國 立 交 通 大 學
財 務 金 融 研 究 所
碩 士 論 文
考慮市場流動性不完全下之選擇權
訂價與避險
Pricing and Hedging Options under Illiquid Markets
.
研 究 生 :
.黃克鈞
指導教授 ::
鍾惠民 博士
考慮市場流動性不完全下之選擇權訂價與避險
Pricing and Hedging Options under Illiquid Markets
研 究 生：黃克鈞 Student：Ke-Chun Huang 指導教授：鍾惠民 博士 Advisor：Dr. Huimin Chung
國 立 交 通 大 學
財務金融研究所
碩 士 論 文
A Thesis
Submitted to Graduate Institute of Finance
College of Management
National Chiao Tung University
in partial Fulfillment of the Requirements
for the Degree of
Master
of
Science in Finance
July 2007
Hsinchu, Taiwan, Republic of China
考慮市場流動性不完全下之選擇權訂價與避險
研究生 : 黃克鈞 指導教授 : 鍾惠民 博士
國立交通大學財務金融研究所
2007 年 7 月
摘要
本篇論文建構在 Frey 與 Patie 2002 年的模型上，考慮流動性為股價所控制 的函數。 Frey 在標準的 Black-Scholes 偏微分方程式中，加入流動性變數，用此 求取選擇權價格。本篇論文的目的為，改善 Frey 偏微分方程式中的人造條件， 使得此非線性偏微分方程更加穩定。因此，利用拔靴法求取波動度的上界，用其 取代原本不合理的波動度上界。數值理論部份，改善了希臘字母(Greeks)在流動 性不佳的情況下，不穩定的狀態。透過希臘字母的趨勢變化，可以幫助交易者更 了解回饋效果在不同流動性市場下的變化。實證部份，我們挑選了CBOE 交易量 前 50 的股票選擇權當作標的物，且用上述的非線性偏微分方程去計算個別選擇 權的價格。結果顯示在不完全流動的市場下，改善後的偏微分方程，可提供更精 確的選擇權價格。 關鍵字: 流動性、選擇權定價、非線性偏微分方程、拔靴法、希臘字母、回饋效 果、有限差分法Pricing and Hedging Options under Illiquid Markets
Student: Ke-Chun Huang Advisors: Dr. Huiming Chung
Graduate Institute of Finance National Chiao Tung University
July 2007
ABSTRACT
In this paper we build on Frey and Patie’s literature (2002), where liquidity is a deterministic function of stock price. Frey implements an important factor, liquidity, into the standard Black-Scholes partial differential equation (PDE) to calculate the option price. The objective of our model is to improve an artificial pattern of Frey PDE to make the nonlinear PDE more reliable. Therefore, we choose bootstrap method to obtain the upper bound of volatility to replace the unreasonable setting. In numerical research, Greeks become smoother than before while using the bigger liquidity parameter. It helps traders to realize the variation of Feedback effect under different liquid markets. In empirical study, we choose the top 50 stock options of CBOE as underlying assets and use the PDE which contains liquidity parameter to solve each option price. The result shows that using the improved PDE offers more precise option prices in illiquid market.
Keyword: Liquidity; Option Pricing; Nonlinear PDE; Bootstrap Method; Greek Letters; Feedback Effect; Finite Difference Method.
誌謝
首先感謝鍾惠民教授在這兩年對學生的關懷與指導，讓我能夠順利的從交 大財金所畢業。另外也對口試委員呂育道、王耀輝、謝文良、戴天時教授敬上最 深的敬意，謝謝諸位教授珍貴的建議，才能使得此篇論文更加的完善。 在交大財金所短短的兩年中，交到了許多的好朋友，生活真的是多彩多姿。 淑惠爽朗的個性、三花超帶衰的生活、胖兔不計較的心態、維峻完美的謀略、殺 手慢步調的幽默、凱秩驚人的天份、阿師超齡的年紀、大爺愛裝娘的怪癖、小郭 多樣化的生活...等。這些點點滴滴都會ㄧ直埋藏在我心中，因為有你們的存在， 我的研究所生活才會如此精采。如果再讓我選擇ㄧ次，我ㄧ定會祈求上帝能夠跟 這群優秀又幽默的同學一起生活，感謝你們帶給我的種種回憶，使我能夠快樂的 學習，在最困苦的時候能不畏懼困難往前走。 最後要感謝我親愛的家人，跟你們生活在ㄧ起是全天下最幸福的事情，也 許我門不是最有錢的家庭，但ㄧ定是最歡樂的家庭。希望我能夠越來越好，永遠 永遠不會讓你們失望！ 黃克鈞 謹誌 國立交通大學財務金融研究所 中華民國九十六年七月Contents
CHINESE ABSRACT ... I ENGLISH ABSRACT ... II ACKNOWLEDGEMENTS ... III 1. INTRODUCTION ... I
2. THE DETERMINISTIC LIQUIDITY MODEL ... 5
2.1BASIC ASSUMPTIONS OF THE DETERMINISTIC LIQUIDITY MODEL... 6
2.2DYNAMIC PROCESS OF ASSET PRICE... 8
2.3DERIVATIVES WITH NONLINEAR PDE... 10
2.4BASIC CONCEPT OF DYNAMIC HEDGING... 11
2.5TRACKING ERROR... 12
2.6NUMERICAL METHOD (FINITE DIFFERENCE METHOD) ... 13
3. NUMERICAL RESULTS ... 16
3.1HEDGING COST IN ILLIQUIDITY... 16
3.2GREEK LETTERS IN ILLIQUID MARKET... 17
4. EMPIRICAL RESULTS ... 19
4.1DATA SELECTION... 19
4.2LEAST SQUARE METHOD AND LOSS FUNCTION... 19
4.3THE PERFORMANCE OF OPTION PRICING MODEL... 21
4.4DYNAMIC HEDGING... 22
5. CONCLUSION... 23
REFERENCES ... 24
1. Introduction
Since the Black-Scholes (BS) formula was innovated, this well-known formula has given traders a benchmark to trade option with appropriate price. Market participants initially take BS price as a standard and adopt hedging strategies to adjust their positions. However, investors gradually find that BS formula offers limited option price because of the restrictive BS assumptions. In order to release the assumptions to provide more reliable option prices, a number of new models have been developed for option valuation.
In this paper, we would like to release a specific assumption that markets are assumed to be completely elastic (perfect liquid market). Specifically, the hypothesis assumes the price do not change no matter how large amounts of an asset are traded. If all of the market participants are small investors, their trading strategies do not influence market prices and the market is perfectly liquid in this scenario. However, many markets are not perfectly liquid due to the presence of large investors. Large investors’ trades have price impacts and thus they face illiquid market. According to Kyle(1985), large investors can use its market power to move prices in a certain direction. Based on Chan and Lakonishok (1995), large institutions with high proportion of trade tend to be associated with larger market impact. Therefore, it obviously violates the truth of reality and conveys an important message that markets are not perfectly liquid.
The absence of liquidity leads markets to face serious problems, such as financial crisis. A well-known event of illiquid problem is Long-Term Capital Management (LTCM) crisis. That is a hedge fund which the company manager longs less liquid instruments and shorts more liquid instruments at the same time. This trading strategy with high leverage initially brings significance profit to LTCM but the
company’s manager does not anticipate that their high risk manipulations would result in financial crisis. When the Russian debt crisis occurs, most of market liquidity suddenly deteriorates. Consequently, the LTCM not only loses all of their properties in a short period but also causes an inevitable financial disaster. Hence, there are numbers of researchers probing into the cause of market illiquidity.
