# Analytic formula for option margin with liquidity costs under dynamic delta
hedging 111This work was supported by “Human Resources Program in Energy
Technology” of the Korea Institute of Energy Technology Evaluation and
Planning (KETEP), granted financial resource from the Ministry of Trade,
Industry & Energy, Republic of Korea. (No. 20184010201680). Kyungsub Lee was
supported by the 2020 Yeungnam University Research Grant.

Kyungsub Lee<EMAIL_ADDRESS>Department of Statistics, Yeungnam
University, Gyeongsan, Gyeongbuk 38541, Korea Byoung Ki Seo 333(corresponding
author<EMAIL_ADDRESS>School of Business Administration, Ulsan National
Institute of Science and Technology, Ulsan 44919, Korea, Tel.: +82 52 217
3150.

###### Abstract

This study derives the expected liquidity cost when performing the delta
hedging process of a European option. This cost is represented by an
integration formula that includes European option prices and a certain
function depending on the delta process. We first define a unit liquidity cost
and then show that the liquidity cost is a multiplication of the unit
liquidity cost, stock price, supply curve parameter, and the square of the
number of options. Using this formula, the expected liquidity cost before
hedging can be calculated much faster than when using a Monte Carlo
simulation. Numerically computed distributions of liquidity costs in special
cases are also provided.

##  1 Introduction

When an option trader has options, s/he may perform dynamic delta hedge to
protect the price risk of the underlying. If the realized volatility is
greater than the volatility s/he expected (i.e., implied volatility s/he
entered into), then the option buyer[seller] will make a profit[loss]; Hence,
option traders are often called volatility traders. It has been established
that the instantaneous profit or loss from an option, and the respective
dynamic delta hedge, is given by the following equation:

$\mathrm{d}\Pi_{t}=\dfrac{1}{2}(\sigma_{r}^{2}-\sigma_{i}^{2})S_{t}^{2}\Gamma_{i}\mathrm{d}t$
(1)

where $\sigma_{r}$, $\sigma_{i}$, $S_{t}$, and $\Gamma_{i}$ is the realized
volatility, the implied volatility, the underlying price, and the gamma,
respectively. This means that if a volatility trader guesses the implied
volatility with relative accuracy (i.e., similarly to the (to-be) realized
volatility), then the option premium s/he paid[got] would be very close to the
profit or loss of the underlying dynamic delta hedge.

However, this equilibrium is only possible based on the assumptions that there
is no transaction cost, and that the trader can buy/sell the underlying at the
price that is taken into consideration when the realized volatility is
calculated (perhaps mid-price). Therefore, transaction cost and liquidity cost
should be counted as margins when a volatility trader considers the price of
an option. As such, the expected liquidity cost for dynamic delta hedging may
also be regarded as a major factor determining the bid/ask spread of option
prices.

In stock markets, traders buy or sell stocks through orders, including limit
and market orders. A limit order is an order to buy or sell a stock at a
specified (or a better) price that remains on an order book (i.e., the list of
all buy and sell limit orders), until it is filled, canceled, or given a
designated time. With limit orders, there is no price risk because the order
is filled at the limit (or a better) price; however, in this case, the risk is
associated with the time until, and uncertainty of, execution, as the limit
orders do not guarantee execution. A market order is an order to buy or sell a
contract at the best current price, which is determined by the outstanding
limit orders. With market orders, one can buy or sell immediately, but there
is a price risk depending on the number of contractible shares and the
liquidity of the market (i.e., the status of existing limit orders).

The distribution of the limit prices and corresponding order sizes determine
the cost of trading, especially when trading is based on market orders. If
there are fewer liquid limit orders, then there are larger bid-ask spreads,
and trading is thus likely to be more expensive. The liquidity cost during
trading also depends on the shape of the limit order curve, which is
determined by the outstanding limit price and the size of orders. This paper
provides a semi-analytic formula for the expected liquidity cost of performing
a delta hedging process on a European option within a presumed limit order
curve.

A growing body of literature has focused on modeling and examining the
dynamics of limit orders; however, this paper cites only a few recent studies.
Lo et al. (2002) proposed an econometric model of the time-to-execution of
limit orders based on survival analysis; the findings show that the
generalized gamma model fits historical data better than the theoretical first
passage time approach. Smith et al. (2003), Cont et al. (2010) and Abergel and
Jedidi (2013) developed a stochastic model for the dynamics of the order
flows, such as the limit orders, market orders, and cancellations based on the
Poisson arrivals, without specific assumptions on the behaviors or preferences
of the market participants. For more information on the statistical property
and modeling of the order book dynamics or the order-driven market, please see
Maslov (2000), Bouchaud et al. (2002), Hollifield et al. (2004), Large (2007),
Toke (2011), Gould et al. (2013), Huang and Kercheval (2012), Malo and
Pennanen (2012), Cont and De Larrard (2013), Xu (2014), and Chiu et al.
(2014).

Despite successfully incorporating the statistical properties of the order
dynamics into a stochastic order flow model, and for the analytical purpose of
the expected liquidity cost, this study employed a deterministic supply curve
of the underlying stock in accordance with Çetin et al. (2004), Çetin et al.
(2006), and Jarrow (2006). The supply curve is the stock price per share that
an investor pays or receives by a market order and is represented as a
function of the quantity of stock executed in a market order. In the
literature, the authors examined the pricing of options in the extended
framework of Black and Scholes (1973), in terms of arbitrage pricing under
illiquid conditions of an underlying asset.

There is numerous literature on computing the additional cost of delta-hedging
under the relaxation of frictionless and competitive hypotheses in the Black-
Scholes framework. It is dated back to Leland (1985), who developed a modified
option replicating strategy in the presence of transaction costs using a
discrete-time replication scheme under the continuous-time framework. Boyle
and Vorst (1992) derived replicating strategies for European options with
transaction costs in a binomial framework of Cox et al. (1979). With
proportional transaction cost, those work provided the European option prices
regardless of the supply of underlying asset, i.e., without the concept of
liquidity risk due to the timing and size of trading. Meanwhile, in the
context of Çetin et al. (2004), liquidity costs depend on the shape of
outstanding limit orders of the time when a trade occurs, and hence are
different from deterministic and proportional transaction costs or fees. The
supply curve of underlying is represented as a function that satisfies certain
conditions such as twice continuously differentiability and non-decreasing in
the trade size. We argue that a linear approximation is sufficient to
calculate the expected liquidity cost in both continuous and discrete trading.
This simplification leads that our result is also related to the classical
work with proportional transaction cost by Leland (1985) and Boyle and Vorst
(1992).

This study shows that the expected liquidity cost of the delta hedging process
depends on the linear approximation of the supply curve, in both continuous
and discrete trading. Take, as an example, a European option with a non-
perfectly liquid underlying stock in an order-driven stock market, where the
exchange is executed using limit and market orders. Furthermore, consider that
the investor who sells the European option wishes to hedge the price risk by
purchasing the delta amounts of the underlying stock. In this case, the delta
is the sensitivity of the option price with respect to the underlying stock
price change, and the expected liquidity cost of the delta hedging is
represented as an integration formula. The total expected liquidity cost of
the delta hedging process is represented by a multiplication of the unit
liquidity cost, current stock price, supply curve parameter, and the square of
the numbers of European options. The unit liquidity cost will then be the
liquidity cost of the delta hedging process and the unit supply curve
parameter.

The integration formula for the expected liquidity cost includes European
option prices with various strike prices and maturities and depends on the
delta function of the option being hedged. The methods calculated by the
options’ values in a static portfolio is an extension of previous work such
as, Demeterfi et al. (1999), Britten-Jones and Neuberger (2000), Carr and
Madan (2001), Carr and Wu (2009), Carr and Wu (2013), Choe and Lee (2014), and
Lee (2014). The numerical computation of the integration is much faster than
the computation of the liquidity cost based on a simulation.

The remainder of the paper is organized as follows. Section 2 derives the
integration formula for the expected liquidity cost of delta hedging processes
under a continuous trading assumption and provides an example in the Black-
Scholes framework. The derivation is extended to the discrete trading case in
Section 3. Section 4 reports the result of numerical and simulation tests for
the expected liquidity cost, and Section 5 shows the distribution of the
liquidity cost in special cases. Section 6 explains the relationship between
an existing queuing limit order model of Cont et al. (2010) and our approach
to calculate the expected liquidity cost. Finally, section 7 concludes the
paper.

##  2 Expected liquidity cost under continuous trading

This section introduces a probability space with a time index set, $[0,T]$,
for some fixed $T>0$. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a filtered
probability space with a filtration, $\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$,
where $\mathcal{F}_{T}=\mathcal{F}$. This space satisfies the usual
conditions. The measure, $\mathbb{P}$, is the physical probability measure and
there exists an equivalent martingale measure, $\mathbb{Q}$, under which all
financial assets are priced (i.e., the discounted asset prices are
martingale). It is worth noting that all the processes introduced in this
paper are defined in the probability space and are adapted to filtration.

