at expiry (see Remark 1).

Example 2. For general intraday trading, T would represent the close of trading. However, thereis an unhedgable potential overnight jump and trading does not resume until the next morning.Therefore, we might choose to benchmark the objective to the opening of trade the next morning.The choice of benchmarking to the morning is natural because the objective can be reset for the


8/30



next trading day. (Of course, the agent may choose to not incorporate a terminal penalty, even onthe day of the options maturity, as in Example 1).

For practical applications, adding a terminal mishedging penalty is more realistic. Without one,the agents incentive to be hedged would (unrealistically) vanish as he approaches the closing bell.This extra penalty reflects the need to be well-hedged for the overnight jump, during which, he is

unable to trade.To make this more precise, let T be the opening of trading the next morning, T T, and let

T = T T denote the overnight time interval. Between T and T, the agent is not allowed totrade but the underlying price continues to evolve. From (4), we obtain

RT = g(T, PT) + XTPT

T0

t0

Pt()ddt

= RT + g(T, PT) g(T, PT) + XT (PT PT)

= RT +

TT

[g(t, Pt) g(T, PT)] dPt + YT (PT PT) .

Then we can set the terms in the terminal penalty to

= TT

[g(t, Pt) g(T, PT)] dPtPT = PT PT .

and the terminal penalty reduces to the quadratic (in YT)

E

2(YTPT )2

=

2E

2TY2T 2E[PT| FT] YT + E

2 | FT

where 2T =

2T. Observe that the distribution of does not depend on PT and so satisfiesAssumption 2.

Example 3. One highly illustrative subcase of Example 2 is when we can approximate the optionas having a constant gamma, i.e.

g(t, Pt) g(t, P0) + (Pt P0) with R constant .

In other words, we assume (t, p) . Under this assumption, the terms in the terminal penaltysimplify down to

= TT

(Pt PT) dPtPT = PT PT .

and so we can use ItoCalculus to derive

E [PT| PT =p] = E

2

TT

g(u, Pu) g(T, PT) du | PT =p

= 0

(see Subsection 3.3 for the full calculation). Following directly from Example 2, the terminal objec-tive would be

E

2(YTPT )2

=

2E

2TY2T +

2

.


9/30



3. Continuous-time Solution

Let J(t,p,y) be a C1,2,2([0, T] RR) function. Then from the Martingale Principle of OptimalControl, we have the HJB equation is

0=inf

2

2y2

y +

22 + J+ ( 1 + )

yJ+

pJ

+ 2ypJ+()

2

22yJ+

2

22pJ

= 12

[(y pJ) (1 + )yJ]2 + 2

2y2 + J

+ 2ypJ+()

2

22yJ+

2

22pJ . (9)

We will not solve the equation in full generality but will stick to two major sub-cases. For these, itsuffices to to make the Ansatz,

J(t,p,y) =

A2(T

t)

2 y2

+ A1(T t, p)y + A0(T t, p) . (10)We will later verify that this Ansatz holds in the proofs of Theorem 1 and Theorem 2. Observethat A2 has no dependence on p. This requires Assumption 2 and will be key to our decomposition.Then the equation breaks up into the following three equations

A2 = 2 1

[(1 A1) (1 + )A2]2 (11a)

A1 =2

2A1 +

1

[(1 A1) (1 + )A2] [A 0 + ( 1 + )A1] (11b)

A0 = 12

[A 0 + ( 1 + )A1]2

+ A12 +

2

2 2A2 + A

0

. (11c)

(Here A and A denote the derivative with respect to p and t, respectively). Observe that theequation for A2 (11a) cannot hold in general: A2 does not have a dependence on p but A1 does.The equation will nonetheless hold in the two cases of interest. These equations are subject to theboundary conditions

A2(0) = E

P2T

A1(0, p) = 2

E [PT| PT =p]A0(0, p) =

2E

2 | PT =p

.

The trading strategy is then

t =Yt (1 + (t, Pt)) (A2(T t)Yt + A1(T t, Pt)) (A1(T t, Pt)Yt + A0(T t, Pt))

. (12)

Hence, the optimal policy comes from solving for A1 and A2. However, they exhibit closed-formsolutions in two important broad cases. Before we move on to these two cases, we first define

K= 1 + =

d =

|K|

2T

K

. (13)


10/30



3.1. Constant Gamma Approximation

On short time scales, we can safely assume that the option has a constant gamma. The approx-imation says that for small intraperiod fluctuations of Pt P0, the options delta varies linearlywith the stock price. We make the the same approximation as in Example 3

g

(t, Pt) g

(t, P0) + (Pt P0) with R constant . (14)As in Example 3, the terminal penalty will be

E

2(YTPT )2

=

2E

2TY2T +

2

and does not depend on PT. Furthermore, the dynamics of (8) are now independent of Pt,

dPt = t dt + dWt

dYt = ( 1 + )t dt + dWt

where is now a constant. The assumption considerably simplifies the problem by dropping thedependence on the state variable Pt and eliminating the function A1.

In this subsection, we will make the additional assumption

Assumption 3. Assume that either1. d 1 holds or2. d < 1 and |K|T + arctanh(d) 0.

Theorem 1. Let Assumption 3 hold. Then the optimal trading intensity is given by

t = h(K(T t))Yt (15)where the function h is defined by

h(x) =tanhx + arctanh(d) |d| < 11 d = 1coth

x + arccoth(d)

|d| > 1 . (16)Under the optimal trading strategy, YT = 0 a.s. Indeed, P[|YT| > M] > 0 for all M > 0. That is,because of the market-impact costs, the position is not perfectly delta hedged, even at the terminaltime and there is a chance (albeit small) that YT is far away from 0.

Proof of Theorem 1 The Martingale Principle of Optimal Control tells us that

Mt =

t0

2

2Y2u Yuu +

22u du + J(t, Yt)

is a submartingale for all t and a martingale under the optimal control. The HJB equation (9)

simplifies to

0 = 2

2y2 + J(t, y) +

()2

2J(t, y) 1

2

y KJ(t, y)2 (17)

(t, y) =1

(y KJ(t, y)) (18)

wehre J and J are derivatives with respect to y. The ansatz (10) then simplifies to

J(t, y) = A2(T t) y2

2+ A0(T t) .


11/30



and (11a) and (11c) simplify to

A2 = 2 1

( KA2)2 ,

A0 =()2

2

A2 .