In academic literature, there are three causes of illiquidity which are transaction costs, asymmetric information of asset and imperfect competition in asset market. In addition, another interesting theory causing market illiquidity is uncertainty. Uncertainty means that traders face a circumstance which they can not anticipate any changes and do not know how to management.
Moreover,many scholars try to explain the financial meaning of liquidity. Kyle
(1985) explains market liquidity from three aspects which are tightness, depth and resilience of the market. Tightness measures the cost between buying and selling a position in a short period. Depth refers to the size of a transaction required to change prices. Resilience measures the speed of prices recovering fundamental after a sudden event occurs.
Schwartz (1988) explains that liquidity is the ability to trade immediately. Amihud and Mendelson (1989) divide the liquidity into two parts including price aspect and time aspect. In point of price aspect, if assets can be traded in specific time and minimize the concession from buyer and his counterparties, assets are more liquid. For time aspect, when buyer and his counterparties make acceptable concession, the less time used for transaction the more liquidity assets have. As I mentioned above, these ideas give us some concepts about market illiquidity.
During recent years, risk management has become a popular issue in the field of finance. Three types of risks are market risk, credit risk and liquidity risk. Bielecki and Rutkowski (2002) have successfully made an exceptional summary regarding
market risk and credit risk. Meanwhile, for liquidity risk, Frey establishes an innovative nonlinear PDE to hedge portfolio perfectly. Frey and Patie (2002) continue this work and make an extension to describe the liquidity as the deterministic function of stock price. In addition, they offer some numerical results to show the illiquidity influence of hedging costs and Greeks’ derivatives.
Furthermore, Esser and Moench (2003) generalize the liquidity model of Frey (2002) and incorporate the stochastic liquidity into stock process. In other words, they use their innovation to analyze the differences among large investors vary from market depth. Then, they make an observation of the variation of large investors’ hedging strategies related to stock prices in illiquid markets as well. In addition, they verify that stochastic liquidity setting is more suitable for financial markets. Overall, their contribution and research provide a flexible model of hedging strategies under illiquid market.
From the above, we can easily say that there are more researchers investigating the modeling and hedging strategies while market illiquidity or large investors exist. In order to appreciate their contribution and dedication, we provide some further related references such as Kyle (1985), Jarrow (1994), Schonbucher and Willmott (2000) and Simona (2006).
Thus, the core of our paper is to analyze the last type of risk, liquidity risk. We capture the notion of liquidity from Frey (2000) and our model builds on Frey and Patie (2002). We take liquidity as a deterministic function of price and implement it into nonlinear PDE. Using finite difference method (FDM), it helps to solve the problem of computing nonlinear PDE to obtain option prices. We choose least square method to estimate the liquidity. In order to make estimation more reliable, we use bootstrap method to calculate the upper bound of volatility of each asset to revise the artificial pattern. Moreover, Esser and Moench (2003) suggest the asymmetry
relationship between the stock price and liquidity should be removed. All of these approaches help us to obtain more precise option prices and apply to dynamic hedging strategy to minimize the tracking error.
When it comes to trading strategy, we must mention feedback effect strategy. There are two types of feedback effect trading strategies. One is called positive feedback effect. For example, investors buy risky assets when the asset price is increasing and sell the risky asset when the asset price is declining. Similarly, there is a negative feedback effect. Its trading strategy is very similar to positive feedback. Large investors buy the risky asset while the asset price is declining and sell the risky asset while asset price is increasing. Those strategies are used to replicate convex payoff, long call, and concave payoff, short call, respectively. We would show these results in numerical section.
The remainder of the paper is organized as follows: in section 2, we introduce the basic assumption of Frey model (2002) and make a description of how to derive the nonlinear PDE. Afterwards, we modify the artificial pattern mentioned in Frey and Patie (2002) and explain why it is not appropriate. In section 3, numerical results would help us to realize the feedback effect. In section 4, estimate option price using data of CBOE top 50 companies is applied to our empirical study. Finally, we summarize the discoveries in this context and offer our conclusions in section 5.
2. The Deterministic Liquidity Model
Frey and Patie (2002) assume that financial markets trade between one riskless
asset (bond) and one risky asset (stock or stock index).They consider bond markets as
perfectly liquid markets. In other words, no matter how large shares of bonds are traded, it does not influence the bond prices. In this circumstance, large investors and small market participants have the same weight in bond markets. On the other hand, stock markets seem to be illiquid markets. For example, a significant trade would cause price impact. This phenomenon is in line with economy intuition that money markets are more liquid than the stock markets.
The cause of the illiquid market is the presence of large investors. Their trading strategies lead to price motion and bring feedback effect to underlying assets. For example, the price of an asset increases (declines) when the institution investors buy (sell) the asset. This assumption is supported by Holthausen and Leftwich (1987) that large transactions are related to price impact. Therefore, Frey and Patie argue that large investors’ manipulations violate the assumption that market is perfectly liquid.
In order to release the above assumption, Frey incorporates the liquidity parameter, ρ, in dynamic process of asset price. Initially, liquidity is treated as a constant which means that the liquidity of the asset does not change through time. Frey and Patie treat liquidity as the deterministic function of time and stock price. Our model builds on this concept and revises an artificial pattern of Black-Scholes PDE.
In the following, we offer the basic assumptions and asset price dynamics of Frey and Patie (2002). First, we introduce the derivation of nonlinear PDE. Second, we provide some essential knowledge of trading strategy. Third, we use the introduction of the tracking error to investigate the performance between perfect and imperfect market liquidity hedging strategy. Finally, we take the advantage of numerical method
to solve the nonlinear PDE and revise the problems of the artificial pattern which may produce irrational volatility.
2.1 Basic assumptions of the deterministic liquidity model
The core of the basic assumption explains the trading strategy which the large investors have followed. Instead of building on microeconomic equilibrium, Frey argues that the variation of asset price dynamics originates from large investor’s stock
holdings or trading strategy
( )
αt . Furthermore, Frey takes the influence of priceimpact as exogenous. Therefore, we illustrate some assumptions of asset trading strategies as follows:
(A1) The stockholdings
( )
αt tare left-continuous (i.e. t lim ss t
α < α
→
= ).
(A2) The right-continuous processα+with t lim
s t s
α+ < α
→
= is a semimartingale.
(A3) The downward-jumps of trading strategy are bounded: αt : αt αt 1/
+ +
Δ = − > − ρ
for some ρ>0.
These three assumptions provide large investors’ trading strategy as a smooth function of the stock price.