If an investor wishes to perform a delta hedge process of an option with an
underlying stock that is not perfectly liquid, they can immediately buy or
sell the necessary number of shares for the process at the best available
price (i.e., a market order) whenever they want. The best price is determined
by the outstanding limit orders, price specifications, and the number of
shares for buying or selling, which is given by liquidity providers.

Table 1 and the blue solid lines of Figure 1 illustrate an example of
outstanding limit orders of Microsoft (left) and Intel (right) observed on
2012.06.21. When the quantity (x-axis) of the graph is negative it is for a
bid order, and when the quantity is positive, it is for an ask order. The
table shows that if the investor wants to buy 150,000 shares of Microsoft by
market orders, they pay 30.14 dollars for 28,632 shares, 30.15 dollars for
83.663 shares and 30.16 dollars for 37,705 shares. Let $m(q)$ be the limit
order curve presented in the figure (blue solid lines) as a function of the
quantity $q$. This function is also called the marginal price curve.

The supply curve of the underlying stock at time $t$ is defined as

$S(t,z)=\frac{1}{z}\int_{0}^{z}m(q)\mathrm{d}q$ (2)

which represents the stock price that an investor pays (or receives) for order
flow $z>0$ (or $z<0$) per share. The terminology “supply curve” comes from
Çetin et al. (2004). In some studies, the quantity,
$\int_{0}^{z}m(q)\mathrm{d}q$, is called the total cost of a market order of
$z$ shares of the stock (Malo and Pennanen, 2012).

When $z=0$, $S(t,z)=S_{t}$, which can be the mid-price of the best bid and ask
prices. In Figure 1, the supply curves are indicated by the dotted red lines
on the corresponding blue solid limit order curves. Practically, due to the
minimum tick size, the plotted supply curve is a piecewise differentiable
curve, and there is a jump at $z=0$. On the other hand, for analysis of
continuous trading, it is assumed that the supply curve is differentiable with
respect to $z$, at least around $z=0$. This theoretical assumption can be
achieved by smoothing the curve around zero; an example of the smoothing
procedure is provided later in the paper. The analysis based on continuous
trading and smooth supply curve is a theoretical issue. From a practical
perspective in the discrete trading case, the differentiability of the supply
curve is not required, as discussed in the next section.

According to Çetin et al. (2004), the supply curve should satisfies:

  1. 1.

$S(t,z)$ is $\mathcal{F}_{t}$-measurable and non-negative.

  2. 2.

$S(t,z)$ is non-decreasing in $z$, i.e., $z_{1}\leq z_{2}$ implies
$S(t,z_{1})\leq S(t,z_{2})$.

  3. 3.

$S(t,z)$ is $C^{2}$ in $z$.

  4. 4.

$S(\cdot,0)$ is a semi-martingale.

  5. 5.

$S(\cdot,z)$ has continuous sample paths for all $z$.

Çetin et al. (2006) assumed an exponential supply curve,

$S(t,z)=\mathrm{e}^{\alpha z}S(t,0),$

which is chosen for simplicity and the ease of generalization. The other
alternative forms are the supply curves with diminishing marginal price
impact, $\alpha\mathrm{sign}(z)\sqrt{|z|}$ and
$\alpha\mathrm{sign}(z)\log(1+|z|)$. The linear supply curve is the first-
order approximation of the exponential form. Blais and Protter (2010) reported
that linear supply curves are fitted well for highly liquid stocks and jump
linear curves are suitable for less liquid equities.

Table 1: Limit orders of Microsoft (MSFT) and Intel (INTC) at a certain moment
on 2012.06.21.

Size | Bid price | Ask price | Size
---|---|---|---
51,326 | 30.13 | 30.14 | 28,632
84,106 | 30.12 | 30.15 | 83,663
8,706 | 30.11 | 30.16 | 66,999
44,038 | 30.10 | 30.17 | 86,886
167,571 | 30.09 | 30.18 | 110,006
14,134 | 30.08 | 30.19 | 30,006
46,380 | 30.07 | 30.20 | 72,106
23,774 | 30.06 | 30.21 | 56,500
23,646 | 30.05 | 30.22 | 36,532
36,675 | 30.04 | 30.23 | 31,600

Size | Bid price | Ask price | Size
---|---|---|---
9,091 | 26.71 | 26.72 | 125,104
34,683 | 26.70 | 26.73 | 174,683
30,295 | 26.69 | 26.74 | 110,674
47,583 | 26.68 | 26.75 | 59,778
58,874 | 26.67 | 26.76 | 60,883
79,774 | 26.66 | 26.77 | 39,946
78,200 | 26.65 | 26.78 | 62,840
11,200 | 26.64 | 26.79 | 28,655
24,200 | 26.63 | 26.80 | 43,600
52,000 | 26.62 | 26.81 | 47,600

Figure 1: Limit order curve (solid blue line) and corresponding supply curve
(dotted red line) of Microsoft (left) and Intel (right) at a certain moment on
2012.06.21.

In some studies, including Çetin et al. (2004), Çetin et al. (2006) and Jarrow
(2006), the liquidity cost during a trading period of $0<u\leq t$ is defined
as

$L_{t}=\sum_{0<u\leq t}\Delta Z_{u}[S(u,\Delta
Z_{u})-S(u,0)]+\int_{0}^{t}\frac{\partial S}{\partial
z}(u,0)\mathrm{d}[Z]_{u}^{c}$ (3)

where $Z_{t}$ is the number of holding shares of the stock at $t$, and
$[Z]^{c}$ is the quadratic variation process of the continuous part of $Z$.
The first term on the right-hand side represents the liquidity cost for
discrete trading. The term $[S(u,\Delta Z_{u})-S(u,0)]$ represents the
additional cost for $\Delta Z_{u}$ shares of the stock due to illiquidity. The
second term, involving an integration with respect to the quadratic variation
process of the continuous part of $Z$, represents the liquidity cost for
continuous trading. Roughly, the integration form in the equation originates
from

$\sum_{0<u\leq t}\frac{S(u,\Delta Z_{u})-S(u,0)}{\Delta Z_{u}}(\Delta
Z_{u})^{2}\rightarrow\int_{0}^{t}\frac{\partial S}{\partial
z}(u,0)\mathrm{d}[Z]_{u}^{c}.$ (4)

In Eq. (3), the trading cost at time 0 is executed, because in a delta hedging
process of a European option, the initial cost is simply calculated by the
initial delta of said option.

As only the first-order partial derivative of $S(u,z)$ with respect to $z$
appears in the integration form in Eq. (3), only the linear approximation of
the supply curve at $z=0$ needs to be considered for analysis in the case of
continuous trading. The linear approximation is represented by $S(t,z)\approx
S_{t}+\alpha S_{t}z$ with some constant $\alpha$.

Consider the liquidity cost due to continuous trading with a continuous delta
process $D(t,x)$ with $x=S_{t}$ at time $t$, where the delta process is
assumed to be a function of $t$ and $S_{t}$, as follows:

$\int_{0}^{T}\frac{\partial S}{\partial
z}(t,0)\mathrm{d}[Z]_{t}^{c}\approx\int_{0}^{T}\alpha
S_{t}\mathrm{d}[D]_{t}=\int_{0}^{T}\alpha S_{t}\left(\frac{\partial
D(t,S_{t})}{\partial x}\right)^{2}\mathrm{d}[S]_{t}$ (5)

where $T$ is the maturity of the option and we use the fact that

$\mathrm{d}D_{t}=\left(\frac{\partial D(t,S_{t})}{\partial
t}+\frac{1}{2}\frac{\partial^{2}D(t,S_{t})}{\partial
x^{2}}\right)\mathrm{d}t+\frac{\partial D(t,S_{t})}{\partial
x}\mathrm{d}S_{t}$

and

$\mathrm{d}[D]_{t}=\left(\frac{\partial D(t,S_{t})}{\partial
x}\right)^{2}\mathrm{d}[S]_{t}.$

The superscript $c$ of $[S]$ is omitted because the continuous underlying
price process is considered. For instance, under the Black-Scholes framework,
$\frac{\partial D(t,S_{t})}{\partial x}$ is simply the Black-Scholes gamma and
$\mathrm{d}[S]_{t}=\sigma^{2}S_{t}^{2}\mathrm{d}t$ with a volatility parameter
$\sigma$.

The aim is to represent the risk-neutral expectation of the above equation in
terms of European option prices with various maturities and strike prices. A
future price process with maturity $t$, defined by
$F^{(t)}_{u}=\mathbb{E}^{\mathbb{Q}}[S_{t}|\mathcal{F}_{u}]$, for $u\leq t$, a
martingale under $\mathbb{Q}$ is introduced. For simplicity, assume that the
instantaneous interest rate is constant at $r$. The future price process can
then be represented simply as $F^{(t)}_{u}=\mathrm{e}^{r(t-u)}S_{u}$.