We can see that A2 is a parabola opening downward in A2 with vertex at the point (K

, 2) andleft and right critical points are A2 and A

+2 , respectively where

A2 =

K |K| .

IfA2(0)= 2T (A2 , A+2 ) then A2(Tt) > 0 and so A2(Tt) A+2 as Tt . This corresponds

to the case when 1 < d < 1 where the solution is given by

A2(T

t) =

K

+

|K|tanh(

|K

|(T

t) + arctanh(d))

A0(T t) = ()2

2K

(T t) +

K

log cosh|K|(T t) + arctanh(d)+ 1

2log(1 d2) . (19)

Similarly, if A2(0)= 2T

A2 , A

+2

then A2 is fixed at the critical point. This corresponds to the

case |d| = 1 and the solution is given by

A2(T t) = K

+d

|K| = 2T

A0(T t) = 22

22T(T t) .

(20)

Finally, if A2(0) > A+2 then A2(T t) < 0 and so A2 A+2 . This corresponds to the case d > 1where the solution is given by

A2(T t) = K

+

|K| coth(|K|(T t) + arccoth(d))

A0(T t) = ()2

2K

(T t) +

K

log sinh|K|(T t) + arccoth(d)+ 1

2log(d2 1) . (21)

These three cases cover case 1 of Assumption 3. Under case 2 of Assumption 3, equation (21) alsoholds and A2(T t) is well-defined for t [0, T] since |K|(T t) + arccoth(d) 0 for all t [0, T].

Thus we have a solution J to the HJB equation (18) and we see that Mt has non-negative drift

for all and is a local martingale for the optimal given in (15). To conclude the proof, we need toshow that Mt is a submartingale for all and a martingale for the optimal . Thus, it sufficesto show E[M]T < for all . Observe that the function A2(t) is uniformly bounded by someC > 0 independent of t [0, T] so that

E[M]T = E

T0

J(t, Yt)2 dt C2E

T0

|Yt|2 dt . (22)


12/30



Itos Lemma gives us

EY2t = Y2

0 + E

t0

2YsdYs + d[Y]s

= Y20 + Et

0

CsYs + 22 ds

Y20 + Et

0

C|Ys|2 + 22 ds

for some C, C > 0 independent of t [0, T]. Thus, a standard application of Gronwalls Lemmayields E[M]T < and we have (15) is the optimal policy.

The second part also follows from the fact that YT is given by a linear SDE. Its solution

Yt = e

t

0Kh(K(Ts))ds

Y0 +

t0

es

0Kh(K(Tu))dudWs

has a density that is non-singular with respect to the Lebesgue measure on R as h is bounded.

In Almgren and Chriss (2001) is an urgency parameter which dictates the speed of liquidation:the higher the faster the initial liquidation. In Garleanu and Pedersen (2009), this term givesthe relative intensity of trading1, that is, the higher , the faster the agent trades towards theMerton-optimal portfolio.

The agents trading target for shares Xt is the Black-Scholes delta hedge,

targett = g(t, Pt) .

He constantly trades towards this target but is prevented from holding the exact Black-Scholesdelta hedge by the convex temporary impact stemming from limited liquidity. Hence, the tradingintensity t is proportional to the degree of mishedge Yt and the urgency parameter . There isa greater penalty to being mishedged with higher underlying volatility and risk aversion so

these parameters increase urgency. Similarly, a more illiquid market (higher ) makes trading morecostly, which decreases trading intensity.

The solution falls into three cases depending on whether h(), the trading intensity proportion,is decreasing, flat, or increasing in time, i.e. whether the agent trades less intensely, is flat, or moreintensely (respectively) towards the end of the trading period. This, in turn, depends on the valueof a constant d which gives the relative size of the overnight jump PT versus permanent impactand noise dPt.

The interpretation of d is clearer when there is no permanent impact ( = 0) and we firsthandle this case. In this case, the cost of being mishedged comes purely from running and terminalmishedging terms. Intraperiod trading can be thought of as primarily hedging out the intraperiodfluctuations dPt while trading near the end of the period is primarily for hedging the terminaljump PT. When d < 1, the majority of the fluctuations occur during the day, and the solution is

marked by t h(K(T t)) decreasing as t approaches T. That is, the agent is more concernedabout hedging intraperiod fluctuations dPt and relaxes hedging intensity to reduce hedging costas the end of the period approaches. This is the case in Example 1 where d = 0. The second caseis when d = 1 and hedging risk for the intraperiod and end-of-day are weighted equally. Then theagents trading intensity is constant in time. The final case is when d > 1 and the terminal jumpis large compared to the daily fluctuations. So the agent trades more intensely with time andt h(K(T t)) increases towards d as t approaches T. This case could occur in Example 2.1 The a/ in Proposition 5 of Garleanu and Pedersen (2009) is equivalent to h in our setup


13/30



Units of Fundamental Quantities

Yt sh 1/$ $/sh2

P $/sh sec $/sh2 sh2/$ sh/sec $/sh/

sec J $

Units of Derived QuantitiesK 1 d 1 1/sec

Table 1 Units for constants: sh is a share, sec is a second, and $ is a dollar.

Remark 1. If we consider the case when [0, T] is the trading day, the last case (21) when d > 1is the most realistic. Options market-makers typically increase their hedging towards the closeof trading to minimize overnight exposure. The trader may choose a lower d as options expiryapproaches, selecting d = 0 on the day of options expiry. However, this strategy may be deemedtoo risky and the trader may choose d > 0 even at expiry to avoid the risk of being mishedged.

When there is permanent impact ( > 0), the running penalty comes from two sources in (6), therunning mishedging term and the permanent impact. The running mishedging term is as above butthe permanent impact term is more subtle. If the trader is in a net long delta position, he sells theposition to remain hedged, incurring a capital loss from the downward pressure on his inventory.A buy to hedge out a net short delta position incurs a similar loss and so the permanent impactcreates a cost for being mishedged in either direction. The terms for A2 in (19), (20), and (21) havean extra term which accounts for the permanent impact. The size of d balances the running costs(both from running fluctuations and the permanent impact effect) with the terminal cost of beingmishedged.