It is obvious that parameter ρ causes the fluctuation from the standard Black-Scholes model. When the parameter equals to zero or large investor maintains
his stock holdings without changing (i.e.αt =0), the asset price dynamics follow the
Black-Scholes model where volatility follows the famous Brownian motion. Then, we will provide the asset price dynamics with considering liquidity which is decided by
large investor’s trading strategyαtdepending on St
(
ρ α,)
.(A4) If the trading strategy follows the previous assumptions, the stochastic differential equation is as follows:
( )
t t t t t
dS =σS dW− +ρλ S− S d− αt+ (2.1)
where implies the left limit lim t
s t
S
<
In this stochastic differential equation, we assume σ to be positive and λ to be a continuous function. The parameter ρ must be greater than zero as the number of
ρ increases, the less liquidity the market becomes. We use 1/(ρλ(St−)St−) to
represent the market depth at time t. From this equation, we realize how much shares could cause the asset price to move by one unit. According to the suggestion of Esser
and Moench (2003), they indicate that continuous functionλdoes not have significant
influence to explain the liquidity tendency. Thus, we do not incorporateλin our model
for the following pages. We transfer the stochastic differential equation transfers to equation (2.2).
t t t t
dS =σS dW− +ρS d− αt+ (2.2)
The following shows an example providing a better understanding of the asset price assumptions. First, we set up a scenario that large investors hold the specific
asset of K shares which K must over zero and less than1/ρ. There is still another
limitation of asset price. When the asset price drops under S, large investors would
sell all of the stock holdings. If there is no transaction obstacle (perfect liquid market),
his portfolio value would be above or at least equals toK S.
After the basic setting, we start to introduce a large investor trading strategy. The
stopping timeτ is set up asτ =inf{t>0,St <S}. The trading strategyα is set up
asαt =K, 0≤ ≤t τ and αt=0 for t >τ . It corresponds to the Assumption A1, A2 and A3.
We can find out that α is left continuous becauseαt =K, 0≤ ≤t τ and α
+
has a
bounded downward jumps
(
− > −K 1/ρ)
. Then, we can derive the portfolio value attimeτ in illiquid markets. When investor’s stock price becomesSτ =Sτ−−ρS Kτ− <Sτ−
his portfolio does not have the perfect protection, i.e. 2
KSτ =KSτ−−ρS Kτ− <KSτ−. It is obvious that the difference between deterministic liquidity model and Black-Scholes model is whether implementing the liquidity pattern into the stochastic
differential equation or not. To make the deterministic liquidity model useable, we need to find out how to determine the parameterρ. As we known, there are two different avenues to determine theρ. In this article, our approach uses the observed data to calculate the implied liquidity. Then, we would mention how to estimate the implied liquidity later.
2.2 Dynamic process of asset price
In this part, we combine the underlying asset’s stochastic differential equation and
the trading strategy α by using the formula. In addition, Frey and Patie
assume that their stockholdings trading strategy is a smooth function
'
Ito s
φ of time and
the current asset price. This assumption plays an important role of replicating in option valuation.
(A5) Trading strategyφis a function of time and current asset.
1,2
[0, ]T ×\+→\ is of class C ([0, ]T ×\+).
t
where (t,S )φ =αt
Moreover, ρ φS s( , ) 1 for all ( , ) t S < t S ∈ [0, ]T ×\+
In the sequel, we derive the asset price dynamics by stochastic differential
equation (2.2) and assumption A5. First, we implement formula into trading
strategy to get the following
' Ito s 2 2 1 ( , ) ( , ) ( , ) ( , ) 2 t s t t t t ss t t t dα φ= t S dS +⎛⎜φ t S + φ t S ν t S S ⎞⎟ ⎝ ⎠dt (2.3)
Hoping to get more realistic asset dynamics, we add the risk free rateγ into
equation (2.2).
t t t t t
dS =rS dt− +σS dW− +ρS d− αt+ (2.4) Then, the further step is to put the equation (2.3) into equation (2.4).
2 2 1 ( , ) ( , ) ( , ) ( , ) 2 t t t t s t t t t t ss t t t dS =rS dt+σS dW+ρ φS t S dS +ρS ⎛⎜φ t S + φ t S ν t S S ⎞⎟dt ⎝ ⎠ (2.5)
2 2 1 (1 ( , )) ( , ) ( , ) ( , ) 2 t s t t t t t t ss t t t t S t S dS S dW S t S t S t S S rS d ρ φ σ ⎡ρ ⎛φ φ ν ⎞ ⎤ t − = +⎢ ⎜ + ⎟+ ⎥ ⎝ ⎠ ⎣ ⎦ .(2.6)
By dividing the coefficient (1−ρ φSt s( ,t St)), we get the new asset price dynamics.
2 2 1 ( , ) ( , ) ( , ) 2 (1 ( , )) (1 ( , )) t t t ss t t t t t t t s t t s t S t S t S t S S rS S dS dW dt S t S S t S ρ φ φ ν σ ρ φ ρ φ ⎡ ⎛ + ⎞+ ⎤ ⎜ ⎟ ⎢ ⎝ ⎠ ⎥ ⎣ ⎦ = + − − .(2.7)
In order to make the asset price dynamics more clearly, we take as the
volatility term of underlying asset and use to represent the drift term of
underlying asset. Finally, we derive the general form of the asset price process under market illiquidity. ( , t) v t S ( , t) b t S ( , ) ( , ) t t t t t dS =v t S S dW +b t S S dtt .(2.8) ( , ) (1 ( , )) t t s t v t S S t S σ ρ φ = − (2.9) 2 2 1 ( , ) ( , ) ( , ) ( , ) (1 ( , )) 2 (1 ( , )) t t t ss t t t t s t t s t r b t S t S t S t S S S t S S t S ρ φ φ ν ρ φ ρ φ ⎛ ⎞ = ⎜ + ⎟+ − ⎝ ⎠ − (2.10)
There is something interesting after we derive this innovative asset price dynamics. The volatility term has transferred from constant into a function which
depends on time and asset price, i.e. . This transition results from large
investors’ trading strategy. In trading markets, large investors have two general trading strategies, positive feedback strategy and contrarian strategy, to protect their position. When large investors follows positive feedback strategy (contrarian feedback strategy), their trading strategy is to buy (sell) additional shares of the assets while the asset price lifts up (drops down).
( , t)
v t S
In other words, the new volatility becomes greater than the constant volatility if the representative investor chose the positive feedback strategy, i.e.
( , t)
v t S >σ if ( ,φs t St)>0. On the contrary, the new volatility is smaller than the
constant volatility if the representative investor chose the contrarian feedback strategy, i.e. v t S( , t)<σ if ( ,φs t St)<0.
2.3 Derivatives with nonlinear PDE
The objection of this part is calculating the derivatives price. Having the new asset price dynamics helps us to replicate the derivatives such as call or put option. In recent research, there are at least three methods to estimate the value of derivatives including binomial tree, Monte Carol simulation and solving the partial differential equation (PDE). Each of approach has its own advantages and disadvantages. In this context, we choose PDE method as an instrument to estimate the price of derivatives in illiquid market.
In the beginning, we assume the derivative is path-independent and set its smooth
payoff as . Before building up the new asset price dynamics (2.8), traders use
the standard Black-Scholes strategy to hedge. However, Black-Scholes theory is only effective in perfectly liquid market. Consequently, traders would face a great loss when they adopt the Black-Scholes trading strategy to replicate the derivatives. Since Frey and Patie have built up the new theory considering market illiquidity, traders can obtain more reliable hedging strategy and value of derivatives by adopting the Frey and Patie theory.