Hereafter, $\mathbb{E}^{\mathbb{Q}}[\int_{0}^{T}\alpha
S_{t}\mathrm{d}[D]_{t}]$ is derived from the European option prices. If
$F^{(t)}$ is continuous, then for a continuous function $g(x)$ and its
antiderivative $G(x)$, it is known that

$\displaystyle\begin{aligned}
\int_{0}^{t}g\left(F^{(t)}_{s}\right)\mathrm{d}[F^{(t)}]_{s}&=2\left\\{\int_{0}^{t}\left(G\left(F^{(t)}_{0}\right)-G\left(F^{(t)}_{s}\right)\right)\mathrm{d}F^{(t)}_{s}\right.\\\
&\left.+\int_{0}^{F^{(t)}_{0}}g(k)\left(k-F^{(t)}_{t}\right)^{+}\mathrm{d}k+\int_{F^{(t)}_{0}}^{\infty}g(k)\left(F^{(t)}_{t}-k\right)^{+}\mathrm{d}k\right\\}.\end{aligned}$
(6)

Please see Choe and Lee (2014) and Lee (2014). Note that $F^{(t)}_{t}$ simply
equals $S_{t}$. This is derived from a comparison of the following two
equations, involving Itò’s formula for $\bar{G}$, which is the second
antiderivative of $g$

$\displaystyle\bar{G}\left(F_{t}^{(t)}\right)-\bar{G}\left(F_{0}^{(t)}\right)=\int_{0}^{t}G\left(F_{s}^{(t)}\right)\mathrm{d}F_{s}^{(t)}+\frac{1}{2}\int_{0}^{t}g\left(F_{0}^{(t)}\right)\mathrm{d}[F^{(t)}]_{s}$
(7)

and Taylor’s theorem with the integral form of the remainder term

$\displaystyle\begin{aligned}
\bar{G}\left(F_{t}^{(t)}\right)-\bar{G}\left(F_{0}^{(t)}\right)&=G\left(F_{0}^{(t)}\right)\left(F_{t}^{(t)}-F_{0}^{(t)}\right)\\\
&+\int_{0}^{F^{(t)}_{0}}g(k)\left(k-F^{(t)}_{t}\right)^{+}\mathrm{d}k+\int_{F^{(t)}_{0}}^{\infty}g(k)\left(F^{(t)}_{t}-k\right)^{+}\mathrm{d}k.\end{aligned}$
(8)

In addition, by taking $\mathbb{Q}$-expectation on both sides of Eq. (6) and
assuming the stochastic integration with respect to $F^{(t)}$ in the equation
is a $\mathbb{Q}$-martingale, we have

$\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{t}g\left(F^{(t)}_{s}\right)\mathrm{d}[F^{(t)}]_{s}\right]=2\mathrm{e}^{rt}\int_{0}^{\infty}g(k)\phi^{(t)}(S_{0},k)\mathrm{d}k$
(9)

or equivalently, under the constant interest rate assumption,

$\displaystyle\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{t}\mathrm{e}^{2r(t-s)}g\left(e^{r(t-s)}S_{s}\right)\mathrm{d}[S]_{s}\right]=2\mathrm{e}^{rt}\int_{0}^{\infty}g(k)\phi^{(t)}(S_{0},k)\mathrm{d}k$
(10)

where

$\phi^{(t)}\left(S_{0},k\right)=\left\\{\begin{array}[]{ll}p^{(t)}\left(S_{0},k\right)=\mathrm{e}^{-rt}\mathbb{E}^{\mathbb{Q}}[(k-S_{t})^{+}],&0<k\leq\mathrm{e}^{rt}S_{0},\\\
c^{(t)}\left(S_{0},k\right)=\mathrm{e}^{-rt}\mathbb{E}^{\mathbb{Q}}[(S_{t}-k)^{+}],&\mathrm{e}^{rt}S_{0}<k<\infty.\end{array}\right.$
(11)

Note that $p^{(t)}\left(S_{0},k\right)$ and $c^{(t)}\left(S_{0},k\right)$ are
the European put and call option prices, with maturity $t$ and strike price
$k$, respectively.

In Eq. (10), the function $g$ in the integrand depends solely on the stock
price process $S$, and we need an extended version of the above argument to a
function of two variables of $t$ and $S_{t}$ to deal with the expected
liquidity cost represented in Eq. (5). The following technical assumption is
for treating a non-differentiable point in European option payoffs.

######  Assumption 1.

Let $\psi(t,x)\in C^{1}$ in both $t$ and $x$. The partial derivatives,
$\frac{\partial\psi}{\partial x}$ and $\frac{\partial^{2}\psi}{\partial
t\partial x}$, are absolutely continuous with respect to $x$, for all $t$,
such that

$\displaystyle\frac{\partial\psi}{\partial
x}(t,b)-\frac{\partial\psi}{\partial
x}(t,a)=\int_{a}^{b}\frac{\partial^{2}\psi}{\partial x^{2}}(t,x)\mathrm{d}x$
(12) $\displaystyle\frac{\partial^{2}\psi}{\partial t\partial
x}(t,b)-\frac{\partial^{2}\psi}{\partial t\partial
x}(t,a)=\int_{a}^{b}\frac{\partial^{3}\psi}{\partial t\partial
x^{2}}(t,x)\mathrm{d}x$ (13)

where the second partial derivatives in the r.h.s. can be the generalized
derivatives in the distributional sense. Let $F^{(T)}$ be a continuous process
and assume that

$\displaystyle\int_{0}^{t}\frac{\partial\psi}{\partial
x}\left(s,F^{(T)}_{s}\right)\mathrm{d}F_{s}^{(T)},\quad t\leq T$ (14)

is a $\mathbb{Q}$-martingale with respect to $\mathcal{F}$. For example, if

$\frac{\partial\psi}{\partial x}\left(t,F^{(T)}_{t}\right)\in
L^{2}_{\mathbb{Q},[F^{(T)}]}([0,T]\times\Omega)$ (15)

where $L^{2}_{\mathbb{Q},[F^{(T)}]}([0,T]\times\Omega)$ is the space of the
adapted process X, such that

$\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}X_{u}^{2}\mathrm{d}[F^{(T)}]_{u}\right]<\infty$
(16)

then the martingale property of the stochastic integral is guaranteed, see Kuo
(2006). In addition, for convenience, let

$f(t,x)=\frac{\partial^{2}\psi}{\partial x^{2}}(t,x).$ (17)

######  Theorem 1.

Under Assumption 1, The risk-neutral expectation of the integration of
$f\left(t,F^{(T)}\right)$ with respect to the quadratic variation of the
futures price process $[F^{(T)}]$ is represented by the weighted European
option prices. That is

$\displaystyle\begin{aligned}
\mathbb{E}^{\mathbb{Q}}&\left[\int_{0}^{T}\mathrm{e}^{2r(T-t)}f\left(t,\mathrm{e}^{r(T-t)}S_{t}\right)\mathrm{d}[S]_{t}\right]\\\
=&-2\int_{0}^{T}\int_{0}^{\infty}\mathrm{e}^{r(2T-t)}\frac{\partial
f}{\partial
t}\left(t,\mathrm{e}^{r(T-t)}k\right)\phi^{(t)}(S_{0},k)\mathrm{d}k\mathrm{d}t\\\
&+2\mathrm{e}^{rT}\int_{0}^{\infty}f(T,k)\phi^{(T)}(S_{0},k)\mathrm{d}k.\end{aligned}$
(18)

###### Proof.

By Taylor’s theorem

$\displaystyle\begin{aligned}
\psi\left(T,F^{(T)}_{T}\right)={}&\psi\left(T,F^{(T)}_{0}\right)+\frac{\partial\psi}{\partial
x}\left(T,F^{(T)}_{0}\right)\left(F^{(T)}_{T}-F^{(T)}_{0}\right)\\\
&+\int_{0}^{F^{(T)}_{0}}\frac{\partial^{2}\psi}{\partial
x^{2}}(T,k)\left(k-F^{(T)}_{T}\right)^{+}\mathrm{d}k\\\
&+\int_{F^{(T)}_{0}}^{\infty}\frac{\partial^{2}\psi}{\partial
x^{2}}(T,k)\left(F^{(T)}_{T}-k\right)^{+}\mathrm{d}k.\end{aligned}$ (19)

In addition, according to Meyer-Itó’s formula (Protter, 2005),

$\displaystyle\begin{aligned}
\psi\left(T,F^{(T)}_{T}\right)={}&\psi\left(0,F^{(T)}_{0}\right)+\int_{0}^{T}\frac{\partial\psi}{\partial
t}\left(t,F^{(T)}_{t}\right)\mathrm{d}t+\int_{0}^{T}\frac{\partial\psi}{\partial
x}\left(t,F^{(T)}_{t}\right)\mathrm{d}F^{(T)}_{t}\\\
&+\frac{1}{2}\int_{0}^{T}\frac{\partial^{2}\psi}{\partial
x^{2}}\left(t,F^{(T)}_{t}\right)\mathrm{d}[F^{(T)}]_{t}.\end{aligned}$ (20)