3.2. Comparative Statics

We give some comparative statics for our solution. The effect of the temporary impact term is the easiest to understand. As the cost of hedging decreases, it becomes optimal to trade more

aggressively to minimize mishedging error (see Figure 2c). The effect of risk aversion is similar.As increases, so does the penalty for being mishedged and there is a stronger incentive to trademore aggressively (see Figure 2e). To understand the effect for intraperiod volatility , recall thatthe agent is trying to hedge both intraperiod volatility (via intraperiod trading) and volatilityfrom the terminal jump T (via trading near the end). As intraperiod volatility increases whilethe terminal jump size T remains constant, the agent optimally trades more aggressively initiallywhile still ending at the same trading intensity at time t = T (see Figure 2b). The effect of increasingterminal jump volatility T while keeping intraperiod volatility constant is roughly the opposite(see Figure 2f): it increases trading near t = T while trading during the rest of the period remainsrelatively unaffected. Note that with higher T, there is a need to trade slightly more intenselyeven at the beginning of the period t = 0 as this impacts the mishedging at the end of the periodt = T.

and the permanent impact have more complex effects, which are qualitatively similar toeach other. There are two competing effects at play, whose importance differs with the time tomaturity. Firstly, there is an adverse-impact effect stemming from the permanent impact. Tosee this, let Yt < 0 so the agent optimally wishes to purchase shares. This purchase will push upeach shares value by , hence resulting in a decrease of Yt < 0 in the stock value of the agentsportfolio. In general, rebalancing under permanent impact always hurts his existing position, givingthe agent an incentive to trade less aggressively in the hope that subsequent price-fluctuations willautomatically rehedge his portfolio. The second effect is dubbed the leverage-effect. With = 0,the purchase of a single share increases the traders net delta position Y by 1. However, if > 0


14/30



(a) varying (b) varying

(c) varying (d) varying

(e) varying (f) varying TFigure 2 Comparative statics of (KA2(T t) )/ as a function of t. Parameters (unless otherwise specified):

= 5.0, = 0.2, = 0.2, = 0.2, = 2.0, T = 0.4, T= 1.0.

then his net delta position actually increases by K= 1 + due to the permanent impacts effecton the options price . When > 0, the effect of a purchase on the portfolios net delta position

is leveraged by the permanent impact on the options delta. We can think of this as effectivelylowering the cost of trading, (i.e. lowering the cost of achieving a fixed change in delta). Hence,the agent optimally increases the intensity of trading. When < 0, the effect of a sale or purchaseis diminished, if not reversed. We can think of this as reducing the effectiveness of trading on thenet delta position or increasing the cost of rebalancing. Either way, the agent optimally decreasesthe intensity of trading.

The relative importance of the two effects can be seen in the expression

t =1

Yt KJ(t, Yt) .


15/30



The leverage-effect acts through KJ while the adverse impact acts through Yt. In Figure 2,d > 1 and the pressure to be hedged (KJ) increases towards the end of the period. Hence, nearthe beginning of the period, the adverse impact is the overriding concern and trading intensity isless aggressive with higher because of the adverse-impact effect (Figure 2d) Near the end, theagent is more concerned about hedging the terminal jump. This combined with the ease of hedging

from the leverage effect acting through KJ

(t, Yt) implies the agent trades more aggressively withhigher near the end. The effect of is similar except that it only directly affects the leverageeffect. With higher , the increase in the leverage effect allows the agent to optimally trade moreaggressively, particularly near the end of the day (Figure 2a). However, like varying T, because affects mishedging near the closing bell, it has an indirect effect on the trading intensity duringthe middle of the day as well.

3.3. General Options, No Permanent Impact

We can relax the constant gamma assumption and allow for general options. However, to make thesolution tractable, we need to dispense with permanent impact (= 0). We are still able to obtaina fairly explicit solution for the control in (12) through A2 and A1. Recall that we have madethe assumptions Assumption 1 on our option.

Theorem 2. The optimal control is given by

t = A2(T t)Yt + A1(T t, Pt)

(23)

with

A2(T t) =

tanh ((T t) + arctanh(d)) d < 1 2T d = 1

coth((T t) + arccoth(d)) d > 1and

A1(T t, p) =

11

d2

sech((T t) + arctanh(d))E[F(p + BTt)] d < 1

exp

1

2T(T t)

E [F(p + BTt)] d = 1

1d2 1 csch ((T t) + arccoth(d)) E [F(p + BTt)] d > 1

where we assume B is a new Brownian Motion, the constants and d are defined by

=

d =

2T

= T

and the function F() is defined by

F(p) =

2EPTPT =p .

Proof In this case, the equations (11) then reduce to

A2 = 2 1

A22

A1 =2

2A1

1

A2A1

A0 = A21

2+ 2A1 +

2

2

2A2 + A

0

.


16/30



The solution for A2 then follows immediately. (Observe that the range of tanh includes (0, 1) andthe range of coth includes (1, )). Feynman Kac then gives us

A1(T t, p) = E

exp

Tt

A2(T s)

ds

F(p + BTt)

(24)

where B is any Brownian Motion. We have by Assumption 1 that g is bounded hence g isLipschitz. Since we have

F(p) = 2

E [PT| PT =p]

=

2E

2TT


we can show that F is Lipschitz as well. Hence A1 given by an expectation in (24) is well-defined.Observe that

sech(arctanh(d)) = 1 d2

d < 1csch(arccoth(d)) =

d2 1 d > 1

from which our expressions for A1 then follow. Finally, since the PDE for A0 is linear, it is straight-forward to verify it exists and so we may define J as in (10).

For the verification argument, define

Mt =2

2Y2u Yuu +

22u du + J(t, Pt, Yt)

Since J solves the HJB equation (9), we have that Mt has non-negative drift for all and zerodrift for the optimal policy given in (23). Again, we shall show that for all , the expectedquadratic variation E[M]T < . We have

E[M]T = E

T0

A2(T t)Yt + A1(T t, Pt)2 dt CE

T0

|1 + Yt + Pt|2 dt , (25)

where C is a positive constant. Since A1 is bounded and A2 is Lipschitz in p, we have for all that

E

T0

|1 + Yt + Pt|2 E|1 + Y0 + P0|2 + 2ET

0

(1 + Ys + Ps) [ 1 + (1+(s, Ps))] s

+ 2(1+(s, Ps))2ds

E|1 + Y0 + P0|2 + CET

0

|1 + Ys + Ps|2 ds (26)

by the definition of and the fact that (t, p) is bounded (1). A standard application of GronwallsLemma yields E[M]T < .