( T)
h S
It is comparatively easy to understand the Frey and Patie PDE by introducing the Black-Scholes PDE at first because we are familiar with the classical theory. The Black-Scholes PDE is presented as follows:
2 2 1 ( , ) ( , ) ( , ) ( , ) 2 t s ss u t S +rSu t S + σ S u t S =ru t S (2.11) where ( , ) is the derivative pice at time T, i.e. (u T S h ST)
To recall the equation (2.8), we find out the volatility term in equation (2.11)
should be replaced as . Then, we obtain the extension of the Black-Scholes
PDE as equation (2.12) ( , t) v t S 2 2 2 1 ( , ) ( , ) ( , ) ( , ) 2 (1 ( , )) t s ss t ss t u t S rSu t S S u t S ru t S S u t S σ ρ + + = − (2.12)
s
where ( ,uss t St)=φ ( ,t St)
From the above equation, the Black-Scholes PDE becomes the special case of the
Frey and Patie PDE. The v t S( , t) is transferred into σ while the parameter ρ
equals zero in the Frey and Patie model. For this reason, the nonlinear PDE is the general form of the Black-Scholes PDE.
The nonlinear PDE provides us an avenue to estimate the value of derivatives, but it is easily to tell that we could not solve it directly. We must take advantage of numerical method to solve the equation such as finite difference method. The numerical method would be presented in the latter part of this article.
2.4 Basic concept of dynamic hedging
In order to understand the performance of different model’s hedging strategy, traders must possess the fundamental knowledge of dynamic hedging. For this reason, it is essential to make detail explanations. We first assume that the representative
hedger is a large investor and his initial stockholdings is α and the share of bond is
β . The hedger uses these two assets to replicate the derivative which the expiration
date is T .
When the market is not perfectly liquid, the value of the hedger’s portfolio is hard to define. The hedger’s position has two kinds of values. One is mark-to-market value and the other is liquidation value. If the stock market liquidity is perfect, these two kinds of value are the same. In other situation, mark-to-market value is higher than liquidation value. Taking the limit order for example, once the stock price declines
below S the hedger sells all of his stockholdings. Theoretically, the value of this
portfolio is VtM =αtSt( , )ρ α +βt, i.e. mark-to market value. However, the hedger only
receives the value (liquidation value) less than the mark-to-market value when the hedger sells his position to the market. Although liquidation value is in line with the real value of the hedger’s position, it is difficult to define a precise liquidation cost.
Therefore, we adopt the mark-to-market value as the hedger’s position value to the following analysis.
Corresponding to the stock price variation, the representative hedger adjusts his stockholdings and bonds to minimize the risk before the expiration date . At time t, the mark-to-market value is
Τ ( , )
M t t t
V =αS ρ α +βt and the initial value is 0M
V .
Generally speaking, delta hedging strategy is widely used in financial market. If there is no external finance to support the hedger, the process of relocation position is called
self-financing. During the period
(
0 ~ T)
, the gains from relocation position aredefined as
0 ( , )
T
t s s
G =
∫
α dS ρ α . Throughout the self-financing strategy, the value of theportfolio at time T is 0 0 0 ( , ) T M T t s s V =V +G =V +
∫
α dS ρ α .2.5 Tracking error
Tracking error is an instrument to measure the performance of the self-financing. In other words, the difference between the derivative’s payoff and the replication value at time T is called tracking error. The figure is positive when we made a loss from our hedging and vice versa. This instrument provides an easy understanding to capture whether the model we used is appropriate or not. Using the above hedging as
an example, we define the tracking error(eTM)as:
(
)
(
,)
(
(
,)
)
( 0 0 T M M T T T T s s e =h S ρ α −V =h S ρ α − V +∫
α dS ( , ))ρ α (2.13)Therefore, we can compare the standard Black-Scholes hedging strategy and the innovative hedging strategy which considers the market is illiquid. If the former tracking error is greater than the latter, it proofs that the Frey and Patie model is a superior model. In the following proposition, we recall the nonlinear PDE (equation (2.12)) to proof the perfect hedging under market illiquidity. The tracking error equals to follows:
( )
( )
( 0 0 ( TM M
T T T T s s
e =h S −V =h S − V +
∫
α dS ρ α, )) (2.14)By using theIto s' formula we can obtain
( )
T 2 2 0 0 0 ( , ) 1 (0, ) ( , ) + ( , ) ( , ) ( , ) 2 T T T S t t t t SS t t t h S u T S u S u t S dS u t S u t S v t S S dt = = +∫
∫
+ (2.15)We continuously transfer the left part pattern T 2 2
0 1 ( , ) ( , ) ( , ) 2 t t SS t t t u t S + u t S v t S S dt
∫
into by using the nonlinear Black-Scholes PDE and then we
derive the tracking error:
0 ( , ) ( , ) T s ru t S −rSu t S dt
∫
( )
0 0 0 0 0 0 0 ( ( , )) (0, ) ( , ) ( , ) ( , ) ( ( , )) = ( , ) ( , ) T M T T s s T T T S t t s s s T s e h S V dS u S u t S dS ru t S rSu t S dt V dS ru t S rSu t S dt α ρ α 0 α ρ α = − + = + + − − + −∫
∫
∫
∫
∫
(2.16)If we assume that the risk-free interest rate equals to zero, the hedger who adopts self-financing hedging leads the tracking error to be zero. In other words, using the hedging strategy derived by nonlinear Black-Scholes PDE makes the perfectly hedging in illiquid market.
2.6 Numerical Method (Finite difference method)
As we known, partial differential equation (PDE) plays an important role in financial engineering and becomes the essential instrument for option pricing. However, the analytical solution is not always available by using the PDE. Taking the nonlinear Black-Scholes PDE for example, it could not be solved analytically. Therefore, numerical method is needed to estimate the value of the derivatives. In this way, we adopt the finite difference method which is widely used in option valuation.
When it comes to finite difference method, there are three kinds of approach such as explicit, implicit and Crank-Nicolson method. Explicit method uses the least time for computation, but it may face the instable problem. Implicit method is robustness
for option valuation and provides the precise option value. However, it does not have the problem of instability. Last, Crank-Nicoleson approach is the hybrid between explicit and implicit method. Because we hope to have a more precise and robust option price, the implicit would be the optimum choice. Therefore, our first step is to make the nonlinear PDE become discrete. Throughout the discreteness, the original equation is transferred as below:
( )
( )
1 1 1 1 1 1 2 1 1 2 1 1 1 2 2 1 2 2 i i i i i i i j j j j i j j j i j j j j U U U U U U U rS S rU t S ν S − − − − − − + − + − − − − − + + + Δ Δ Δ = (2.17)i is the mesh grid of time. i=0, t,2 t,...,N t=T
j is the mesh grid of asset price. j=0, ,2 ,...,M =Smax
where
S S S
δ δ δ
δ δ δ
After arrangement, we have
(
)
(
)
(
)
2 2 1 2 2 1 2 2 1 1 0.5 * * * 0.5 * ( ) * * 1 ( ) * * * + -0.5*( ) * * 0.5 * * * i i i i i1 j j j j i i j j U r j t j t U j t r t j t r j t U ν ν ν j U − − − − + = Δ − Δ + + Δ + Δ Δ − Δ (2.18) That is equal to(
)
(
(
)
1 1 1 1 j 1 2 2 2 2 2 2 j +c 0.5 * * * 0.5 * ( ) * * 1 ( ) * * * c -0.5*( ) * * 0.5 * * * i i i i j j j j j j i j i j j i j U a U b U U a r j t j b j t r t j t r j t ν ν ν − − − − + = + = Δ − = + Δ + Δ = Δ −)
j Δt Δ (2.19)This equation means that we need the three previous option prices to get the present option price. There are M option prices which belongs to different stock price needed to be solved simultaneously. Then, we have to solve the tridiagonal matrix as below:
1 1 2 2 2 3 3 3 2 2 2 1 1 . . . M M M M M b c a b c a b c A b c a a b − − − − − ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎟
)
(2.20)Since we have already known the terminal value, the previous option value can be derived by backward method. In addition, we also apply Thomas algorithm which is designed to solve the tridiagonal matrix. In addition, Frey and Patie (2002) set up an artificial pattern to smooth the volatility term.
