the market price of liquidityrisk from the prices of traded European plain vanilla put options.
The chapter is organized as follows: Section 1.2 summarizes the main ideasof Frey’s model. Section 1.3 motivates the introduction of liquidity as anautonomous source of risk. In a clinical study we show that modeling liquidityas a constant parameter or a deterministic function of the stock does notseem appropriate. Our general framework is presented in Section 1.4. InSection 1.5 we derive the effective dynamics for the underlying asset withSL. We use these results to compare the stock price dynamics in our SLmodel with the benchmark cases of geometric Brownian motion (as in thecase of Black and Scholes (1973) (henceforth BS)) and constant liquidity (as12 1 Modeling Feedback Effects with Stochastic Liquidity
 
in Frey (2000)). In Section 1.6 we exemplify the effects of SL by simulatingsample paths of the asset price. Section 1.7 describes the general setup fora liquidity derivative, to illustrate how the SL model can be used to buildpricing tools. The model is calibrated for an example using real-world limitorder book data so that one gets an impression of the magnitude of theliquidity effect. Furthermore, we present an innovative approach to estimatethe market price of liquidity risk from the prices of European put options.The chapter concludes in Section 1.8 with a brief summary and a discussionof issues for further research.
1.2 The Deterministic Liquidity Model
We now describe the setup proposed by Frey (2000). Assume there exists arisky asset S (the stock) and a risk-free investment earning a zero interest rate(the bond). There are no liquidity effects on the bond; only the underlyingasset S is affected by this source of risk. Furthermore, there is a single largeinvestor whose trading strategy influences the price process of the underlyingasset. The risky asset follows the stochastic differential equation
dS, = 0 S;.dW3 + pS,_d¢y,
where ¢ denotes the trading strategy of the large investor, i.e. the number ofstocks held by him or her. ¢* denotes the right-continuous version of ¢, andp = 0 is a constant liquidity parameter. An increase in p means a decliningliquidity in the market. For g = 0 the model represents the standard BSsetup with zero drift. The quantity 1/(pS) is called the market depth, i.e.the order size that moves the price by one unit. Furthermore, we need theassumption that pd¢f > —1 in order to ensure non-negativity of the assetprice.
Frey (2000) discusses the impact of the trading strategy on the price processfor the case of a smooth strategy (¢ = (¢,S) € C!)!. The partial deriv-
1¢12 denotes the set of functions in two variables that are once continuously differen-tiable in the first variable and twice continuously differentiable in the second variable.1.3 Is Market Liquidity Indeed Stochastic? 13
 
atives of ¢ are denoted by subscripts for ease of notation. This yields theeffective dynamics for the underlying asset,
dS, = b(t, S)dt + Su(t, S)\dw5,
whereots) = —“—(65) 1— pS¢s_ ps. 1 2,2bt, S) = T— pSds (#1 + Zas05%?)
assuming pS¢s < 1. Note that volatility has changed from o to a/(1 — pS¢s)compared with a perfectly liquid market.
As discussed in Frey (2000) there are two basic types of trading strategies:On the one hand, the large investor can use a positive feedback strategy, i.e.és > 0. That means he or she buys the risky asset when the price isincreasing, and he or she sells when the price is declining, thus reinforcingthe effect of rising or falling prices. For example, in a standard BS model onewould use such a strategy to duplicate a convex payoff like a long call. Onthe other hand, the large trader can employ a contrarian feedback strategy,ie. dg < 0, which means buying stocks when prices drop and vice versa.This would be the strategy used to duplicate a concave payoff, like a short
call.
The basic model of Frey (2000) is extended in the paper of Frey andPatie (2001) by introducing a deterministic liquidity function p(S). Theextension does not change the fundamental properties of the model, sinceliquidity is not an autonomous source of risk as it is perfectly correlatedwith the risky asset.
1.3 Is Market Liquidity Indeed Stochastic?
In this section we briefly discuss whether variations in liquidity can be ex-plained empirically by variations in the asset price or whether they can be14 1 Modeling Feedback Effects with Stochastic Liquidity
 