The optimal control differs from the constant gamma case in that the target is no longer thedelta-neutral portfolio. There is a new term A1(T t, Pt), which biases the target portfolio toaccount for the third-derivative of the option price, or the delta of the gamma. If this quantity


17/30



vanishes, then F(p) vanishes and so does A1. To see this, consider the setup of Example 2 andobserve that by Itos Isometry that

F(p) =

2E

2TT


.

F has an interpretation as the average change in delta of the option over the period T to T. Wesaw in Example 3 that F and hence A1 vanish for options with constant gamma.

More generally, if the option has vanishing delta of gamma (that is, g vanishes) then for t > T,we have by Itos Lemma that

E [g(t, Pt) g(T, PT) ] = Et

T

g(u, Pu)dWu +1

2g(u, Pu)

2 du

= 0 .

Hence, F and A1 also vanish. For an option with positive delta of gamma (g > 0), an agent who

is delta-neutral at the close T will actually have a positive expected overnight delta exposure F.He corrects for this by adding a negative bias to his trading target for shares Xt is

targett = g(t, Pt) A1(T t, Pt)A2(T t) .

The effect is the opposite if g < 0. Observe that for constant p, the magnitude of A1 increases ast T (see Figure 3). This follows since the bias corrects for an overnight delta slippage and is lessrelevant far away from the closing bell.

We plot the offset A1 in Figure 3. Compared with Figure 3 we omit the plots for (as = 0)and (as can only affect A1 through a nonzero ). As expected, a higher reduces the trackingbias A1 as it increases the penalty for not being delta-hedged, although this difference disappearsas t T (see Figure 3a.) Varying and have inverse effects: increasing is similar to increasing (increasing the penalty for mishedging) while increasing has the inverse effect (increasing themarket impact costs). The results are plotted in Figure 3b and Figure 3c. Finally, increasing Talso increases the running costs of being mishedged (it increases A2 in Figure 2f). This in turndecreases A1 (see (24) and Figure 3d).

4. Asymptotics and Applications

4.1. Small Market-Impact Cost Limit

We are interested in the limit as both the temporary and permanent impacts vanish. For instance,in the constant gamma case (Theorem 1) the trading intensity t is proportional to the distancefrom being delta hedged Yt = Xt + g

(t, Pt) as given in

t = coth

K(T t) + arccoth(d)

Yt .

We observe in Figure 2d that the profile of trading intensity flattens as a function of time withdecreasing and from Figure 2c, we see that the trading intensity shifts upward with decreasing .Hence, we observe the trading intensity both increasing and flattening as market-impact decreases.

In the limit as 0 and 0, we have K 1, . If T > 0 then d as well. ThereforeA2(T t) and t/Yt and the process Yt is driven back to zero very quickly. That is,the agent aggressively drives Xt towards g

(t, Pt). Hence, we recover the Black-Scholes equation inthe limit of vanishing transaction costs. Similarly, for general options and no permanent impact(Theorem 2), we still have A2(T t) and A1(T t, p) 0 for all p and t < T and so againt/Yt . The following theorem makes this idea precise.


18/30



(a) varying (b) varying

(c) varying (d) varying TFigure 3 Comparative statics ofA1(T t) as a function oft. Parameters (unless otherwise specified): = 0.2, =

0.2, = 2.0, T = 0.4, T= 1.0. We normalize E[F(p+BTt)] to be 1.

Theorem 3. Assume that the assumptions of either Theorem 1 or Theorem 2 hold. In the limitof vanishing market-impact costs, we recover the Black-Scholes delta hedge. That is

Y

0 as

, 0 where is the L2(P )-norm and is the Lesbegue measure on [0, T].Proof If we make the assumptions of Theorem 1, we have from (26) that

E

Y2t Y0 + 22T expt

0

Kcoth

K(T s) + arccoth(d)ds .and the result then follows from noting that , K 1, and d in our limit. In the setup ofTheorem 2, we the result follows similarly by (26) and the fact that A2(Tt) and A1(Tt, p) 0 in the limit.

4.2. Small Intraperiod Mishedging Penalty Limit

We consider the case when the penalty for the terminal jump is large compared to the intraperiodhedging penalty. Mathematically, this is the limit when 0 and and T are held constant.The objective in this limit is given by

inf

E

2T

2Y2T

T0

Yuu +

22u du

(27)

where we have removed the intraperiod mishedging penalty. We include this result because (28)clearly demonstrates the increasing nature of the functions t/Yt as t T in terms of algebraic,rather than transcendental, functions.


19/30



Theorem 4. We assume that the option has constant gamma as in Theorem 1. Then for smallintraperiod mishedging, the optimal trading intensity t can be written as

t = d1 + Kd(T t) Yt . (28)

Proof If we make the Ansatz (10) then again A1 = 0 and we separate the resulting HJB bypowers of y. This yields

(t, y) =1

( KA2(T t))y

and

A2 = 1

(1 + )A2

2A2(0)=

2T

A0 =22

2A2 A0(0)= 0 .

By a similar argument as in Theorem 1, we obtain that

A2(T t) =1

K+ d1 + dK(T t)

A0(T t) = 22

2K

(T t) +

Klog

1 + dK(T t) .The result then follows.

For a sense of how the execution of the policy in Theorem 4 compares to the full case in Theo-rem 1, see Figure 4. Observe that without an intraperiod mishedging penalty, the trading intensityratio t/Yt (see (28)) is lower than with the intraperiod penalty (see (15)), especially during thebeginning of the period. The extra trading comes from hedging out fluctuations during the period.Intuitively, the optimal investor trades less aggressively when only faced with a terminal and norunning mishedging penalty. As in the d > 1 case in Theorem 1, trading becomes more aggressivetowards the end of the period.

If we take the interpretation of [0, T] as the trading day, then this approximation is valid whenthe the overnight jump is large compared to the daily fluctuations. This may be the case in advanceof a major post closing-bell announcement.

Figure 4 Comparison of trading intensity with and without intraperiod mishedging penalty. The parameter valuesare defined as in Figure 2

.


20/30



4.3. Restriction on the direction of trading

If executed through a broker, the delta-hedging problem is more challenging. Regulatory policymandates that brokers can only either buy or sell for any given client order. This is to preventmarket-manipulation and to protect clients from potentially unscrupulous dealers. We solve theoptimal delta-hedging problem under these constraints.