{
}
(
2 2 0 2 1 0 1 max , 1 min , 0.02, 0.85 SS Su where σ ν α α ρ α α ⎧ ⎫ ⎪ ⎪ = ⎨ ⎬ − ⎪ ⎪ ⎩ = = ⎭ (2.21)After incorporating this smooth pattern, the derivatives’ payoff truly becomes smooth
than before. However, it causes the volatility instable. If we assume that σ =0.4 and
1 SuSS
α >ρ , the volatility will turn into 0.4 2.667
1 0.85− = . It is definitely unreasonable for
economic intuition. Consequently, we use bootstrap method to obtain the ninety-nine percentage of the volatility to control the volatility upper bound.
{
}
(
)
2 2 0 2 99% 1 0 1 99% ˆ min max , , 1 min , ˆ0.02, 0.85, is the 99% of the volatility distribution
SS Su where σ ν α σ α ρ α α σ ⎧ ⎡ ⎤ ⎫ ⎪ ⎢ ⎥ ⎪ = ⎨ ⎬ ⎢ − ⎥ ⎪ ⎣ ⎦ ⎪ ⎩ ⎭ = = (2.22)
The conversion of the volatility makes the Greeks of derivatives more stable. In addition, it also provides a better performance of option valuation. All of the above numerical methods help us to estimate the option value more feasible and reliable.
3. Numerical Results
In this section, we use a specific setting to show the hedging costs (option values) through different kinds of liquidity. From this experiment, we would show the tendency of hedging cost when the market liquidity is decreasing. In addition, we compare the Greeks of the derivative to show the difference between the Frey model and the improved Frey model which revises the volatility smooth function. Before analyzing, there is one thing needed to be informed that liquidity is the opposite of liquidity parameter. In other words, the higher (lower) the liquidity, the smaller (bigger) the liquidity parameter is.
3.1 Hedging cost in illiquidity
Six factors needed to estimate option values include the stock price, exercise price, risk free rate, maturity, sigma and liquidity. The underlying asset prices are between 0 and 100. The exercise price, risk free rate, maturity and sigma are 50, 0.05, 0.25 and 0.25 respectively. The range of the liquidity is among 0 ~ 1.
From figure 3-1, the hedging costs of call options have increased obviously when the liquidity decreases. This phenomenon indicates that the representative hedger has to pay additional cost because the low liquidity enlarges the volatility. According to the understanding of standard Black-Scholes formula, the option price goes up when the volatility goes up. Therefore, the reason of raising hedging cost is the decreasing liquidity which results in higher volatility. In addition, the hedging cost around the exercise price is higher than any other price.
[figure 3-1]
On the other hand, we find hedging cost of put option in the same situation from figure 3-2. The put option’s hedging cost raises when the liquidity becomes worse. Consequently, the results show that the representative hedger has to pay more money
than they are in the perfect liquidity.
[figure 3-2]
3.2 Greek letters in illiquid market
When it comes to dynamic hedging, the Greek letters are the most popular hedging strategies. Greek letters include delta, gamma, vega …etc. In this part, we are going show the influence of Greek letters resulting from various liquidities. Furthermore, we imply the volatility smooth function into the extension Black-Scholes PDE to get the steady Greek letters throughout the different liquidities. In the following subsection, we assume a situation when the representative hedger sells either a call or a put option. Using the simulation method observes the tendency and variation of Greek letters.
Delta ratio measures the sensitivity of the option price corresponding to the variation of underlying asset price. In figure 3-3, the call delta is increasing from 0 to 1. The representative hedger needs to buy additional shares of stock to protect his position while underlying asset price goes up. In particular, the call delta turns into flatness when the market becomes illiquid.
[figure 3-3]
In figure 3-4, the put delta is from -1 to 0 while the underlying asset price increases. The large investor could sell his shares of stock when the asset price is higher than strike price, and the put option would not be exercised. The put option turns into flatness when the market becomes illiquid. In other words, the illiquid underlying asset flattens the delta hedging strategy no matter what kinds of option are sold by the representative hedger. If the hedger does not adopt the correct liquidity parameter, he would face the over hedge or under hedge problem.
[figure 3-4]
Gamma hedging to measure the sensitivity of the delta ratio throughout the asset prices. Figure 3-5 and 3-6 represent call Gamma and put Gamma. Both of Gamma tendency shows that market liquidity lowers the height of Gamma and makes Gamma title to left. In the illiquid market, the representative hedger relocates stock shares before asset price approaches exercise price.
[figure 3-5]、 [figure 3-6]
If we use the original Frey model without improving the volatility function, the Gamma shows instability. Therefore, we implement the improved volatility smooth function into Frey model. Figure 3-7 and 3-8 shows that the delta ratio is originally stable in specific illiquidity (0-0.4). However, after we implement the smooth volatility function showing in figure 3-3 and 3-5, the delta and Gamma have become smooth and stable. After applying the improved volatility function, we release the limitation of liquidity to get the stable Gamma hedging and delta ratio.
[figure 3-7]、 [figure 3-8]
Vega is used to measure the sensitivity of sigma corresponding to asset price variation. According to equation (2.21), the volatility term increases because of the worse liquidity. The figure 3-9 is in line with the theory of volatility function.
4. Empirical Results
The objection of this part is to investigate the performance of option valuation and dynamic hedging. We choose the standard Black-Sholes model as our benchmark and also compare the traditional Frey model.
4.1 Data selection
In order to select the representative data, we decide to use OptionMetrics as our database which belongs to Wharton university of USA. Because CBOE is famous for trading option, we choose the CBOE’s top 50 equity options ranked by volume to be the research data. The data period is from 2000/01/01 to 2004/12/31 and risk free rate is the three years treasure bill of America. Option price is the average of the bid and ask. We also take the implied volatility as daily volatility of each asset. Furthermore, underlying asset price is the close price. Throughout the arrangement, we use these data to verify the performance of option pricing and dynamic hedging.
4.2 Least square method and loss function
In the improved Frey model, the only unknown factor is the liquidity parameter. Since there is no general definition of liquidity, we follow Bakshi, Cao and Chen (1997) to use the least square method to estimate the liquidity for each trading day. Before introducing the estimation procedure, we set up the daily close option price as
which i is the ith option contract and j is the jth day. In addition,
, i j
C Ci represents
the option value estimated by the improved Frey model. In the improved Frey model,
the volatility smooth function needs a 99th of historical volatility to limit the upper
bound of volatility. Therefore, we use the bootstrap method and repeat 100 times to calculate the volatility upper bound for each underlying asset.