represented as proposed in Frey and Patie (2001). If this approach doesnot work and if liquidity turns out not to be constant, we take this as anindication that liquidity should be modeled as an autonomous source of risk.
From an economic point of view one may argue that investors have differentmotives for submitting orders to the stock market. On the one hand, theremay be a close relationship between the trading activity and the dynamics ofthe stock prices, for example if investors follow feedback strategies or tradefor speculative reasons. On the other hand, so-called noise traders buy andsell assets to invest cash not needed for consumption or to meet cash needsin unforeseen situations. Imagine that a lucky retail investor has won thenational lottery and now wants to buy stocks, or that an insurance companyhas to sell shares after a major damaging event to compensate clients forlosses suffered. Individual liquidity shocks that occur independently of thestock price dynamics induce stochastic changes in market liquidity. As atheoretical reference we can take, for example, Ericsson and Renault (2003).The authors explicitly model individual liquidity shocks to investors in amarket for defaultable bonds and investigate optimal liquidation strategiesif liquidity is stochastic.
From an empirical perspective one can state that a growing branch of liter-ature provides empirical evidence that market liquidity exhibits an intradayU-shaped or a J-shaped pattern, which is remarkably stable over time. Areview of related literature can be found in Coughenour and Shastri (1999)or in Ranaldo (2000). For a discussion of a functional form that is able to
reproduce an intraday U-shape pattern see, for example, Ménch (2003).
However, the intraday patterns can explain variations in market liquidityonly to a certain extent, otherwise a time series of daily data should notexhibit any variations in market liquidity. We analyzed daily limit orderbook data collected at 12.00 a.m. for every trading day from January 3 toMarch 28, 2002 for Medion AG, which was one of the most heavily tradedshares at Neuer Markt, the former market segment for growing technologycompanies at the German stock exchange up to spring 2003. Figure 1.1 showsa scatterplot for best bid prices S and percentage liquidity discounts p when1.3 Is Market Liquidity Indeed Stochastic? 15
 
 
 
Fig. 1.1: Scatterplot for best bid prices S and percentage liquidity discounts pif 5,000 Medion shares are sold in a single trade (daily data collectedat 12.00 a.m. CET for every trading day from January 3 to March 28,2002). The dotted line represents the fitted function as proposed byFrey and Patie (2001). The dark solid line shows a fitted cubic splinefor 6 intervals (5 interior knots). The light solid line represents a fittedcubic spline for 10 intervals (9 interior knots).
5,000 Medion shares are to be sold in a single trade. At first sight it is evidentthat daily liquidity discounts are not constant over time. Furthermore, onecan state that there is no significant relationship between the stock price andthe relative liquidity discounts. There is no clear pattern that would motivatea certain functional form between asset price and liquidity risk. For example,the fit of the function proposed in Frey and Patie (2001) given by
p(S) = pOMst . [1 — (S — Sp)? (arIps<so) + aalys>50))]
provides an R? of only 0.03. The fitted function is plotted as a dotted linein Figure 1.1. One may argue that other specifications of the function mightbetter explain variations in market liquidity. However, even cubic splinesthat allow for a high degree of flexibility cannot adequately represent the16 1 Modeling Feedback Effects with Stochastic Liquidity
 
data. For example, for 6 intervals (5 interior knots) we obtain an R? of 0.05,for 10 intervals (9 interior knots) the R? is 0.06. The calibrated splines areplotted as a dark and a light solid line in Figure 1.1. Of course, one couldincrease the number of knots excessively to improve the fit of the underlyingdataset. In the limit all data points may be taken as spline knots. In thiscase the spline interpolates the data points. However, such a model is notrobust concerning the time window of the sample data used for calibration.Thus, its use would be questionable from an economic point of view.
1.4 The Stochastic Liquidity Model
In our model the underlying asset price process is assumed to follow thestochastic differential equation
dS; = wy S,.dt + oS,-dW + p.S,d¢y. (1.1)
In order to be more general than Frey (2000) we relax the assumption ofzero drift and interest rate. Furthermore, we now assume that the liquidityp follows a stochastic process with dynamics given by
dps = B(t, p)dt + v(t, p)d Ww? (1.2)
with the correlation specificationdW e dW? = ydt. (1.3)A sensible choice for the stochastics of p could be a mean-reversion processwith a natural long-run level of liquidity in the market. We further assume
that the process stays strictly positive for gp > 0. This restricts the choicesfor the volatility function v(t,e). For example, one might use functions of
the type v(t, p) = ¢/p.Rewriting the above dynamics using a Cholesky decomposition of the covari-ance matrix of dS, and dp, we obtain
dS, = ppSpdt + 0S,dW; + ppSi_-ddf (1.4)B(t, p)dt + v(t, pyydW, + v(t, p)/1 — Pa, (1.5)
Il
dor1.5 The Effective Price Process in the Stochastic Liquidity Model 17
 