Without loss of generality, we impose a buy-only restriction on trading. That is, we add theconstraint t 0 a.s. to the minimization of our objective (6),

inf+

E

2(YTPT )2 +

T0

2

2Y2u Yuu +

22u du

, (29)

where + = { , 0 a.s.}. While the problem is still Markovian, a closed-form solution is no-longer readily available. We use a policy-improvement algorithm that assumes a Markovian optimalpolicy t = (t, Yt) to solve this problem. As a check, the same policy-improvement code was usedto solve the Simplified Model in Section 4.2 and compared against the analytic solution found inSection 4.2. It obtained continuation-value functions J accurate to within .1%.

A comparison of the restricted and unrestricted cases are plotted in Figure 5. Observe theasymmetry in trading policy (t, y) in Figure 5c. For negative values of y, (t, y) is positive toreduce the delta-hedging error, that is the agent still purchases stock as in the unrestricted case(compare with Figure 5a). For positive values of y, an unrestricted agent would sell shares but arestricted agent cannot do so, and the binding constraint forces to be zero (see Figure 5c). Thereis a no-trade curve y(t) 0 such that the agent trades if and only if he his current net delta positionis below this level, i.e. t = 0 when Yt y(t) and t > 0 when Yt < y(t). This curve is marked in theplot in Figure 5c. Observe that y(0) < 0 and t y(t) is an increasing function. Hence, far from theterminal time (t T), the agent does not purchase stocks even when he is net short delta. Theresult is interpreted thus: a selling-restricted agent is hesitant to purchase stock only to see hisdelta position become positive before T due to stock-price fluctuations. However, y(T) = 0 so thiseffect disappears as t T: as the time remaining for Yt to fluctuate above 0 runs out, the agentsrule becomes buy if I am net short delta.In the restricted case, the asymmetry in execution strategy for translates into an asymmetry forthe value function J (compare the restricted case Figure 5d with the unrestricted one Figure 5b).For negative values ofYt, J behaves similarly in both the selling-restricted and non-restricted casesas the optimal strategies are similar. For positive values of Yt, J is significantly higher in theselling-restricted case as the control cannot be exercised to reduce the hedging error. The differencebetween the two cases represents the premium of being able to sell.

4.4. Stock Pinning

The increase in trading intensity for high gamma near the expiry is the mechanism behind stockpinning. A put or call option that is about to expire with a large outstanding institutional interestand with a stock price near a strike level can induce a so-called pinning effect whereby the stock

price at the close of trading on expiry is pinned to the strike level. Empirically, this manifestsitself in the observation that the distribution of end-of-day stock prices on these days deviatessubstantially from the distribution of end-of-day stock price on any other days. That is, the stockprice clumps around the strike level at the close of such an option expiry (Avellaneda and Lipkin2003).

A call or put option near expiry with the underlying prices near its strike exhibits a large positivegamma. When institutional investors are net short gamma (perhaps because they collectively shortthe put or call) option market makers are net long gamma. The market-makers will hedge theirposition by buying stock as the price dips and selling it as the price increases, thus stabilizing


21/30



(a) Unrestricted (t, y) (b) Unrestricted J(t, y)

(c) Restricted (t, y) (d) Restricted J(t,y)

Figure 5 Plots of the continuation-value function J and the optimal policy for the unrestricted case (Figure 5band Figure 5a) and the case when trading is restricted to purchases 0 (Figure 5d and Figure 5c). In

the restricted case, we also plot the no-trade boundary in black, above which (t, y) = 0 (see Figure 5c).Parameters: = 5., = .4, = .2, = .2, = 2., T = .4, T= 1.

the price. Then as Figure 2a shows, a large gamma implies the agent trades intensely as expiryapproaches and as his trades are stabilizing, the permanent impact pins the stock price at thestrike. Since the gamma peaks near expiry with the underlying price near the strike, the effectmanifests itself as pinning near the strike.

Our model provides some insight into this model. Observe that for a call option, the delta isgiven by

C(t, p) = E [ 1PT>0|

Pt =p] = N pT tand so the gamma is given by

C(t, p) =1

22 (T t) exp

p2

2 (T t)

. (30)

For small fluctuations of p, the gamma appears to be flat. Let g = NC where N is a large positivenumber representing the number of call options held. We make the constant gamma assumptionsof Theorem 1, where now the gamma will be N C(t, p) for the gamma of the entire position.


22/30



It turns out that this assumption is valid for large N and moderately large values of T t. Forsimplicity, we take d = 1 so that t = Yt and we rewrite (7) and (8) as

dPt =1

(dYt t dt)

dYt =

(1 + )Yt dt + dWt .

Hence, we have that the stationary distribution of Yt is

Yt N

0,()2

2(1 + )

.

That is, both t and Yt are distributed with standard deviation proportional to

N in the limit asN . Hence, we have that the standard deviation of PT 1/

N as N . In other words, if

P0 starts close to the strike then Pt will remain not deviate far and the price is pinned to the strike.We can see from (30) that near the strike (p = 0) the gamma of the call option is flat in moneyness.Hence, our constant gamma approximation is valid in the limit as N as the pinned stock onlysees a flat gamma surface.

4.5. Intraday Trading Patterns

If we continue with the interpretation of the trading period as the trading day, we see that it isconsistent with the intraday trading pattern. The typical daily profile is U-shaped, with the mostintense trading occurring during the opening and close (see Figure 6a). Compare Figure 6a withthe graph of the average trading volume from delta hedging in Figure 6b as predicted by our model,

E|t|with Y0 N(0, 0). The distributional assumption on Y0 comes from the overnight fluctuationsin the underlying affecting the delta of the option. The expression for E |t| is given in (31). Theparameters for the plot are chosen to reflect realistic values and are explained in Example 4.

We see that our model predicts a trading rate that is roughly 20% of the trading for (NYSE:BA).More importantly, the trading volume profile roughly corresponds to the double-humped tradingpattern that is observed in the empirical data. Near the open, the high E |t| comes from thehigh initial variance in Yt as the trader trades down his net delta position from the previousevenings jump. Near the close, the high E|t| comes from the increase in h near t = T, whichrepresents aggressive trading in anticipation of the coming evenings overnight jump. Clearly theintraday trading volume is determined by many additional factors. However, on stocks with a largeoutstanding interest, Figure 6 illustrates how delta hedging may account for a significant fractionof the trading.