By combining the above results, we have

$\displaystyle\begin{aligned}
\frac{1}{2}\int_{0}^{T}\frac{\partial^{2}\psi}{\partial
x^{2}}&\left(t,F^{(T)}_{t}\right)\mathrm{d}[F^{(T)}]_{t}=\int_{0}^{T}\left(\frac{\partial\psi}{\partial
t}\left(t,F^{(T)}_{0}\right)-\frac{\partial\psi}{\partial
t}\left(t,F^{(T)}_{t}\right)\right)\mathrm{d}t\\\
&+\int_{0}^{T}\left(\frac{\partial\psi}{\partial
x}\left(T,F^{(T)}_{0}\right)-\frac{\partial\psi}{\partial
x}\left(t,F^{(T)}_{t}\right)\right)\mathrm{d}F^{(T)}_{t}\\\
&+\int_{0}^{F^{(T)}_{0}}\frac{\partial^{2}\psi}{\partial
x^{2}}(T,k)\left(k-F^{(T)}_{T}\right)^{+}\mathrm{d}k\\\
&+\int_{F^{(T)}_{0}}^{\infty}\frac{\partial^{2}\psi}{\partial
x^{2}}(T,k)\left(F^{(T)}_{T}-k\right)^{+}\mathrm{d}k.\end{aligned}$ (21)

To deal with the integration term with respect to $t$ of the r.h.s. of the
above equation, recall Taylor’s theorem

$\displaystyle\begin{aligned} \frac{\partial\psi}{\partial
t}\left(t,F^{(T)}_{0}\right)-&\frac{\partial\psi}{\partial
t}\left(t,F^{(T)}_{t}\right)=-\frac{\partial^{2}\psi}{\partial t\partial
x}\left(t,F_{0}^{(T)}\right)\left(F_{t}^{(T)}-F_{0}^{(T)}\right)\\\
&-\int_{0}^{F^{(T)}_{0}}\frac{\partial^{3}\psi}{\partial t\partial
x^{2}}(t,k^{\prime})\left(k^{\prime}-F_{t}^{(T)}\right)^{+}\mathrm{d}k^{\prime}\\\
&-\int_{F^{(T)}_{0}}^{\infty}\frac{\partial^{3}\psi}{\partial t\partial
x^{2}}(t,k^{\prime})\left(F_{t}^{(T)}-k^{\prime}\right)^{+}\mathrm{d}k^{\prime}.\end{aligned}$
(22)

Putting $k=k^{\prime}\mathrm{e}^{-r(T-t)}$ and taking the
$\mathbb{Q}$-expectation on both sides, we obtain

$\displaystyle\begin{aligned}
\mathbb{E}^{\mathbb{Q}}&\left[\frac{\partial\psi}{\partial
t}\left(t,F^{(T)}_{0}\right)-\frac{\partial\psi}{\partial
t}\left(t,F^{(T)}_{t}\right)\right]\\\
&=-\mathrm{e}^{r(2T-t)}\int_{0}^{\infty}\frac{\partial f}{\partial
t}\left(t,\mathrm{e}^{r(T-t)}k\right)\phi^{(t)}(S_{0},k)\mathrm{d}k.\end{aligned}$
(23)

Therefore, by taking $\mathbb{Q}$-expectation on Eq. (21) and noting that the
stochastic integration with respect to $F_{t}^{(T)}$ (the second term of the
r.h.s. of the equation) is a $\mathbb{Q}$-martingale, we have

$\displaystyle\begin{aligned}
\mathbb{E}^{\mathbb{Q}}&\left[\int_{0}^{T}f\left(t,F^{(T)}_{t}\right)\mathrm{d}[F^{(T)}]_{t}\right]\\\
=&-2\int_{0}^{T}\int_{0}^{\infty}\mathrm{e}^{r(2T-t)}\frac{\partial
f}{\partial
t}\left(t,\mathrm{e}^{r(T-t)}k\right)\phi^{(t)}(S_{0},k)\mathrm{d}k\mathrm{d}t\\\
&+2\mathrm{e}^{rT}\int_{0}^{\infty}f(T,k)\phi^{(T)}(S_{0},k)\mathrm{d}k.\end{aligned}$
(24)

Finally, $F_{t}^{(T)}=\mathrm{e}^{r(T-t)}S_{t}$ and
$\mathrm{d}[F^{(T)}]_{t}=\mathrm{e}^{2r(T-t)}\mathrm{d}[S]_{t}$ is used. ∎

In the above theorem, the expected liquidity cost for a delta hedging process
is represented by a Riemann integration formula. By putting

$f(t,x)=\mathrm{e}^{-3r(T-t)}x\left(\frac{\partial D}{\partial
x}\left(t,\mathrm{e}^{-r(T-t)}x\right)\right)^{2},$ (25)

and applying the above theorem,

$\displaystyle\begin{aligned} \mathbb{E}^{\mathbb{Q}}&\left[\int_{0}^{T}\alpha
S_{t}\mathrm{d}[D]_{t}\right]=\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}\alpha
S_{t}\left(\frac{\partial D(t,S_{t})}{\partial
x}\right)^{2}\mathrm{d}[S]_{t}\right]\\\
={}&\alpha\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}\mathrm{e}^{2r(T-t)}f\left(t,\mathrm{e}^{r(T-t)}S_{t}\right)\mathrm{d}[S]_{t}\right]\\\
={}&-2\alpha\int_{0}^{T}\int_{0}^{\infty}\mathrm{e}^{r(2T-t)}\frac{\partial
f}{\partial
t}\left(t,\mathrm{e}^{r(T-t)}k\right)\phi^{(t)}(S_{0},k)\mathrm{d}k\mathrm{d}t\\\
&+2\alpha\mathrm{e}^{rT}\int_{0}^{\infty}f(T,k)\phi^{(T)}(S_{0},k)\mathrm{d}k.\end{aligned}$
(26)

The expected liquidity increases linearly with $\alpha$ and quadratically with
the number of options being hedged. The expected liquidity cost can be
conveniently represented in the following way.

######  Theorem 2.

Consider the delta hedging process of $N$ numbers of European options with a
strike $K$, maturity $T$, and current underlying price $S_{0}$. Let
$M=K/S_{0}$ and define the unit liquidity cost by

$\displaystyle\begin{aligned}
I={}&-2\int_{0}^{T}\int_{0}^{\infty}\mathrm{e}^{r(2T-t)}\frac{\partial
f}{\partial
t}\left(t,\mathrm{e}^{r(T-t)}k;M,T\right)\phi^{(t)}(1,k)\mathrm{d}k\mathrm{d}t\\\
&+2\mathrm{e}^{rT}\int_{0}^{\infty}f(T,k;M,T)\phi^{(T)}(1,k)\mathrm{d}k\end{aligned}$
(27)

where

$f(t,x;M,T)=\mathrm{e}^{-3r(T-t)}x\left(\frac{\partial\bar{D}}{\partial
x}\left(t,\mathrm{e}^{-r(T-t)}x;M,T\right)\right)^{2}$ (28)

and $\bar{D}(t,x;M,T)$ is the delta function of the European option with a
strike $M$, maturity $T$, and current underlying price 1; in other words, the
values are denominated by the current underlying stock price, $S_{0}$. The
expected liquidity cost of the delta hedging process of $D=N\bar{D}$ can then
be expressed as

$\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}\alpha
S_{t}\mathrm{d}[D]_{t}\right]=\alpha N^{2}S_{0}I.$ (29)

###### Proof.

Let $\bar{S}_{t}=S_{t}/S_{0}$. Then

$\displaystyle\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}\alpha
S_{t}\mathrm{d}[D]_{t}\right]=\alpha
N^{2}S_{0}\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}\alpha\bar{S}_{t}\mathrm{d}[\bar{D}]_{t}\right]$
(30)

and use Theorem 1. ∎

######  Example 1.

Suppose that the Black-Scholes delta function is used to hedge the option. As
an exemplary case, the trading strategy is the trader’s own choice and does
not need to be an optimal replication. The delta process of the European call
option with strike $K$ and maturity $T$ is

$D(t,S_{t})=\Phi(h_{t}),\quad
h_{t}=\frac{\log\frac{S_{t}}{K}+\left(r+\frac{\sigma^{2}}{2}\right)(T-t)}{\sigma\sqrt{T-t}},$
(31)

for $0\leq t<T$, where $\Phi$ denotes the standard cumulative normal
distribution. Therefore, the gamma of the option $x=S_{t}$ is

$\frac{\partial D(t,x)}{\partial
x}=\frac{1}{x\sigma\sqrt{2\pi(T-t)}}\exp\left[-\frac{1}{2}\left\\{\frac{\log\frac{x}{K}+\left(r+\frac{\sigma^{2}}{2}\right)(T-t)}{\sigma\sqrt{T-t}}\right\\}^{2}\right]$
(32)

and

$f(t,x)=\frac{1}{2\pi
x\sigma^{2}(T-t)}\exp\left[-r(T-t)-\left\\{\frac{\log\frac{x}{K}+\frac{\sigma^{2}}{2}(T-t)}{\sigma\sqrt{T-t}}\right\\}^{2}\right].$
(33)

and

$\displaystyle\begin{aligned} \frac{\partial f}{\partial
t}(t,x)={}&\frac{\sigma^{2}\left\\{4+4r(T-t)+\sigma^{2}(T-t)\right\\}(T-t)-4\log^{2}\frac{x}{K}}{8\pi\sigma^{4}(T-t)^{3}x}\\\
&\times\exp\left[-r(T-t)-\left\\{\frac{\log\frac{x}{K}+\frac{\sigma^{2}}{2}(T-t)}{\sigma\sqrt{T-t}}\right\\}^{2}\right].\end{aligned}$
(34)

For the above formula to be valid, we assume that there are no additional
costs such as transaction costs. With an infinite variation hedging strategy,
the theoretical super-replication price of an option is explained in Çetin et
al. (2010).