with a two-dimensional standard Brownian motion (W;,W;). We now as-sume that ¢ - the number of shares held by the large investor — does notonly depend on S and t, as is the case in the model with deterministic liqui-dity, but also on the SL factor p. The effects on the trading strategy are nowtwofold: First, ¢ is influenced by changes in S. Second, ¢ varies with chang-ing liquidity. The impact of S on ¢ can be modeled by the two basic typesof trading strategies. For a positive feedback strategy, ¢ is an increasingfunction of S for all ; for a contrarian feedback strategy, ¢ is a decreasingfunction of S for all p. To characterize the impact of » on ¢ consider thefollowing scenario. The more illiquid the market, the fewer shares the largetrader will hold due to external or internal regulations, no matter whethera positive or contrarian feedback strategy is considered. Thus, a reasonablechoice would be a decreasing absolute ¢-value with respect to p (for all S).
1.5 The Effective Price Process in the StochasticLiquidity Model
In this section we derive the effective price process for the SL modeland analyze it in detail. Consider a smooth trading strategy(@ = (t,S:, 1) € Ch?)?, An application of Ité’s formula leads to thefollowing proposition.
Proposition 1.1 (Dynamics of the State Variables) Suppose the trad-ing strategy of the large trader is given by $(t,S,p) € Ch??. Then, underthe assumption pSds <1 for any point in time, the solution to the systemof stochastic differential equations (1.4) and (1.5) satisfies
dS, = b(t, S,p)dt + u(t, S, p)SAW, + O(t, S, p) SAW, (1.6)dp, = Blt, p)dt + v(t, p)ydW + v(t, p) /1— Pd, (1.7)
2C1:2.2 denotes the set of functions in three variables that are once continuously differ-entiable in the first variable and twice continuously differentiable in the second and thirdvariable.18 1 Modeling Feedback Effects with Stochastic Liquidity
 
where= 7 n_ PbpYv(6$.0) = pag ta (18)ot, S,p) = Tae (1.9)S 1bt, S,p) = a [(B + 4+ Alen tds + 54m”) (1.10)
+ 5ossH(0? +87) + VSdos (ve +V1- ?)| .
The proof is given in Appendix 1.9.1.
In the case of p = 0 or ¢ = constant we are in the classical BS scenario withdrift yz, and constant volatility o. If p 4 0 and ¢ 4 0 the trading strategyof the large trader has an effect on the instantaneous volatilities v and @, aswell as on the total volatility and the correlation between the two processes.
In the special case where liquidity has no impact on the strategy of the largetrader, ie. ¢, = 0, we get close to the scenario of Frey (2000). Then,0 and the second summand of v will vanish and feedback effects are onlyincorporated due to the term pS¢g. However, since p is stochastic in ourmodel, the sensitivity of the stock price w.r.t. the trading activity of thelarge investor will now vary, in contrast to the deterministic liquidity model.
The analysis is more complex for ¢, # 0. We assume p > 0, in order todiscuss how variations in liquidity influence the trading strategy of the largeinvestor. Assume that when illiquidity increases, the large trader has toreduce the position in the stock. Thus, the investor sells shares if he orshe has a long position, or he or she buys back shares if a short position isconsidered. In the first case, ¢ is monotonically decreasing in p, starting witha positive ¢. In the latter case, ¢ is monotonically increasing in p, startingwith a negative ¢. Thus, ¢ is approaching zero in absolute value (i.e. thelarge investor has closed the position in the stock almost completely) as ptends to infinity.
We assume a positive ¢ in the following so that ¢, should be negative, nomatter if a positive feedback or a contrarian trading strategy is considered.1.5 The Effective Price Process in the Stochastic Liquidity Model 19
 