Example 4. We choose = .04 based on the corresponding value for Boeing (NYSE:BA) in Fig-ure 2 (see the chart for units). We set 0 =

(1.5

T) with T from Figure 2. This corresponds

to the change in the options delta from an overnight jump in the stock of size 1 .5 T. If the bookwere perfectly hedged the night before, we would see a jump of size T in the option value. Thefactor 1.5 accounts for the book not being perfectly hedged at the close of the previous evening.We set T = 6.5 hours, the length of the trading day. = 104 corresponds to a temporary impactof 10 cents for trading 20% of ADV or 32 shares per second. We choose the permanent impact= 105 corresponds to a permanent impact of 1 cent for trading 32 shares. For the computationof d, we choose T = 25 to penalize not just for the unhedgable overnight fluctuation but for thehedging costs to be incurred for future days starting the next morning. This penalty accounts forWe choose = 105 to corresponds to the gamma position of the entire book. In the expression


23/30



for the gamma of an option (30), (3.5 hours, 0) .1 is the gamma of an At-The-Money optionon Boeing (NYSE:BA) about 3 hours from expiry. Hence, for nearest month options we expectthe gamma of an individual option gamma to be between .01 and .1 (compare with Figure 2).Therefore, our assumption on the net book corresponds to a position with one to ten millionAt-The-Money contracts. Finally, we choose 108. This can be thought of as corresponding toa relative risk aversion of 1 for an agent with a portfolio of a million dollars.

We now give the derivation of E|t| based on the assumptions of Theorem 1. It is clear that Ytis normally distributed with mean zero. From (8) and (15), we obtain a PDE for the variance ofYt as

d

dtEY2t = 2Kh(K(T t))EY2t + ()2 .

We are interested in the more realistic case when d > 1 (see Remark 1) whose solution is given by

EY2t = e2H(t) (T)

2+ ()

2e2H(t)

t0

e2H(s) ds

where

H(t) = kKt

0

coth(kK(T s) + arccoth(d)) ds= log [sinh(kK(T t) + arccoth(d))csch(kK T + arccoth(d))] .

Since we havet0

e2H(s) ds = sinh2(kK T+ arccoth(d))

t0

csch2(K(T s) + arccoth(d)) ds

=sinh2(kK T + arccoth(d))

K[coth(K(T t) + arccoth(d)) coth(KT + arccoth(d))]

we can compute the variance of Yt as

EY2t =sinh2(kK(T t) + arccoth(d))

sinh2(kK T + arccoth(d))

(T)2

+()

2

Ksinh2(kK T+ arccoth(d))

coth(K(T t) + arccoth(d)) coth(KT + arccoth(d))

.

Since t is normally distributed, we have that

E [|t|] =2 coth(K(T t) + arccoth(d))EY2t . (31)This is plotted in Figure 6b with parameters chosen as in Example 4.

5. Discrete-Time Formulation and Solution

In this section, we assume a constant model as in Theorem 1. We need a discrete-time formulationand solution to the hedging problem in order to perform the simulations on TAQ data in Section 6.The simplest discrete strategy would be to evaluate the continuous-time strategy at each timepoint in question, buying tt shares over the interval [t, t + t]. In effect, this is a forward Euler


24/30



(a) Average Minutely Trading Volume of Boeing (NYSE:BA) in theperiod April 3rd, 2000 to March 30th, 2001.

(b) Intensity of trading E|t| for one trading day with parameters givenin Example 4. In our example, the trading predicted by our modelaccounts for 20% of trading flow.

Figure 6 Intraday trading intensity, actual and projected in our model from delta hedging.

discretization of the underlying dynamics. Unfortunately, this strategy results in unrealistic andcostly overshooting.

To illustrate this, consider the mean value of Yt, Yt = E [Yt]. Under the continuous-time policyof Theorem 1, it would behave as

Yt = Y0 t

0

Kh(K(T s))Ys ds

and converges exponentially towards 0 but remains positive if Y0 > 0. On the other hand, under adiscrete Euler policy where we purchase 0t shares, Yt would behave as

Yt = Y0 (1 Kh(KT)t) .


25/30



Depending on the size of t, this may not be positive for fixed Y0 > 0. So while this strategyconverges to the continuous strategy as t 0, for fixed t, it may overshoot zero, a result whichis known in the literature (see, for example, Ascher and Petzold (1998)). Indeed, simulations usingthe discretized Euler-method exhibited this overhedging. In order to obtain well-behaved discretetime solutions we must pose the discrete-time problem directly. The overshooting is so sever as to

lead to unstable solutions! It is especially prominent when temporary impact is small compared toK2A2(T t) (see the denominator of discrete-time solution (33)). The continuous-time solutionis important as it is closed-form and provides important intuition for understanding the problemdynamics. The discrete-time solution resembles the discretized continuous solution but makes animportant correction for overshooting (33) that is important for practical applications.

5.1. Discrete-Time Formulation

We will discretize the dynamics by imposing a lattice TN of N points of width t = T /N. Thatis TN =

TN

(Z [0, N)). At each time t TN, the agent sets his strategy for the entire timestep[t, t + t). Under such conditions, it is only necessary to consider the implied discrete-time process.Hence, the agents stock position is given by

Xt = X0 + uTn,u


26/30



PT T

BA 53.27084 0.0433451 0.8270762 0.0515023BAC 49.5763872 0.0243846 0.7866773 0.0780991MSFT 64.9043047 0.0122172 1.4496147 0.1173921PFE 43.5416817 0.0164636 0.6695676 0.1008441

WMT 52.5306878 0.0208141 0.9813519 0.0773531Table 2 Summary Statistics of Stocks: the average closing stock price PT, absolute volatility in dollars per share

per square-root seconds , the average overnight volatility in dollars per share T, and the average of acall-option that matures in 5 days and was At-The-Money at the previous close.

6. Simulation Using TAQ Data

To test the hedging strategy, we use stock TAQ Data to simulate delta hedging a call optionusing our trading strategy and compare that to the benchmark Black-Scholes strategy. We choosefive companies: Boeing (NYSE:BA), Bank of America (NYSE:BAC), Microsoft (NASDAQ:MSFT),Pfizer (NYSE:PFE), and Wal-Mart (NYSE:WMT), each representing a different sector of theeconomy. Daily NBBO data was collected for the 251 trading days from April 3rd, 2000 to March

30th, 2001 (inclusive) and the mid-price prevailing at each minute of each trading day (9:30AM to4:00PM) was struck (computed). Summary statistics of the data are given in Figure 2.We use the price data to calibrate our the discrete-time model (see Section 5) under minutely

rehedging. We then use the data to simulate delta hedging a derivatives position with a flatgamma. We chose to corresponded to that of one million call options that were At-The-Moneyat the previous trading days close and mature 5 days after that days close. These quantities arecomputed using the Black-Scholes formula.