i l N 2 * 99 , 0 1 1 1 option price price calculate from the improve Frey model
arg min ( , , , , , , , , ) j M N implied obs th N j i j j j j initial i j j i observed option C S K T r sigma C ρ ρ ρ α α = = ⎛ ⎞ ⎜ ⎟ = − ⎜ ⎟ ⎝ ⎠
∑∑
σ , (4.1)In equation (4.1), the first step is using all of the option contracts in the first day to estimate the appropriate liquidity parameter by least square method. The range of the
initial
ρ is from 0 to 1. Pattern search algorithm chooses ρinitial corresponding to the
minimum value of least square automatically. The parameter N represents the length of moving window and n represents the length of date. In this article, we choose N
equals to 1 as daily estimation. In the successive days, the ρinitial is replaced by the
T ρ as shown in equation 4.2. i l N 2 * 99 , 1 0 1 1 1 option price price calculate from the improve Frey model
arg min ( , , , , , , , , ) , 1, 2,..., j M T implied obs th T j i j j j j T i j j T N i observed option C S K T r sigma C T N N N ρ ρ ρ − α α = − + = ⎛ ⎞ ⎜ ⎟ = ⎜ − ⎟ ⎝ ⎠ = + +
∑ ∑
{
n−1}
, σ (4.2)As a result, we obtain series of liquidity throughout n days. Having the liquidity time series helps to calculate the at-the-money option contracts. In order to compare the performance of option pricing between the Black-Sholes model and improved Frey model, we illustrate different kinds of loss function such as $MSE, %MSE and IVMSE. The definition of $MSE, %MSE and IVMSE are displayed in equation 4.3, 4.4 and 4.5 respectively. i
(
2 1 1 $ n obs i i i MSE C C n = =∑
−)
.. (4.3) i 2 1 1 % obs n i i obs i i C C MSE n = C ⎛ − ⎞ = ⎜⎜ ⎝ ⎠∑
⎟⎟ .(4.4) i(
2 1 1 n i i i IVMSE n = σ σ)
=∑
− .(4.5) i i 1 i 1and are the implied volatility ( , , , , ) ( , , , , ) i i i i i i i i obs i i i i i i where BS C S K r T BS C S K r T σ σ σ σ − − = =
4.3 The performance of option pricing model
There are three tables comparing the performance of different models. Table 1 takes $MSE as the loss function. In table 1, most of the assets in the improved Frey
model have the smallest values, and only four option contracts do not fit well. This
result seems that the improved Frey model is the best option pricing model in this inspection. However, table 2 exhibits the top 50 options’ %MSE, and the performance of improved Frey model do not defeat the traditional Frey model.
[Table 1]、[Table 2]
The difference between these two loss functions comes from the different criteria. In other words, we may have inconsistent solutions if we adopt the distinct loss function. According to Engle (1993), the choice of the loss function is important in model evaluation because it exits particular error structure. Taking $MSE for example, $MSE gives more weight for higher value option contracts such as in-the-money and long time-to-maturity contracts. On the other hand, %MSE also has the heteroskedastic error structure. Although %MSE has the advantage to give $1 error less weight in $100 option value than in $10 option value, it overcorrects the error structure. It faces the instable problem when the option contract is out-of-the money and short time-to-maturity. In Pan (2002), Pan takes IVMSE to instead of traditional loss functions. The advantage of IVMSE is that IVMSE does not have heteroskedastic error structure. Therefore, we apply IVMSE to investigate the performance of various models.
Table 3 shows that the improved Frey model seems better than the original Frey model. There are ten option contracts are not over the 95% t test but all of the figures are smaller than the traditional Frey model.
4.4 Dynamic hedging
As a representative hedger, the performance of dynamic hedging is as important as option valuation. Since we have already known how to estimate the option value, it is naturally to implement the improved Frey model into hedging manipulation. The tracking error is a powerful instrument to verify the difference between the estimated option value and the observed option value. The tracking error method is shown in equation (2.14) and then we also compare the performance of the Black-Scholes and the traditional Frey model. In equation (2.14), we adopt the self-financing strategy to
calculate the option value at time T ( M
T
V ). The below equation (4.6) shows the process
of self-financing. i i 1
(
( )
(
)
)
(
i 1(
)
)
1 1 : 1 ( 1) 1 M Frey Frey i i i i i i V C =C− +delta− ∗ S i −S i− +r i− ∗ C− −delta− ∗S i− ∗ Δt ..(4.6)We also take the new pattern to replace the observed option value . For
this reason, we have the total tracking error of specific option contract which time to maturity is n. obs i C h S
(
T)
C( )
i 2 2 T T M M obs i T i i i i e h S V C = =∑
− =∑
− (4.7)Therefore, we choose top 20 option contracts to test the improved Frey model dynamic hedging ability. The maturity time is around one year which is from 2000/01/01 to 2001/1/20. In particular, we only pick up the at-the-money option contracts to be our trading assets. As shown in table 4, the results of improved Frey model is better than Black-Scholes model in dynamic hedging. Through the statistic analysis, the difference between these two models is significant.
[Table 4]
From the above results, the improved Frey model not only provides more precise option values but also accurate dynamic hedging strategies.
5. Conclusion
This article follows Frey and Patie’s (2002) research which incorporate the liquidity parameter into asset price dynamic process. The nonlinear PDE is derived from the stochastic process of underlying asset price. In order to solve the nonlinear partial difference equation, we adopt the finite difference method which is generally used to calculate the option price. Besides, due to the fact that Thomas algorithm is famous for tridiagonal computation, we apply it to reduce the computation time.
The most significant difference between the traditional Frey model and the improved Frey model is the improved volatility smooth function. Without correcting the volatility upper bound, option volatility could be irrational. The irrational volatility results in inconsistent and instable Greek letters. Thus, we release the limit of liquidity range for Greek letters. Originally, the Greek letters was turned into instability while liquidity parameter is over 0.4. Since we have improved the volatility smooth function, the limited liquidity problem no longer exists.
In empirical results, we use three different kinds of loss functions to investigate the performance of various models. The comparison shows that the improved Frey model is superior when we choose the reliable IVMSE as the loss function. On the other hand, tacking error exhibits the model’s hedging ability. The improved Frey model still has the outstanding hedging performance.
In further research, we would use time series model to estimate the liquidity process. The time series of liquidity help market participants to forecast the market liquidity in advance. Therefore, option values are easily calculated by the improve Frey model. Investors take the estimated option value as the benchmark to deicide their trading strategy. Moreover, the exotic option value is also derived by the stochastic process of underlying asset.
References
Amihud, Y., and H. Mendelson. 1989. The effect of computer base trading on
volatility and liquidity. The challenge of information technology for the securities
markets, liquidity, volatility, and global trading: 59-85
Bakshi, G., C. Cao, and Z. Chen. 1997. Empirical performance of alternative option pricing models. Journal of Finance 52: 2003-2049.
Barndimarte, Paolo. 2002. Option valuation by finite difference methods. Numerical
Methods in Finance, 347-364.
Brennan, J. Michael, and E. S. Schwartz. 1978. Finite difference methods and jump processes arising in the pricing of contingent claims: A synthesis. Journal of
Financial and Quantitative Analysis 13(Sep): 461-474
Bielecki, T., and M. Rutkowski. 2002. Credit Risk: Modeling, Valuation and Hedging.
Springer.
Cetin, U., R. Jarrow, P. Protter, and M. Warachka. 2006. Pricing options in an
extended Black Scholes economy with illiquidity: Theory and empirical evidence.
Oxford Journals 19: 493-529
Chan, L. K. C., and J. Lakonishok 1995. The behavior of stock prices around institutional trades. Journal of Finance 50: 1147-1174
Esser, A. and B. Moench. 2003. Modeling feedback effects with stochastic liquidity.
Goethe University, Frankfurt am Main, Germany.