From (1.8) we can see that the sign of u(t, S, 9) depends on y. For y = 0 thevalue of v is the same as in the deterministic liquidity model. Nevertheless,in the SL model we have an additional volatility parameter @ contributingto total volatility. The parameter 0 is negative since ¢, is negative. Theinstantaneous quadratic variation d[S] is given by (see Appendix 1.9.1)a+ peg? + ey dt(1 — pS¢s)?
so that the total instantaneous volatility vz, is equal to
Vio = VPP = VO FV pb, t 270U bp (1.11)
1— pS¢s
The instantaneous covariation of the two processes d[S, p] is given by
d[S,p], = Sv (ve +/1- 70) dtvp aSv | ——2— + ro! dt. 1.12(ee "T= p8ds (1.12)
Thus, the instantaneous correlation 7 equals
_ US _ wtryl-7s
© VS v,o4dt Vv? + 02
UpPbp + YO (1.13)
Jor + pd + 2y0Vpb,
In order to compare the formal setup of Frey (2000) and Frey and Patie (2001)with the SL model, we analyze the volatility and correlation structure for
dS]; = S? (
ll
different specifications of the respective liquidity-related parameters. Anoverview is given in Table 1.1. For p = 0 or ¢ = constant, respectively,we are in the standard BS model with drift y, resulting ind =0 and v=a.The first additional feature is included in Frey’s approach where liquidity isrepresented by a constant p (i.e. do = 0) implying 6 = v = 0. This yields
«=O anda
1— pSds°The correlation parameter 7 must be zero in this scenario, since all terms
v= Vtot =
containing the partial derivative ¢, vanish (so that the dependence of thestrategy on p is of no interest for the effective stock price dynamics).20 1 Modeling Feedback Effects with Stochastic Liquidity
 
Table 1.1: Liquidity-related parameters in different models.
 
 
 
 
 
Model 7 v v Vtot 7BS 0 o 0 v 0Frey 0 TopSds 0 v 0Frey — Patie 1 i=pSde 0 v 1SL 0 (uncorr.) 23, oes P+P>yv ares <0su #0 (com) sie Moe verre oe
 
 
 
Now we take a closer look at the model of Frey and Patie (2001), where isa deterministic function of S. Therefore, the dynamics of p are only drivenby the first component W of the two-dimensional Brownian motion, whichimplies ¢ = 0 and y = 1. Since ¢ depends only on S, the volatility is thesame as in the Frey setup. From (1.13) we can deduce that 7 = 1 in thiscase. Thus, the approach of Frey and Patie (2001) is a special case of ourgeneral framework with y = 1, allowing the coefficients in the dynamics of pto depend explicitly on S.
We now consider the SL model and start with the correlation structure.
1.5.1 Impact on Correlation
It is important to note that y = 0 does not imply that the increments ofthe effective stock price process and the liquidity process are uncorrelated,1.5 The Effective Price Process in the Stochastic Liquidity Model 21
 
as one can see in equation (1.13). In the SL setup the situation where 7 = 0cannot be obtained for a deterministic choice of y. Even for y = 0, there isstill some correlation 7 between dS and dp induced by p¢p:
QR= ee <0.
The parameter 7 is negative since ¢ < 0. This term ultimately modelsliquidity feedback effects. It vanishes for a trading strategy independent ofp, ie. for ¢, = 0. The negative value of 7 can be interpreted in the followingway: If liquidity decreases over a longer period the trader will be forcedto close the position, which will cause the stock price to drop. For y 4 0the numerator of the correlation 7 in equation (1.13) carries an additionalsummand yo. For ¢, = 0 the correlation 7 between dS' and dp is equal to y,the correlation between the increments of the components of the Brownianmotion. Thus, the difference between and 7 is an exclusive result of theliquidity feedback effect.
Next, we discuss the impact of SL on volatility.
1.5.2. Impact on Volatility
First, consider the case y = 0. Then, v is the same as in the determinis-tic liquidity model, but in our framework there is the additional volatility
parameterVpbp——— <0.1— pS¢sThe variable & incorporates the liquidity feedback effect. It contributes to
v=
total volatility if and only if the trading strategy depends on the liquidity
parameter p.
In this case total volatility increases, compared with the deterministic liqui-dity model, and we obtain a total volatility of
Jor vp ee
1— pS¢s
Vtot =22 1 Modeling Feedback Effects with Stochastic Liquidity
 