######  Remark 3.

To represent the expected liquidity cost in terms of the European option
prices, the $\mathbb{Q}$-expectation is taken as the liquidity cost in this
study. On the other hand, for example, under the Black-Scholes framework
(i.e., if the underlying stock price process follows geometric Brownian
motion), one can simply change $r$ as the drift of the stock price process in
order to derive the expectation in terms of physical probability.

##  3 Expected liquidity cost under discrete trading

The previous section considers the expected liquidity cost under the
continuous trading assumption (with an infinite variation trading strategy).
However, the application of continuous trading is not practical here. This
section examines the expected liquidity cost under the discrete time trading
of the delta hedging process. The methodology is similar to that of the
continuous case.

In Figure 2, an example of the supply curve is provided and its linear
approximations plotted. The straight line represents the first-order
approximation around zero, and was used to calculate the expected liquidity
cost under continuous time trading in the previous section. For discrete time
trading, assume that the hedger rebalances the portfolio when the necessary
change in delta exceeds a certain fixed quantity, for example, 3,000 shares.
According to the given supply curve in the figure, the first-order
approximation of the supply curve for the discrete trading (the dashed line in
the figure) is different from the one of continuous trading. Therefore, the
liquidity supply curve parameter, $\alpha^{\prime}$, is used for discrete
trading, which may differ from $\alpha$.

The slope of the supply curve is assumed to be symmetrical as Biais et al.
(1995), Bouchaud et al. (2002), and Potters and Bouchaud (2003) reported no
significant difference between the mean bid and ask side depths of the
outstanding limit orders. In addition, studies have also reported on the
asymmetrical curve shape; see Gu et al. (2008) and Malo and Pennanen (2012).

Figure 2: Example of the supply curve and first-order approximation for
continuous trading (straight line) with a slope $\alpha$, and for discrete
trading (dashed line) with slope $\alpha’$

In addition, it is assumed that the limit order book is perfectly resilient
and that the supply curve is exogenous. If the hedger buys or sells certain
amounts of underlying stock to rebalance the delta, which results in a
depletion of existing limit orders, then the limit orders are replenished
immediately so that there are no effects on the following trades or the other
investors’ trades.

With a delta process, $D(t,S_{t})$, the discrete trading liquidity cost term
in Eq. (3) is rewritten as

$\sum_{0<u\leq t}\Delta D(u,S_{u})[S(u,\Delta D(u,S_{u}))-S_{u}]=\sum_{0<u\leq
t}\alpha^{\prime}S_{u}(\Delta D(u,S_{u}))^{2}$ (35)

where $\alpha^{\prime}S_{u}\Delta D(u,S_{u})$ is a linear form of $S(u,\Delta
D(u,S_{u}))-S_{u}$. Note that the above is not an approximation, but exactly
equivalent under the discrete trading strategy with a fixed $\Delta D$ and
supply curve. If $\Delta D$ or the supply curve is not fixed, then the
liquidity cost cannot be represented by a single parameter $\alpha^{\prime}$.
In addition, by taking the first-order approximation of a delta function, the
liquidity cost can be approximated as

$\sum_{0<u\leq t}\alpha^{\prime}S_{u}\left(\frac{\partial D(u,S_{u})}{\partial
x}\right)^{2}(\Delta S_{u})^{2}$ (36)

and our goal is to compute the expectation of the above equation.

To treat discrete trading, it is convenient to consider a pure jump process
version of the underlying price and the corresponding futures prices (even
though the actual price process is continuous) in the following way. Let
$0=\tau_{0}<\cdots<\tau_{i}<\tau_{i+1}<\cdots$ be the stopping time such that
each $\tau_{i}$ is the rebalancing time of the delta hedging portfolio, by
changing the delta with some fixed $\Delta D$. In addition, let
$\bar{F}^{(t)}$ be the pure jump version of the futures process $F^{(t)}$,
such that $\bar{F}^{(t)}_{\tau_{i}}=F^{(t)}_{\tau_{i}}$ and
$\bar{F}^{(t)}_{u}=\bar{F}^{(t)}_{\tau_{i}}$ for $\tau_{i}<u<\tau_{i+1}$. In
particular, in $\bar{F}^{(T)}$, with the maturity $T$ of the option being
hedged, let $\bar{F}^{(T)}_{T}=F^{(T)}_{T}$.

Suppose that $f(t,x)$ and $\psi(t,x)$ satisfy the assumption 1; then,
according to Itò’s formula for the pure jump process, $\bar{F}^{(T)}$,

$\psi\left(T,\bar{F}^{(T)}_{T}\right)=\psi\left(0,\bar{F}_{0}^{(T)}\right)+\int_{0}^{T}\frac{\partial\psi}{\partial
t}\left(t,\bar{F}_{t}^{(T)}\right)\mathrm{d}t+\sum_{0<t\leq
T}\Delta\psi\left(t,\bar{F}_{t}^{(T)}\right).$ (37)

Compared to the Taylor expansion in Eq. (19), we have

$\displaystyle\begin{aligned} \sum_{0<t\leq
T}\Delta\psi\left(t,\bar{F}_{t}^{(T)}\right)={}&\int_{0}^{T}\left(\frac{\partial\psi}{\partial
t}\left(t,\bar{F}^{(T)}_{0}\right)-\frac{\partial\psi}{\partial
t}\left(t,\bar{F}^{(T)}_{t}\right)\right)\mathrm{d}t\\\
&+\int_{0}^{F_{0}}\frac{\partial^{2}\psi}{\partial
x^{2}}(T,k)\left(k-\bar{F}^{(T)}_{T}\right)^{+}\mathrm{d}k\\\
&+\int_{F^{(T)}_{0}}^{\infty}\frac{\partial^{2}\psi}{\partial
x^{2}}(T,k)\left(\bar{F}^{(T)}_{T}-k\right)^{+}\mathrm{d}k.\end{aligned}$ (38)

By Taylor’s expansion and Eq. (23),

$\displaystyle\begin{aligned} &\mathbb{E}^{\mathbb{Q}}\left[\sum_{0<t\leq
T}\left\\{\frac{\partial\psi}{\partial
x}\left(t,\bar{F}_{t-}^{(T)}\right)\Delta\bar{F}_{t}^{(T)}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial
x^{2}}\left(t,\bar{F}_{t-}^{(T)}\right)\left(\Delta\bar{F}_{t}^{(T)}\right)^{2}+\cdots\right\\}\right]\\\
=&\mathbb{E}^{\mathbb{Q}}\left[\sum_{0<t\leq
T}\Delta\psi\left(t,\bar{F}_{t}^{(T)}\right)\right]\\\
=&-\int_{0}^{T}\int_{0}^{\infty}\mathrm{e}^{r(2T-t)}\frac{\partial f}{\partial
t}\left(t,\mathrm{e}^{r(T-t)}k\right)\phi^{(t)}(k)\mathrm{d}k\mathrm{d}t+\mathrm{e}^{rT}\int_{0}^{\infty}f(T,k)\phi^{(T)}(k)\mathrm{d}k.\end{aligned}$
(39)

Note that

$\displaystyle\mathbb{E}^{\mathbb{Q}}\left[\sum_{0<t\leq
T}\frac{\partial\psi}{\partial
x}\left(t,\bar{F}_{t-}^{(T)}\right)\Delta\bar{F}_{t}^{(T)}\right]=0$ (40)

Assuming that the higher-order terms of the jump in Eq. (39) are negligible,
the following approximation can be derived:

$\displaystyle\begin{aligned} \mathbb{E}^{\mathbb{Q}}&\left[\sum_{0<t\leq
T}\frac{\partial^{2}\psi}{\partial
x^{2}}\left(t,\bar{F}_{t-}^{(T)}\right)\left(\Delta\bar{F}_{t}^{(T)}\right)^{2}\right]\\\
\approx&-2\int_{0}^{T}\int_{0}^{\infty}\mathrm{e}^{r(2T-t)}\frac{\partial
f}{\partial
t}\left(t,\mathrm{e}^{r(T-t)}k\right)\phi^{(t)}(k)\mathrm{d}k\mathrm{d}t\\\
&+2\mathrm{e}^{rT}\int_{0}^{\infty}f(T,k)\phi^{(T)}(k)\mathrm{d}k.\end{aligned}$
(41)

or

$\displaystyle\begin{aligned} \mathbb{E}^{\mathbb{Q}}&\left[\sum_{0<t\leq
T}\mathrm{e}^{2r(T-t)}f\left(t,\mathrm{e}^{r(T-t)}\bar{S}_{t-}\right)(\Delta\bar{S}_{t})^{2}\right]\\\
\approx&-2\int_{0}^{T}\int_{0}^{\infty}\mathrm{e}^{r(2T-t)}\frac{\partial
f}{\partial
t}\left(t,\mathrm{e}^{r(T-t)}k\right)\phi^{(t)}(k)\mathrm{d}k\mathrm{d}t\\\
&+2\mathrm{e}^{rT}\int_{0}^{\infty}f(T,k)\phi^{(T)}(k)\mathrm{d}k.\end{aligned}$
(42)

As in the continuous trading case, if

$\displaystyle f(t,x)=\mathrm{e}^{-3r(T-t)}x\left(\frac{\partial D}{\partial
x}\left(t,\mathrm{e}^{-r(T-t)}x\right)\right)^{2},$ (43)

is chosen, then the expected liquidity cost can be approximated using the
integration formula in Eq. (42).

We examine whether the higher terms in Eq. (39) are negligible. The third-
order term is represented by

$\displaystyle\frac{1}{6}\frac{\partial^{3}\psi}{\partial
x^{3}}\left(t,\bar{F}_{t-}^{(T)}\right)\left(\Delta\bar{F}_{t}^{(T)}\right)^{3}=\frac{1}{6}\left(\bar{F}_{t-}^{(T)}\right)^{3}\frac{\partial
f}{\partial
x}\left(t,\bar{F}_{t-}^{(T)}\right)\left(\frac{\Delta\bar{F}_{t}^{(T)}}{\bar{F}_{t-}^{(T)}}\right)^{3}.$
(44)

Under the Black-Scholes framework, as in Eq. (34), $x^{3}\frac{\partial
f(t,x)}{\partial x}$ is continuous and converges to zero as $x$ goes to
infinity for every $t$, and is hence bounded over $(0,\infty)$. The same
argument is applied to the other higher terms. Therefore, the higher-order
terms are represented by
$O\left(\left(\Delta\bar{F}_{t}^{(T)}/\bar{F}_{t-}^{(T)}\right)^{3}\right)$
and, in this study, it is assumed that the third moment of the return is
negligible. Figure 3 gives an example of $x^{3}\frac{\partial f(t,x)}{\partial
x}$ with $T=0.1,t=0,K=1,r=0.05,\sigma=0.3,\alpha=1$.

In summary, with discrete time trading of delta hedging, the expected
liquidity cost can be calculated using the same method as in the continuous
trading case, if the hedger rebalances the portfolio and the change in delta
is equal to a fixed value.

Figure 3: Example of $x^{3}\frac{\partial f(t,x)}{\partial x}$ with
$T=0.1,t=0,K=1,r=0.05,\sigma=0.3,\alpha=1$

##  4 Numerical and simulation study

This section calculates the expected liquidity cost of the delta hedging
process of European options using the above formulas and the Monte Carlo
simulation. The Black-Scholes delta and the Black-Scholes option prices are
used for the hedging process and the weighted numerical integration,
respectively. According to Theorem 2, it is important to examine the property
of the unit liquidity cost $I$ to calculate the expected liquidity cost of the
delta hedging, because the cost is represented by $\alpha S_{0}N^{2}I$.

First, the numerical integration of $I$ was compared with the simulation
results for the delta hedging processes of the European call options with
maturity $T$ and strike price $K$. This is because at time $t=T$, the delta
function of a European call option is not continuous with respect to the
underlying price; hence, $f$ defined by Eq. (25) or (28) is also
discontinuous. For the numerical integration, the expected liquidity cost is
approximated by taking $T^{\prime}$ to be very close to $T$, where $f$ is
continuous and differentiable with respect to $x$. The approximation to the
numerical integration of the unit liquidity cost is

$\displaystyle\begin{aligned}
I\approx&-2\int_{0}^{T^{\prime}}\int_{0}^{\infty}\mathrm{e}^{r(2T^{\prime}-t)}\frac{\partial
f}{\partial
t}\left(t,\mathrm{e}^{r(T^{\prime}-t)}k;M,T\right)\phi^{(t)}(1,k)\mathrm{d}k\mathrm{d}t\\\
&+2\mathrm{e}^{rT^{\prime}}\int_{0}^{\infty}f(T^{\prime},k;M,T)\phi^{(T^{\prime})}(1,k)\mathrm{d}k.\end{aligned}$
(45)

For the numerical integration, this study used the approximation formula of
Eq. (45) with $T^{\prime}=T-0.004$ (i.e., up to one day before maturity). In
the same way as the Monte Carlo simulation, the delta hedge process stops at
$T^{\prime}$. Table 2 lists the numerical integration of $I$ with various $T$
and $K$, and the corresponding simulation results are reported in the
parentheses. For the simulation and numerical integration, $\sigma=0.3,r=0.05$
($S_{0}=1$ and $\alpha=1$ are assumed implicitly). A time grid $t=[0,T’]$ with
interval size $\Delta t=0.0001$, and a strike price grid $k=[0.5,1.5]$ with
interval size $\Delta k=0.001$, are constructed in the numerical integration.
For $\phi$, the Black-Scholes option prices are used.

In the first simulation, the delta hedging portfolio is rebalanced every hour,
and the number of total sample paths is $10^{4}$; this procedure mimics
continuous trading. In the second simulation, a threshold is fixed to $\Delta
D=0.05$, and the portfolio is rebalanced whenever the change in the delta
process exceeds the threshold. Therefore, this procedure is for discrete
trading. The values reported in the table show that the numerical integrations
of $I$ are similar to the means of corresponding simulation results within the
entire range of $K$ and $T$. The table also shows the 99% confidence intervals
and almost all confidence intervals calculated by simulation results contain
the numerical integration values.

Second, the properties of the unit liquidity cost are examined by plotting
various surfaces of $I$ with respect to the volatility, maturity, and
moneyness. Figures 4, 5 and 6 plot the surface of $I$ as a function of
$\sigma$ and $T$ with fixed $K=0.9,1.0$, and $1.1$, respectively. If a
European call option is at-the-money (i.e., $K=1$), the unit expected
liquidity cost, $I$, does not vary significantly throughout $\sigma$ and $T$,
and values around 0.21 are obtained. If the volatility and maturity are
relatively small, then the expected liquidity cost increases with increasing
volatility and maturity, which is consistent with our expectation.

In contrast to the previous case, if the volatility and maturity are
relatively large, then the expected liquidity cost decreases as volatility and
maturity increase. The delta of a European option varies mostly around at-the-
money, similar to the expected liquidity cost. An at-the-money European option
with a larger maturity and volatility has a higher probability of ending up
out-of-the-money or in-the-money than an at-the-money European option with
near maturity and low volatility. Therefore, an at-the-money European option
with large maturity and volatility might have a lower liquidity cost than an
at-the-money European option with smaller maturity and less volatility.

In addition, although the European call option is not at-the-money (i.e.,
$K=0.9$ or $1.1$, if $\sigma$ is sufficiently large), then the unit expected
liquidity cost, $I$, does not vary much, in the same way as maturity $T$. As
it is a European call option, delta is more sensitive closer to its
expiration, and the delta hedging liquidity cost is concentrated near the
maturity (i.e., the liquidity cost when it is far from maturity is relatively
insignificant).

Figure 7 plots the surface of $I$ as a function of $K$ and $T$ with fixed
$\sigma=0.3$. The expected liquidity cost increases with increasing maturity
for all values over $K$. On the other hand, if a European option is at-the-
money, the expected liquidity cost increases slowly compared to the cost of
the in-the-money or out-of-the-money European options. For in-the-money and
out-of-the-money options, the expected liquidity cost increases significantly
as the maturity $T$ increases. In addition, the expected liquidity cost of the
at-the-money option is generally larger than the cost of the in-the-money or
out-of-the-money options. The reason is that the moneyness of the deep in-the-
money and out-of-the-money European options with a short time to maturity have
little probability of changing; therefore, the deltas of the options do not
change significantly, as a small change in delta implies a low liquidity cost.