We can think of the flat gamma derivative as approximating that of an At-The-Money call optionunder Approximation (14). A call-options gamma profile becomes very narrow in moneyness nearthe options expiry so that small fluctuations in the stock price will greatly affect the gamma, thusviolating the assumptions of the model. This feature is not unique to our setup but stems fromthe kink in the hockey-stick shape payoff of the call and the associated difficulty of hedging

At-The-Money call options near expiry is a widely recognized problem among practitioners. Wesidestep this issue in our simulation by choosing a call option that expires in 5 days.

We set the initial position X0 so that the option is initially delta hedged at the previous close.This (optimistically) simulates the position of a trader who hedges an option across multiple days.His initial position in the morning may not be hedged due to the overnight fluctuation.

We follow the logic of Example 4 in choosing our parameter values. We choose 106. Weassume no permanent impact = 0 and a temporary impact of the form = p where p > 0is a proportionality constant and is the absolute volatility of the stock. This accounts for thewell-known stylized fact that, caeteris paribus, market impact is higher for stocks with greatervolatility (Ane and Geman 2000, Cont 2001, Jain and Joh 1988). We choose p = 10

2 so that 104 as in Example 4. The parameter was chosen based on empirical values while T waschosen to be 10 times its empirical value to induce the agent to hedge aggressively towards theclose. Given our choice of parameter values, the characteristic time-scale at which trading intensitypeaks near the market close is

1

=

p

1

which is on the order of 10 to 20 minutes. In other words, for our liquid stocks, the agent tradeswith a constant proportional intensity until the last hours of the trading day.

We track the performance of two different agents, one using the proposed hedging intensity ofSection 5 (Model) and another whose strategy is to maintain a Black-Scholes delta hedged (BS).


27/30



The Black-Scholes trader will not be perfectly hedged due to fluctuations in the subsequent minuteinterval before rehedging. However, (BS) will likely maintain a tighter hedge on average as hewould be perfectly hedged if the stock did not fluctuate in the subsequent interval whereas theother trader (Model) is not even hedged at the start of the interval to begin with.

The dollar terminal exposure, dollar running exposure, and dollar impact costs are given in

Figure 3. The formulas for these quantities are given by

terminal exposure = |TYT|intraday exposure =

tT

2Y2t t (34)

impact cost =tT

2t t .

Each term has dimension of dollars to make them directly comparable. Hedging according to themodel in Section 5, (Model) saves $10K a day in market impact costs, while taking on a comparableamount of risk.

7. Conclusion

Market-impact from hedging a non-trivial outstanding position can greatly increase costs. Thispaper develops a highly-tractable framework for analyzing optimal hedging of options for traderswith large positions who face market impact. We use a linear-quadratic framework and find thatthe optimal solution for options hedging is for the agent to trade towards a target portfolio.The trader finds it prohibitively expensive to hold exactly the Black-Scholes hedge portfolio dueto the market-impact costs and can only trade towards it to minimize the turnover cost. We alsoshow that our results can be interpreted as a kind of Merton problem where the Merton-optimalportfolio is the Black-Scholes hedge portfolio.

We show that in the limit of vanishing market-impact costs, we recover that the agent engagesin Black-Scholes hedging. We apply our result to stock-pinning and our predictions are consistentwith the U-shaped intraday trading pattern. Finally, we present the results of numerical simula-tions which demonstrate the practicality of this technique even for real-world intraday stock-pricefluctuations.


28/30



Black-ScholesTerminal Intraday Impact

BA 9.965e+02 9.661e+03 1.235e+04BAC 1.283e+03 9.300e+03 1.865e+04MSFT 5.758e+03 1.387e+04 6.022e+04

PFE 1.736e+03 7.984e+03 1.583e+04WMT 2.142e+03 1.114e+04 2.564e+04

Model, = 0.5 106Terminal Intraday Impact

BA 1.086e+03 2.751e+04 5.036e+02BAC 1.392e+03 3.095e+04 6.780e+02MSFT 5.858e+03 4.969e+04 2.019e+03PFE 1.838e+03 2.572e+04 5.163e+02WMT 2.242e+03 3.641e+04 9.844e+02

Model = 1.0

106

Terminal Intraday ImpactBA 1.043e+03 2.359e+04 6.779e+02BAC 1.339e+03 2.641e+04 8.921e+02MSFT 5.808e+03 4.267e+04 2.585e+03PFE 1.789e+03 2.205e+04 6.686e+02WMT 2.193e+03 3.116e+04 1.281e+03

Model, = 2.0 106Terminal Intraday Impact

BA 1.020e+03 2.034e+04 9.253e+02BAC 1.311e+03 2.258e+04 1.192e+03

MSFT 5.783e+03 3.667e+04 3.401e+03PFE 1.763e+03 1.894e+04 8.820e+02WMT 2.167e+03 2.672e+04 1.700e+03

Table 3 Summary Statistics of Simulation

Summary Statistics of Simulation: Average dollar exposure versus dollar market impact by stockfor a simulated Black-Scholes trader and one that follows our Model. See (34) for details of theterms.

References

A. Alfonsi and A. Schied. Optimal trade execution and absence of price manipulations in limit order book

models. Working Papers, Apr 2009.

A. Alfonsi, A. Fruth, and A. Schied. Optimal execution strategies in limit order books with general shape

functions. Quantitative Finance Papers 0708.1756, arXiv.org, Aug. 2007. URL http://ideas.repec.

org/p/arx/papers/0708.1756.html.

R. Almgren and N. Chriss. Value under liquidation. Risk, Jan 1999.

R. Almgren and N. Chriss. Optimal execution of portfolio transactions. Journal of Risk, Apr 2001.

T. Ane and H. Geman. Order flow, transaction clock, and normality of asset returns. Jour-

nal of Finance, 55(5):22592284, October 2000. URL http://ideas.repec.org/a/bla/jfinan/

v55y2000i5p2259-2284.html.