Here we see a key result of our approach: For 7 = 0 the total volatility ofthe SL model is greater than the volatility in the constant liquidity setup, nomatter which strategy is used.
For 7 4 0 the direction of the changes in total volatility (see equation (1.11))depends on the sign of y. The expression for v contains an additional sum-mand describing the liquidity feedback effect, and @ is reduced in absolutevalues due to y # 0 (see equations (1.8) and(1.9)). The total volatility inequation (1.11) is a decreasing function of 7, since ¢, < 0. Thus, even for7 <0 the total volatility is greater than the volatility in Frey’s deterministicliquidity setup (2000). Only for large positive values of + is it possible thatthe total volatility is lower than the volatility in Frey’s (2000) model. Thiswill be the case if
112 7> svpldyl (1.14)
For a strategy ¢ independent of p the total volatility in the SL model is equalto the volatility in the constant liquidity model, again reflecting the lack ofthe liquidity feedback effect in this case.
Finally, we look at the two basic types of trading strategies with respect to S.
1.5.2.1 Positive Feedback Strategy
A positive feedback strategy, i.e. dg > 0, leads to
Utor > 1/0? + pPv? G2 + 2yaVpby.
The expression on the right-hand side is greater than or equal to o if and
only if11S sev Plll- (1.15)
Importantly, wo is greater than the BS volatility o for y < 0. This is similarto the result derived in Frey (2000).1.6 Numerical Results 23
 
1.5.2.2 Contrarian Feedback Strategy
A contrarian feedback strategy (ie. gs < 0) implies that the total volatility
Vor < 4/0? + pPv*g2 + 2ZyoV pb
which is less or equal to ¢ if and only if (1.14) holds. Thus, for non-positive
satisfies
values of y the instantaneous volatility in the SL model is not lower thanthe BS volatility in this case. Note that this is opposite to the result inFrey’s (2000) model.
1.6 Numerical Results
In order to visualize the formal analysis of the previous section we presentsome simulation-based results. We compare a sample path of stock prices inthe BS model with stock prices in the SL setting for both a positive and acontrarian feedback strategy.
1.6.1 Parameter Specification
In order to ensure non-negativity and stationary behavior of the liquidityprocess we specify the dynamics for p in equation (1.2) as a square-rootprocess with a mean-reverting drift component (see Cox, Ingersoll, andRoss (1985) (henceforth CIR)):
B(t,p) = (8 —p), v(t,p) = CV.To create paths of the underlying and the liquidity parameter, Monte Carlosimulation techniques are used. For a fixed realization of (W;,W,) we have
used the dynamics of equations (1.4) and (1.5) for Sp = 80.0, ¢ = 0.1 and
1. with p = 0 in the BS setting and24 1 Modeling Feedback Effects with Stochastic Liquidity
 
2. for the SL setup: «=0.35, @=p) =0.05, ¢=0.2, y=0.0.
The stochastic processes are discretized using an Euler scheme with N = 4000steps and time intervals of length At = 1/360.
Figure 1.2 shows the stock holdings of the large investor as a function of pand S for a positive (left graph) and a contrarian feedback strategy (right
graph).
 
Fig. 1.2: Positive and contrarian feedback strategy.
The shape of the functions can be explained by the following intuition. Ifliquidity drops, i.e. p increases, the large trader is forced to sell shares.Therefore, ¢ is assumed to be monotonically decreasing in p for all S and forany feedback strategy, i.e. the derivative of ¢ with respect to p is negative.It seems reasonable to assume that for very small and for very large valuesof p the absolute value of ¢, is small. In the first case, the asset still has asufficient market depth. In the latter case, the large trader has already soldalmost all of his holdings in the stock. Thus, in both scenarios, the largetrader adjusts the position in the stock by only a small amount.
In order to characterize the relationship between ¢ and S' we have to dis-tinguish between the positive feedback and the contrarian feedback strategy.In the first (second) case, the large trader buys (sells) assets as the stockprice increases and sells (buys) when the stock price declines. Thus, ¢ ismonotonically increasing (decreasing) in S for all p in the case of the posi-tive (contrarian) feedback strategy. For very small and very large values of S1.6 Numerical Results 25
 
       
 
 
 
~— stock price, if rho stoch., pos. strategy (1) 1~~ stock price in Black-Scholes (3)100 +——-—rho (4) 080.670 | 0.44 0.249 Pe an o,0 2,000 4,000
Fig. 1.3: Sample paths for the stock price in the BS model and the SL setup ifthe large trader follows a positive feedback strategy.
the changes of the stock holdings of the large trader are negligible when theasset prices vary (similar to the relationship between p and ¢). However, forasset prices in between, the absolute value of dg increases when S increasesand a positive (contrarian) feedback strategy is considered.
There are a variety of functional forms for ¢ = ¢(S,p) that are able toreproduce those features described above. We use the incomplete gammafunction to model this scenario. The exact functional form for ¢(S, e) canbe found in Appendix 11.9.2.
1.6.2 Positive Feedback Strategy
In Figure 1.3 we compare stock price paths generated by the BS model withthose produced by our SL model in the case of a positive feedback strategy.
As shown analytically in equation (1.15) one can see that for the given choiceof parameters, the volatility of the stock price in the SL model is increasedcompared with BS. If the large investor follows a positive feedback strategy26 1 Modeling Feedback Effects with Stochastic Liquidity
 