Figure 8 plots the surface of $I$ as a function of $\sigma$ and $K$ with fixed
maturity $T=0.1$. Similar to the previous cases, if a European option is at-
the-money, the expected liquidity cost is not significantly affected by
$\sigma$. For an at-the-money European option, delta is sufficiently sensitive
to the movement of the underlying price, regardless of the volatility level.
As a result, the liquidity costs of at-the-money European options are large
for all $\sigma$. For in-the-money and out-of-the-money options, the liquidity
cost increases with increasing $\sigma$. When there is more volatility, the
moneyness is more likely to be altered from in-the-money to out-of-the-money,
or vice versa. In addition, the expected liquidity cost of the at-the-money
option is larger than the cost of the in-the-money or out-of-the-money
options.

Table 2: Comparison of the numerical integration $I$ and the corresponding simulation test with various $K$ and $T$ with $\sigma=0.3,r=0.05$ with mean values of simulations and the half-length of 99% confidence interval (C.I.) |  |  |  | $K$ |  |
---|---|---|---|---|---|---
method | $T$ | 0.8 | 0.9 | 1.0 | 1.1 | 1.2
numerical | 0.1 | 0.0049 | 0.0791 | 0.2040 | 0.1195 | 0.0245
simul. 1 |  | 0.0046 | 0.0775 | 0.2039 | 0.1213 | 0.0243
C.I. |  | $\pm 0.0006$ | $\pm 0.0024$ | $\pm 0.0030$ | $\pm 0.0033$ | $\pm 0.0017$
simul. 2 |  | 0.0045 | 0.0772 | 0.2035 | 0.1207 | 0.0231
C.I. |  | $\pm 0.0006$ | $\pm 0.0024$ | $\pm 0.0030$ | $\pm 0.0033$ | $\pm 0.0017$
numerical | 0.2 | 0.0283 | 0.1259 | 0.2165 | 0.1774 | 0.0844
simul. 1 |  | 0.0275 | 0.1268 | 0.2168 | 0.1793 | 0.0838
C.I. |  | $\pm 0.0016$ | $\pm 0.0032$ | $\pm 0.0035$ | $\pm 0.0039$ | $0.0033$
simul. 2 |  | 0.0263 | 0.1260 | 0.2161 | 0.1787 | 0.0835
C.I. |  | $\pm 0.0016$ | $\pm 0.0031$ | $\pm 0.0035$ | $\pm 0.0039$ | $0.0033$
numerical | 0.5 | 0.0834 | 0.1672 | 0.2259 | 0.2256 | 0.1797
simul. 1 |  | 0.0833 | 0.1674 | 0.2279 | 0.2228 | 0.1793
C.I. |  | $\pm 0.0028$ | $\pm 0.0036$ | $\pm 0.0040$ | $\pm 0.0043$ | $\pm 0.0046$
simul. 2 |  | 0.0826 | 0.1669 | 0.2272 | 0.2223 | 0.1784
C.I. |  | $\pm 0.0028$ | $\pm 0.0036$ | $\pm 0.0040$ | $\pm 0.0043$ | $\pm 0.0046$
numerical | 1.0 | 0.1194 | 0.1829 | 0.2297 | 0.2428 | 0.2293
simul. 1 |  | 0.1215 | 0.1821 | 0.2290 | 0.2423 | 0.2303
C.I. |  | $\pm 0.0033$ | $\pm 0.0038$ | $\pm 0.0043$ | $\pm 0.0046$ | $\pm 0.0051$
simul. 2 |  | 0.1203 | 0.1811 | 0.2289 | 0.2412 | 0.2293
C.I. |  | $\pm 0.0033$ | $\pm 0.0038$ | $\pm 0.0043$ | $\pm 0.0046$ | $\pm 0.0050$
Figure 4: Unit expected liquidity cost $I$ for various $\sigma$ and $T$ with
$K=0.9$ Figure 5: Unit expected liquidity cost $I$ for various $\sigma$ and
$T$ with $K=1$ Figure 6: Unit expected liquidity cost $I$ for various $\sigma$
and $T$ with $K=1.1$ Figure 7: Unit expected liquidity cost $I$ for various
$K$ and $T$ with $\sigma=0.3$ Figure 8: Unit expected liquidity cost $I$ for
various $\sigma$ and $K$ with $T=0.1$

##  5 Distribution of liquidity cost

This section examines the distribution of the liquidity cost in the special
case of discrete trading and linear supply curve. Delta hedging is performed
with a fixed time interval, and this setting is different from Section 3. The
computation of the distribution is based on a recursive formula for a
numerical integration, and it takes a longer time than the computation of an
expected liquidity cost, but a shorter time than a Monte Carlo simulation.

Recall that the liquidity cost of a European option with strike $K$, under a
discrete delta hedging and the above condition, is

$L=\sum_{i=1}^{N}\alpha^{\prime}S_{i-1}\left(\frac{\partial
D_{N-i+1}(S_{i-1},K)}{\partial x}\right)^{2}(S_{i}-S_{i-1})^{2}$ (46)

which is similar to Eq. (36) but over an equidistant partition
$0=t_{0}<t_{1}<\cdots<t_{N}=T$ and the notation of the delta function is
slightly different from the previous sections. The subscript of $D$ denotes
the time to maturity, and $S_{i-1}$ is at the time $t_{i-1}$ stock price. The
partial derivative $\partial x$ implies the partial derivative with respect to
the first argument of $D$, the stock price (i.e., the Gamma of the option
price). Calculating the distribution of $L$ is equivalent to calculating the
expectation of $(L-\xi)^{+}$ for some constant $\xi$. The calculation of the
expectation of $(L-\xi)^{+}$ is similar to a derivation of an Asian-type call
option, and the distribution of $L$ is derived by differentiating the call
option price twice with respect to $\xi$ (Breeden and Litzenberger, 1978). To
calculate the distribution of $L$, a recursive method, which was introduced in
Lee (2014), Park et al. (2016), and Park and Lee (2020) is extended and
applied.

The numerical method to compute the distribution of the liquidity cost is
explained briefly. Let $g_{n}^{N}(x_{0},\cdots,x_{n},K,\xi)$ represent the
$\mathcal{F}_{n}$-conditional expectation of $L$ such that

$\displaystyle\begin{aligned}
&g_{n}^{N}(S_{0},\cdots,S_{n},K,\xi)=\mathbb{E}^{\mathbb{Q}}[(L-\xi)^{+}|\mathcal{F}_{n}]\\\
&=\mathbb{E}^{\mathbb{Q}}\left[\left.\left(\sum_{i=1}^{N}\alpha^{\prime}S_{i-1}\left(\frac{\partial
D_{N-i+1}(S_{i-1},K)}{\partial
x}\right)^{2}(S_{i}-S_{i-1})^{2}-\xi\right)^{+}\right|\mathcal{F}_{n}\right].\end{aligned}$
(47)

The superscript, $N$, of $g$ denotes the total number of terms inside the
summation, and the subscript $(n)$ denotes the time when the expectation is
performed. As in the previous section, the risk-neutral probability is used,
but the choice of the measure depends simply on the practitioner’s purpose.

The goal is the time 0 expectation, for example,

$g_{0}^{N}(S_{0},K,\xi)=\mathbb{E}^{\mathbb{Q}}[(L-\xi)^{+}].$ (48)

Since

$g_{n}^{N}(S_{0},\cdots,S_{n},K,\xi)=\mathbb{E}^{\mathbb{Q}}[g_{n+1}^{N}(S_{0},\cdots,S_{n},S_{n+1},K,\xi)|\mathcal{F}_{n}],$
(49)

the following integration form can be derived:

$g_{n}^{N}(x_{0},\cdots,x_{n},K,\xi)=\int_{0}^{\infty}g_{n+1}^{N}(x_{0},\cdots,x_{n},x_{n+1},K,\xi)p(x_{n+1};x_{n})\mathrm{d}x_{n+1}$
(50)

where $p$ is the transition function for the stock price from $x_{n}$ to
$x_{n+1}$ over the time interval $[t_{n},t_{n+1}]$ under the corresponding
measure. By rearranging the strike price of the ‘Asian-type option’,

$g_{n}^{N}(x_{0},\cdots,x_{n},K,\xi)=g_{0}^{N-n}(x_{n},K,\xi_{n})$ (51)

where

$\xi_{n}=\xi-\sum_{i=1}^{n}\ell_{i}:=\xi-\sum_{i=1}^{n}\alpha^{\prime}x_{i-1}\left(\frac{\partial
D_{N-i+1}(x_{i-1},K)}{\partial x}\right)^{2}(x_{i}-x_{i-1})^{2}.$ (52)