Engle, R. 1993. A comment on Hendry and Clements on the limitations of comparing mean square forecast errors, Journal of Forecasting 12: 642-644
Frey, Rudiger. 2000. Market illiquidity as a source of model risk in dynamic hedging.
Model Risk.
Frey, R., and P. Patie. 2002. Risk management for derivatives in illiquid markets: A simulation study. RISK Publications.
Henry-Labordere, P. 2004. The feedback effect of hedging in portfolio optimization. Working paper.
Hull, John .C. 2006. Wiener Process and Ito’s Lemma. Options, Futures, and Other
Derivatives, 263-277.
Kyle, A. S. 1985. Continuous auctions and insider trading. Journal of Finance 53: 1315-35
Lyukov, A. 2004. Option Pricing with Feedback Effects. International Journal of
Theoretical and Applied Finance.
Merton, C. R. 1975. Option pricing when underlying stock return are discontinuous.
Pritsker, M. 2002. Large investors, implications for equilibrium asset returns, shock absorption, and liquidity. mimeo. Federal Reserve Board.
Sanfelici, S. 2006. Calibration of a nonlinear feedback option pricing model. Working paper.
Schwartz, R. A. 1988. Equity markets: Structure, trading, and performance. New York:Harper and Row, inc.
Schonbucher, P., and P. Wilmott. 2000. The Feedback Effect of Hedging in illiquid Markets. SIAM Journal on Applied Mathematics 61: 232-272.
Vijh, A. M. 1990. Liquidity of the CBOE Equity Options. Journal of Finance 45: 1157-79.
Tables
Table 1. $MSE of top 50 option contracts
BS_$MSE Frey_$MSE SFrey_$MSE Std T statistic BS_$MSE Frey_$MSE SFrey_$MSE Std T statistic
BAC 0.6462 0.1028 0.0586 0.2585 6.0327 BGEN 0.2499 0.1920 0.1065 1.2797 2.1128 CE 0.0799 0.0460 0.0054 0.4354 2.9986 CIEN 0.3258 0.3107 0.0190 2.0548 4.9374 Citigroup 0.4749 0.0729 0.0514 0.0915 8.2656 CNXT 0.4773 0.4505 0.4154 4.2624 0.2637 CSCO 0.1728 0.0925 0.0114 0.7412 3.8628 CY 0.0548 0.0427 0.0046 0.3131 4.3111 DELL 0.1674 0.0306 0.0133 0.0658 9.2973 F 0.3221 0.2535 0.0614 1.7475 3.8721 EP 0.1230 0.0226 0.0086 0.0708 6.9729 GLW 1.0047 0.5415 0.1921 3.3707 3.6430 GE 0.3543 0.0343 0.0060 0.0884 11.3110 JDSU 0.4287 0.3933 0.1085 4.1517 2.4262 GM 0.4106 0.0438 0.0222 0.1052 7.2409 JNPR 4.4466 2.2138 1.1251 7.0796 5.4415 HPQ 0.1513 0.0968 0.0113 0.7791 3.8703 KO 0.4183 0.0616 0.0454 0.0967 5.9051 IBM 1.4252 0.2776 0.1284 0.9060 5.8101 LSI 0.5545 0.1708 0.1189 0.3956 4.6454 INTC 0.2007 0.0573 0.0079 0.3757 4.6343 MOT 0.2139 0.1609 0.0894 1.1449 2.2096 JPM 0.2574 0.0426 0.0209 0.1501 5.0863 MRK 0.7456 0.1042 0.0777 0.3236 2.8984 MO 0.4706 0.0666 0.0473 0.1136 6.0048 NOK 0.8593 0.1705 0.0591 1.3737 2.8710 MSFT 0.9610 0.0917 0.0618 0.1469 7.1862 NT 1.5271 0.8537 0.7427 1.2478 3.0815 ORCL 0.0799 0.0694 0.0114 0.4593 4.4566 NVLS 0.1029 0.0659 0.0130 0.2959 6.3384 QCOM 0.3350 0.1899 0.0324 1.1130 4.9897 NXTL 0.3046 0.0864 0.0207 0.6391 3.6428 TWX 0.2235 0.0408 0.0103 0.1483 7.2501 PALM 0.2100 0.1596 0.1160 4.9042 0.2998 WMT 0.5858 0.1010 0.0667 0.1727 6.9888 PFE 0.3186 0.0431 0.0293 0.0623 7.8249 XMSR 0.0371 0.0194 0.0112 0.0779 3.4163 Q 0.0476 0.0347 0.0261 0.4398 0.6861 YHOO 1.3244 0.3440 0.0575 3.4969 2.8907 TMX 2.6397 2.2634 2.2261 3.4922 0.3700 AMAT 0.1528 0.1426 0.0595 0.9712 3.0283 TXN 0.1566 0.1048 0.0143 0.8391 3.8169 AMCC 0.9761 0.5449 0.1604 2.4070 5.5336 TYC 0.3295 0.1167 0.0861 0.4394 2.4659 AMD 0.1044 0.0892 0.0071 0.7409 3.9249 VRTS 1.2432 1.1119 0.6541 3.8894 4.1632 AMZN 0.5425 0.0840 0.0327 0.4201 4.3242 XLNX 0.1238 0.0920 0.0216 0.6157 4.0473 BMY 0.2618 0.0510 0.0241 0.2786 3.4197 XRX 0.0307 0.0076 0.0030 0.0217 7.3817
Table 2. %MSE of top 50 option contracts
BS_%MSE Frey_%MSE SFrey_%MSE Std T statistic BS_%MSE Frey_%MSE SFrey_%MSE Std T statistic
BAC 4.8578 0.9935 0.7814 1.7760 4.1370 BGEN 0.0636 0.0404 0.0105 0.3270 2.8962 CE 0.6891 0.1548 0.0969 1.4218 1.3073 CIEN 1.3462 0.5409 0.4422 3.2971 1.0469 Citigroup 2.2746 0.4884 0.4811 2.6146 0.0926 CNXT 3.0225 3.3146 3.2971 3.7965 0.1222 CSCO 1.8752 0.2883 0.2521 2.4506 0.5184 CY 0.1997 0.0481 0.0135 0.3175 3.8589 DELL 0.6919 0.0955 0.0837 0.9362 0.4483 F 7.6450 4.6748 4.5001 2.0530 2.1062 EP 0.4258 0.0801 0.0373 0.4625 3.2608 GLW 3.9990 0.6790 0.1215 2.9232 6.7119 GE 6.6343 1.3486 0.9224 11.4000 1.3224 JDSU 1.0806 0.5012 0.4213 0.4320 6.4038 GM 0.9656 0.1390 0.1113 0.8188 1.1897 JNPR 1.7166 0.4091 0.0673 3.2655 3.7095 HPQ 1.6184 0.2403 0.2271 1.7321 0.2644 KO 3.9112 0.5617 0.4310 4.0797 1.1379 IBM 1.3129 0.2419 0.2213 1.7488 0.4125 LSI 13.4340 4.5765 1.6858 