29/30



U. M. Ascher and L. R. Petzold. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, PA, 1998.

M. Avellaneda and M. D. Lipkin. A market-induced mechanism for stock pinning. Quantitative Finance, 3:417425, Sep 2003.

K. Back. Insider trading in continuous time. Review of Financial Studies, Jan 1992.

P. Bank and D. Baum. Hedging and portfolio optimization in financial markets with a large trader. Mathemat-ical Finance, 14(1):118, 2004. URL http://ideas.repec.org/a/bla/mathfi/v14y2004i1p1-18.html.

D. Bertsimas and A. W. Lo. Optimal control of execution costs. Journal of Financial Markets, 1(1):150,1998. URL http://EconPapers.repec.org/RePEc:eee:finmar:v:1:y:1998:i:1:p:1-50.

F. Black and M. Scholes. The pricing of options and corporate liabilities. The Journal of Political Economy,Jan 1973.

U. Cetin, R. Jarrow, and P. Protter. Liquidity risk and arbitrage pricing theory. Finance and Stochastics, 8(3):311341, 08 2004. URL http://ideas.repec.org/a/spr/finsto/v8y2004i3p311-341.html.

U. Cetin, R. Jarrow, P. Protter, and M. Warachka. Pricing options in an extended black scholes economywith illiquidity: Theory and empirical evidence. Review of Financial Studies, 19(2):493529, 2006. URL

http://ideas.repec.org/a/oup/rfinst/v19y2006i2p493-529.html.U. Cetin, H. M. Soner, and N. Touzi. Option hedging for small investors under liquidity costs. Finance and

Stochastics, 14:317341, Jan 2010.

L. Clewlow and S. Hodges. Optimal delta-hedging under transactions costs. Journal of Economic Dynam-ics and Control, 21(8-9):13531376, June 1997. URL http://ideas.repec.org/a/eee/dyncon/v21y1997i8-9p1353-1376.html.

R. Cont. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1:223236, 2001.

J. Cvitanic. Theory of portfolio optimization in markets with frictions, 1999. URL http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.9078.

J. Cvitanic and I. Karatzas. Hedging and portfolio optimization under transaction costs: A martingaleapproach12. Mathematical Finance, 6(2):133165, 1996. ISSN 1467-9965. doi: 10.1111/j.1467-9965.

1996.tb00075.x. URL http://dx.doi.org/10.1111/j.1467-9965.1996.tb00075.x.

J. Cvitanic and H. Wang. On optimal terminal wealth under transaction costs. Journal of MathematicalEconomics, 35:223231, Jan 2001.

M. H. A. Davis and A. R. Norman. Portfolio selection with transaction costs. Mathematics of OperationsResearch, 15(4):676713, Nov 1990.

M. H. A. Davis, V. G. Panas, and T. Zariphopoulou. European option pricing with transaction costs. SIAMJ. Control Optim., 31(2):470493, 1993. ISSN 0363-0129. doi: http://dx.doi.org/10.1137/0331022.

F. Fabozzi. Robust portfolio optimization and management. Frank J. Fabozzi series. John Wiley, 2007. ISBN9780471921226. URL http://books.google.com/books?id=9QnuJqeXSmQC.

N. Garleanu and L. Pedersen. Dynamic trading with predictable returns and transaction costs. NBERWorking Paper, Mar 2009.

G. Hernandez-Del-Valle and C. G. Pacheco-Gonzalez. Optimal execution of equity with geometric priceprocess, 2009.

R. Hodges and K. Neuberger. Optimal replication of contingent claims under transaction costs. Rev. FuturesMarkets, 8:222239, 1989.

U. Horst and F. Naujokat. Illiquidity and derivative valuation. Quantitative finance papers, arXiv.org, 2008.URL http://econpapers.repec.org/RePEc:arx:papers:0901.0091.

P. C. Jain and G.-H. Joh. The dependence between hourly prices and trading volume. The Journal ofFinancial and Quantitative Analysis, 23(3):pp. 269283, 1988. ISSN 00221090. URL http://www.jstor.org/stable/2331067.


30/30



K. Janecek and S. E. Shreve. Asymptotic analysis for optimal investment and consumption with transactioncosts. Finance and Stochastics, 8:181206, Jan 2004.

P. Kratz and T. Schoneborn. Optimal liquidation in dark pools. papers.ssrn.com, Mar 2010.

A. Kyle. Continuous auctions and insider trading. Econometrica, 53(6):13151335, Jan 1985.

H. E. Leland. Option pricing and replication with transactions costs. Journal of Finance, pages 12831301,

1985.R. Merton. Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4

(1):141183, Jan 1973.

A. Obizhaeva and J. Wang. Optimal trading strategy and supply/demand dynamics. Working Paper 11444,National Bureau of Economic Research, June 2005. URL http://www.nber.org/papers/w11444.

S. Predoiu, G. Shaikhet, and S. Shreve. Optimal execution in a general one-sided limit-order book.math.cmu.edu, Feb 2010.

L. C. G. Rogers and S. Singh. The cost of illiquidity and its effects on hedging. Preprint, Aug 2007.

A. Schied and T. Schoneborn. Optimal portfolio liquidation for cara investors. Mpra paper, UniversityLibrary of Munich, Germany, 2007. URL http://econpapers.repec.org/RePEc:pra:mprapa:5075.

T. Schoeneborn and A. Schied. Liquidation in the face of adversity: stealth vs. sunshine trading.

papers.ssrn.com, Mar 2010.S. Shreve and H. Soner. Optimal investment and consumption with transaction costs. The Annals of Applied

Probability, 4(3):609692, Aug 1994.

H. M. Soner and G. Barles. Option pricing with transaction costs and a nonlinear black-scholes equa-tion. Finance and Stochastics, 2(4):369397, 1998. URL http://ideas.repec.org/a/spr/finsto/v2y1998i4p369-397.html.

H. M. Soner, S. E. Shreve, and J. Cvitanic. There is no nontrivial hedging portfolio for option pricing withtransaction costs. The Annals of Applied Probability, 5(2):327355, Jan 1995.

A. E. Whalley and P. Wilmott. An asymptotic analysis of an optimal hedging model for option pricing withtransaction costs. Mathematical Finance, 7(3):307324, 1997. URL http://ideas.repec.org/a/bla/mathfi/v7y1997i3p307-324.html.

almgren, li - closed-form solutions for option hedging with market impact

Documents