By denominating the Asian-type option price by $x_{n}$,

$g_{0}^{N-n}(x_{n},K,\xi_{n})=x_{n}g_{0}^{N-n}\left(1,\frac{K}{x_{n}},\frac{\xi_{n}}{x_{n}}\right),$
(53)

and by Eq. (50),

$x_{n}g_{0}^{N-n}\left(1,\frac{K}{x_{n}},\frac{\xi_{n}}{x_{n}}\right)=\int_{0}^{\infty}x_{n+1}g_{0}^{N-n-1}\left(1,\frac{K}{x_{n+1}},\frac{\xi_{n+1}}{x_{n+1}}\right)p(x_{n+1};x_{n})\mathrm{d}x_{n+1}.$
(54)

Since the above equation holds for all $x_{n}>0$, by putting $x_{n}=1$, and
using the relation between $\xi_{n}$ and $\xi_{n+1}$,

$g_{0}^{N-n}\left(1,K,\xi_{n}\right)=\int_{0}^{\infty}x_{n+1}g_{0}^{N-n-1}\left(1,\frac{K}{x_{n+1}},\frac{\xi_{n}-\ell_{n+1}}{x_{n+1}}\right)p(x_{n+1};1)\mathrm{d}x_{n+1}.$
(55)

In particular, when $n=N-1$, by Eq. (51) and the definition of $g$,

$\displaystyle\begin{aligned}
g_{0}^{1}(1,K,\xi_{N-1})&=g_{N-1}^{N}(x_{0},\cdots,x_{N-1},K,\xi)\\\
&=\int_{0}^{\infty}\left(\alpha^{\prime}\left(\frac{\partial
D_{1}(1,K)}{\partial
x}\right)^{2}(x_{N}-1)^{2}-\xi_{N-1}\right)^{+}p(x_{N};1)\mathrm{d}x_{N}.\end{aligned}$
(56)

Repeating the numerical integration from $g_{0}^{1}$ to $g_{0}^{N}$ and
applying $g_{0}^{N}(x,K,\xi)=xg_{0}^{N}(1,K/x,\xi/x)$ the Asian-type option
price, the distribution of the liquidity cost can be calculated.

In Figure 9, 10, 11 the comparison between the distribution calculated using
the numerical method (solid line) and the Monte Carlo simulation (bar graph)
is presented. The numerical results are quite close to the Monte Carlo
simulation results over all $\sigma$, strike $K$ and maturity $T$.

Figure 9: Distribution of liquidity errors with $\sigma=0.8$, $T=0.1$ and
$K=0.9$ (left) $K=1$ (center) $K=1.1$ (right)

Figure 10: Distribution of liquidity errors with $\sigma=0.4$, $T=0.5$ and
$K=0.9$ (left) $K=1$ (center) $K=1.1$ (right)

Figure 11: Distribution of liquidity errors with $\sigma=0.2$, $T=1$ and
$K=0.9$ (left) $K=1$ (center) $K=1.1$ (right)

##  6 Supply curve and order book model

The expected liquidity cost depends on $\alpha$, and in practice, $\alpha$
depends on the estimation procedure of the supply curve. If the supply curve
is discontinuous, then continuous hedging is excluded, and discrete trading
should be considered.

This section discusses how to apply the existing order book model to our
method for calculating the liquidity cost. If the limit order model is assumed
to be stochastic, then the supply curve is also stochastic. The dynamics of
the supply curve, which in this approach are governed by parameter
$\alpha_{t}$ or $\alpha^{\prime}_{t}$, are independent from the stock price
and the investor’s hedging process. The expected liquidity cost for a
continuous hedging process is then represented by

$\mathbb{E}\left[\int_{0}^{T}\alpha_{t}S_{t}\mathrm{d}[D]_{t}\right]=\int_{0}^{T}\mathbb{E}[\alpha_{t}]\mathbb{E}\left[\sigma^{2}_{t}S^{3}_{t}\left(\frac{\partial
D(t,S_{t})}{\partial x}\right)^{2}\right]\mathrm{d}t$ (57)

Here, it is assumed that
$\mathrm{d}[S]_{t}=\sigma^{2}_{t}S^{2}_{t}\mathrm{d}t$ for some volatility
process and $\sigma_{t}$ for convenience. A similar argument can be made for
discrete trading. The formula shows that only the expected slope of the supply
curve $\mathbb{E}[\alpha_{t}]$ needs to be considered when calculating the
total expected liquidity cost. The relationship between the existing
stochastic queuing model of the limit order book and the expected supply curve
is explained.

Cont et al. (2010) proposed a stochastic model for order book dynamics where
the limit and market orders and the cancellation of limit orders are described
by independent Poisson processes. In this framework, the limit orders are
placed on a discrete price grid and the limit order arrivals depend on how far
the price of the limit order is from the best quote (i.e., the relative
price). Biais et al. (1995), Bouchaud et al. (2002), and Potters and Bouchaud
(2003) also showed that the order arrival rates depend on relative price
rather than actual price.

The limit buy and sell orders arriving at a distance of $i$ ticks from the
opposite best quote, or simply the $i$-th tick, are assumed to have the same
intensity rate $\lambda(i)$. The market buy and sell orders, which occur in
only the best quotes of ask and bid, respectively, are assumed to arrive at
independent and exponential times at a rate of $\mu$. The rate of cancellation
of the limit orders at a distance of the $i$-th tick from the opposite best
quotes is proportional to the existing sizes of the limit orders. If the total
number of the outstanding limit orders is $y$, then the cancellation rate is
$\theta(i)y$. Cont et al. (2010) chose the size unit as the average size of
the limit orders, but in this section, the size unit is assumed to be one
without a loss of generality.

As the arrival rates of the bid and ask sides are assumed to be equal to each
other, it is sufficient to consider only one side of bid and ask. Let
$N_{t}^{\ell(i)}$ be the total number of limit orders (of bid or ask side) at
the $i$-th tick up to time $t$; let $N_{t}^{c(i)}$ be the total number of
cancellations of limit orders at the $i$-th tick up to time $t$, and
$N_{t}^{m}$ the total number of market orders arrived up to $t$. Under this
assumption, the number of outstanding limit orders at time $t$ at the $i$-th
tick is represented by three Poisson processes:

$Y^{1}_{t}=N_{t}^{\ell(1)}-N_{t}^{m}-N_{t}^{c(1)}\quad\mathrm{and}\quad
Y^{i}_{t}=N_{t}^{\ell(i)}-N_{t}^{c(i)},\quad\textrm{for }i>1.$ (58)

For $i=1$, based on the above equation, the expected number of outstanding
limit orders at time $t$, denoted by $y^{1}_{t}$, is represented by an
integration equation

$y^{1}_{t}=\lambda(1)t-\mu t-\theta(1)\int_{0}^{t}y^{1}_{s}\mathrm{d}s$ (59)

By solving the above equation, the expected outstanding limit orders is

$y^{1}_{t}=\frac{\lambda(1)-\mu}{\theta(1)}\left(1-\mathrm{e}^{-\theta(1)t}\right)$
(60)

where$y^{1}_{0}=0$. As $t\rightarrow\infty$,

$y^{1}_{t}\rightarrow y^{1}:=\frac{\lambda(1)-\mu}{\theta(1)}$ (61)

which is the expected number of outstanding limit orders at the first tick at
the stationary state.

$y^{i}_{t}\rightarrow y^{i}:=\frac{\lambda(i)}{\theta(i)}.$ (62)

With these results, the expected stationary state limit order curve $m(q)$ and
corresponding supply curve are defined as Eq. (2). We have

$m(q)=\mathrm{sign}(q)\sum_{i=1}^{j}y^{i}$ (63)

if $y^{j-1}<q\leq y^{j}$, for $j\geq 1$, and $y^{0}=0$. Therefore, with the
given exponential rates $\lambda(i),\theta(i)$, and $\mu$, one can derive the
limit order and supply curves. The rates can be estimated from the data
consisting of the sequences of limit and market orders by counting the number
of orders as described in Cont et al. (2010). In some studies, the limit order
arrival function is fitted by a power law function of the form

$\lambda(i)=\frac{k}{i^{a}}$ (64)

for some $a$ and $k$ (Bouchaud et al., 2002; Zovko and Farmer, 2002).

Once the rates are estimated, the expected supply curve is derived; the
remaining part of the calculation of the expected liquidity cost is
straightforward, as described in Sections 2 and 3.

##  7 Conclusion

An integration formula was derived for the expected liquidity cost when
performing the delta hedging process of a European option. The liquidity cost
is represented by the multiplication of the unit liquidity cost, current stock
price, liquidity parameter, and the square of numbers of options being hedged.
The numerical procedure of the integration is correct and much faster than
using the Monte Carlo method. Interestingly, the expected liquidity cost of
at-the-money European call options is significantly affected by volatility and
maturity. In general, the expected liquidity cost increases with increasing
volatility and maturity. This paper provided a numerical method for
calculating the distribution of the liquidity cost. Furthermore, the
relationship between the queuing modeling of the order book and the present
approach was explained.

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