13.1750 7.7731 INTC 0.7016 0.0594 0.0591 0.4783 0.0224 MOT 8.4932 6.3928 4.3343 18.0287 3.9553 JPM 1.8600 0.3501 0.3024 2.7427 0.6099 MRK 2.3723 0.4813 0.2280 2.2138 4.0548 MO 4.5312 0.7967 0.5740 7.2656 1.0847 NOK 11.0760 2.5643 0.1311 17.7740 4.8518 MSFT 5.8402 1.3535 1.3390 7.2604 0.0651 NT 0.8694 0.3591 0.1399 1.5942 4.3812 ORCL 0.4689 0.0764 0.0391 0.3342 3.9310 NVLS 0.1538 0.1207 0.0113 3.7652 1.0287 QCOM 0.0178 0.0142 0.0104 0.1402 0.9576 NXTL 2.5488 0.3918 0.1085 3.2336 3.1064 TWX 0.7891 0.1601 0.0961 1.7236 1.3079 PALM 5.4323 3.4556 3.0202 4.2098 3.5827 WMT 5.4230 2.8050 2.6561 10.6040 0.4864 PFE 1.9348 0.3154 0.3110 1.1111 0.1326 XMSR 0.1107 0.0430 0.0344 0.1114 2.4963 Q 0.5569 0.3315 0.2888 3.3644 0.4518 YHOO 0.6596 0.0928 0.0235 0.6080 4.0264 TMX 3.5201 0.6696 0.5760 6.7867 0.4915 AMAT 0.1764 0.0186 0.0082 0.0529 6.9188 TXN 0.3653 0.0376 0.0224 0.3972 1.3515 AMCC 1.3001 1.1951 1.1161 6.1653 0.4591 TYC 19.2760 9.4093 7.9265 15.2347 3.3716 AMD 0.0862 0.0398 0.0103 0.2283 4.5669 VRTS 1.3394 0.6759 0.6279 1.0503 1.5845 AMZN 0.4214 0.0364 0.0236 0.1614 2.7968 XLNX 0.0851 0.0126 0.0045 0.0442 6.4591 BMY 5.6361 0.9886 0.9101 6.1192 0.4585 XRX 0.5667 0.0957 0.0564 0.3518 3.9497
Table 3. IVMSE of top 50 option contracts
BS_IVMSE Frey_IVMSE SFrey_IVMSE Std T statistic BS_IVMSE Frey_IVMSE SFrey_IVMSE Std T statistic
BAC 0.0255 0.0038 0.0028 0.0040 8.5148 BGEN 0.0181 0.0061 0.0026 0.0202 5.5510 CE 0.0202 0.0082 0.0021 0.0310 6.3222 CIEN 0.0461 0.0185 0.0050 0.0549 8.5637 Citigroup 0.0313 0.0050 0.0039 0.0047 8.5596 CNXT 0.2317 0.0932 0.0764 0.8845 0.6726 CSCO 0.0307 0.0071 0.0030 0.0171 8.5861 CY 0.0153 0.0058 0.0018 0.0162 8.7490 DELL 0.0218 0.0038 0.0021 0.0060 10.4730 F 0.2148 0.1987 0.1063 0.6634 4.4998 EP 0.0284 0.0113 0.0083 0.0671 1.5692 GLW 0.1453 0.0333 0.0271 0.1221 1.7969 GE 0.4560 0.0387 0.0051 0.9117 1.2766 JDSU 0.1080 0.0243 0.0136 0.0758 5.3311 GM 0.0254 0.0027 0.0020 0.0044 5.6405 JNPR 0.4542 0.3966 0.3228 0.4667 5.4778 HPQ 0.0307 0.0052 0.0030 0.0090 8.4035 KO 0.0281 0.0041 0.0037 0.0033 4.4664 IBM 0.0257 0.0042 0.0030 0.0054 7.7903 LSI 0.2597 0.0523 0.0122 0.0729 19.5150 INTC 0.0234 0.0037 0.0016 0.0072 10.0160 MOT 0.0200 0.0076 0.0033 0.0382 3.9597 JPM 0.0254 0.0037 0.0027 0.0071 5.1765 MRK 0.0446 0.0082 0.0041 0.0194 7.6500 MO 0.0357 0.0052 0.0042 0.0065 5.4128 NOK 0.1493 0.0251 0.0042 0.0460 16.1130 MSFT 0.0595 0.0093 0.0082 0.0056 6.9705 NT 0.2156 0.1864 0.1251 0.7437 2.8553 ORCL 0.0249 0.0061 0.0030 0.0204 5.4018 NVLS 0.0121 0.0040 0.0016 0.0225 3.7463 QCOM 0.3340 0.0589 0.0344 0.3259 2.6431 NXTL 0.0534 0.0101 0.0063 0.0369 3.6554 TWX 0.0305 0.0051 0.0027 0.0141 5.9668 PALM 0.5290 0.4697 0.4594 0.3654 0.9765 WMT 0.0339 0.0059 0.0049 0.0057 5.6724 PFE 0.0362 0.0051 0.0043 0.0038 6.9891 XMSR 0.0731 0.0606 0.0466 0.1375 3.3046 Q 0.4222 0.2132 0.2107 1.0535 0.0771 YHOO 0.0895 0.0070 0.0059 0.0235 1.5719 TMX 0.4910 0.4425 0.4269 0.4880 1.2062 AMAT 0.0133 0.0038 0.0015 0.0097 8.6067 TXN 0.0180 0.0032 0.0011 0.0058 13.0120 AMCC 0.0579 0.0401 0.0346 0.0915 2.1170 TYC 0.0437 0.0189 0.0169 0.1025 0.6955 AMD 0.0159 0.0092 0.0047 0.0316 5.0388 VRTS 0.0343 0.0166 0.0138 0.0559 1.7829 AMZN 0.0631 0.0104 0.0047 0.0290 6.9593 XLNX 0.0094 0.0038 0.0017 0.0223 3.4277 BMY 0.0330 0.0059 0.0044 0.0182 2.9514 XRX 0.0277 0.0092 0.0065 0.0281 3.3986
BS Improved_Frey Std T P BS Improved_Frey Std T P BAC 0.5630 0.5541 0.0309 4.1085 0.00006 INTC 1.3557 1.3212 0.1558 3.5956 0.00039 CE 0.2116 0.1128 0.3738 4.3070 0.00003 JPM 1.9558 1.9443 0.0505 3.7037 0.00026 Citigroup 2.2183 2.2129 0.0181 4.7660 0.00000 MO 0.3734 0.3692 0.0172 3.9696 0.00009 CSCO 5.2731 4.5691 2.3742 4.8261 0.00034 MSFT 0.9737 0.9448 0.1000 4.7002 0.00000 DELL 0.6709 0.6486 0.0530 6.8437 0.00000 ORCL 6.6402 6.5256 0.3659 5.0891 0.00000 EP 0.4940 0.4857 0.0260 5.1133 0.00000 QCOM 1.9295 1.6568 0.5394 8.2168 0.00000 GE 1.1087 1.1042 0.0236 3.0769 0.00231 TWX 1.4726 1.3915 0.2329 5.5221 0.00000 GM 0.8154 0.8102 0.0179 4.7172 0.00000 WMT 0.7225 0.7092 0.0383 5.6426 0.00000 HPQ 1.6919 1.6312 0.1687 5.8378 0.00000 XMSR 0.3280 0.3131 0.0619 3.1040 0.00225 IBM 1.2586 1.2374 0.1103 3.1343 0.00192 YHOO 2.9279 2.3269 1.6026 6.0930 0.00000
Figures
[figure 3-1] Call option prices under different illiquid markets
[figure 3-3] Call option deltas under different illiquid markets
[figure 3-5] Call option Gammas under different illiquid markets
[figure 3-7] Traditional option deltas under illiquid markets