in a bullish market, the stock price dynamics in the SL model exceed thecorresponding stock prices in a BS world. Rising stock prices motivate thelarge investor to buy additional stocks, which will cause the stock price togrow even further in an illiquid market. One can notice the opposite effectfor a decreasing S' since in this case the large investor wants to get rid of the
holdings, which will accelerate the decline in the stock price.
The role of the liquidity parameter p is more subtle. In fact it can have twodifferent implications: First, all else equal, the trader has to sell shares if theybecome more and more illiquid. Second, if p is very high, the stock becomesmore volatile so that a large trader who follows a positive feedback strat-egy can cause the stock price to rise to tremendously high values in bullishmarkets. However, when illiquidity exceeds a certain threshold (p = 0.3 inFigure 1.3), the large trader is forced to close the position, and the marketcollapses. These features, which distinguish the SL model from the BS andthe deterministic liquidity model, become evident around the time step 3,500in Figure 1.3.
When p approaches zero the stock price in the SL model runs parallel to the
stock price in the BS model, as one can observe around time step 500.
1.6.3 Contrarian Feedback Strategy
In Figure 1.4 we contrast stock prices simulated in the BS model with those
generated in the SL model in the case of a contrarian feedback strategy.
In general, volatility is reduced compared with BS. However, there are alsoexceptions, as one can observe around time step 3,500. When the assetbecomes very illiquid the trading activities of the large trader can dominatethe stock price dynamics and so have a destabilizing effect. Again, this isa unique feature of the SL setup that cannot be reproduced in the constantliquidity framework.1.7 Application: Liquidity Derivative 27
 
—stock price, if rho stoch., contr, strategy (2)
400 | — stock price in Black-Scholes (3)
— tho (4) 7 0.8
 
 
 
 
0 2,000 4,000
Fig. 1.4: Contrarian feedback strategy versus Black-Scholes.1.7 Application: Liquidity Derivative
After having introduced the general framework of our model, we now presentan application that deals with the pricing of a liquidity derivative on anunderlying asset that is traded in an illiquid market. To ease the expositionwe take a step backward and restrict the general framework so that thetrading strategy of the large investor depends only on the asset price S,not on liquidity p. The derivative under consideration compensates the largeinvestor following a stop loss strategy for the liquidity discount if the stop lossorder is executed. We provide an example of such a derivative, the pricingof which is based on our sample data from the German electronic limit orderbook XETRA for the technology company Medion.
In contrast to retail investors whose individual trading volume is too smallto affect prices adversely, large traders have to consider liquidity aspects be-fore an investment is made. If the degree of illiquidity in a certain market israpidly changing over time one may hesitate to invest substantial amounts,although the risk-reward profile of the investment per se might be promis-28 1 Modeling Feedback Effects with Stochastic Liquidity
 
ing. A long position in the proposed liquidity derivative hedges against theliquidity risk and enables a large trader to act like a small retail investor. Cur-rently such liquidity derivatives are not yet actively traded in the market.The lack of appropriate pricing tools and order book data needed for cali-bration may partly explain this situation. However, as competition amongelectronic trading platforms sharpens, many exchanges might provide clients
with real-time access to data on market liquidity.
Trading strategies that limit the downside risk of a portfolio and that dependonly on the asset price are still heavily used by institutional investors. Con-sidering a simple stop loss strategy, however, one might argue that a largetrade in the stock market is usually broken up into smaller packages to min-imize the adverse impact on the overall transaction price. We neverthelessfocus on this application since, despite its simplicity, it already incorporates abasic structure and can thus serve as a guideline to construct more elaborate
pricing tools.
1.7.1 Pricing Formulas
Assume that ¢ is a stop loss trading strategy. The random variable7 := inf{t|S, < S} denotes the stopping time when the underlying assetfalls below a certain level S for the first time, implying S$, = 5. The initialprice of the underlying asset is assumed to be Sp > S. Up to the hittingtime 7