Financial Mathematics: Pricing in Illiquid Markets Research Paper

Introduction

Economy is the basis for the development of the modern human society. However, it is at the same time one of the most complicated spheres of knowledge. Thus, to be able to see the implications that this or that economic activity or phenomenon has, or will have, it is necessary to operate properly with the major terms and definitions and be able to apply the respective theoretical knowledge in practice. Drawing from these considerations, the proposed paper will focus on such an important aspect of the modern economy as illiquid markets, pricing policies observed in such markets, and portfolio choices that are often conditioned by illiquidity. What makes some assets illiquid? Is illiquidity of markets dangerous? How is pricing affected by illiquidity levels? These are some of the questions that the proposed paper will make an attempt to answer.

Literature Review

General Notions

Thus, the major topic of the proposed paper is the analysis of pricing policies observed in the illiquid markets in the contexts of portfolio choices and overall economic conditions. The analysis of this topic becomes especially relevant in the light of the current global economic recession, during which major companies consider the alternatives of turning their illiquid and non-marketable assets into the liquid ones (Longstaff, 2009, p. 1120). Accordingly, the illiquid market, and all phenomena associated with it, is perceived by scholars as a rather dangerous environment. In such a market, trading becomes more time-consuming and marketability of the major assets is rather doubtful, which makes pricing policies in the illiquid market considerably cautious (Schönbucher and Wilmott, 2000, pp. 232 – 233; Brenner et al., 2001, pp. 789 – 790).

Needless to say, give the crucial importance of the topic of illiquid markets and their pricing policies, this topic has been heavily discussed by various scholars. The major ideas examined range from such seemingly minor issues as the speed of transactions and infrequent adjustment in illiquid markets (Garleanu, 2008, pp. 535 – 536; Capozza et al., 2003, pp. 8 – 9) to such drastically important aspects as asset market functioning in the context of illiquidity and pricing policies developed in respect of the portfolio choices often observed in illiquid markets (Longstaff, 2009, pp. 1123 – 1124; Frey et al., 2002, pp. 3 – 4). In addition, scholarly attention in the works concerning illiquid markets, their prices and portfolio choices made often touch upon the issues of hedging (Frey et al., 2002, p. 2), various risks associated with liquidity, or illiquidity, levels (Janosi et al., 2002, pp. 4 – 5), and asset pricing (Chan and Faff, 2002, p. 20).

Major Theories

Naturally, scholarly discussions of illiquid markets and phenomena associated with them come from generalized ideas to more specific notions, theories, and models that allow better illustrating the functioning of economy under the conditions of illiquidity. One of the most prominent models among them is the two-asset economy model developed by Garleanu (2008, pp. 535 – 536) and Brenner et al., (2001, p. 792). The essence of this model is expressed by two equations that follow (equations 1 and 2):

• dD (t) = m_Ddt + σ_Dd B (t),
• dηⁱ (t) = m_ηdt + σ_ηd Bⁱ (t)

In these equations, m_D and σ_D serve as constants, while B marks the standard Brownian motion, while t refer to time of the liquidity change. At the same time, the concept of i in the second equation denotes an agent of asset trading, while ηⁱ is the cumulative endowment process this agent goes through. Thus, the very two-asset economy model is based on the idea that there are two asset types, one of which has no risks and pays considerable interest on the regular basis, while another one displays a cumulative nature of dividends and can be calculated only under the condition of the asset market liquidity equilibrium (Garleanu, 2008, p. 536; Brenner et al., 2002, p. 792), expressed by the equation (equation 3):

• Σμ_kE_t [1 p(s) = p_j I p(t) = p_k] = Σ E_t [1 p(s) = p_j I p(t) = p_k] = μ_j

Interestingly, Longstaff (2009, pp. 1122 – 1123) and Muzziolli et al. (2001, pp. 4 – 5) continue developing the two-asset model for illiquid market stock pricing, but they do it naming this model in different ways. If Longstaff (2009) refers to it as the “two asset version of the standard Robert E. Lucas (1978) pure exchange economy” (p. 1122), Muzziolli et al. (2001) call the model a “binomial tree” (p. 4). These two theoretical models are not accidentally considered together. The reason for this is the similar consideration of the models that Longstaff (2009) and Muzziolli et al. (2001) present irrespective of the chronological and geographical differences.

In more detail, the point that makes the ideas by Longstaff (2009) and Muzziolli et al. (2001) similar, and that differentiates those ideas from the views by Garleanu (2008) and Brenner et al., (2001) is the possibility of two homogeneous asset types functioning with the market. At the same time, Longstaff (2009, p. 1123) notices that under the condition of illiquidity of one of the assets, the previously dynamically complete market becomes incomplete, and this sutation can be expressed by the following relations (equations 4 and 5):

• dX / X = μ_X dt + σ_XdZ_X
• dY / Y = μ_Ydt + σ_YdZ_Y

Similarly to the ideas by Garleanu (2008) and Brenner et al., (2001), Longstaff (2009) and Muzziolli et al. (2001) in the above equations focus on the ideas of the constant asset values expressed as μ_X, σ_X, μ_Y, σ_Y, placed in the context of dividends generated by the assets X and Y, one of which is liquid and another one is illiquid (Longstaff, 2009, p. 1123).

So, one can see that rather often the scholarly ideas in consideration of market illiquidity and asset-pricing were conditioned by such a dual paradigm, in which there is always a complete element, i. e. a liquid asset that is subject to few or no risks and pays dividends, and an incomplete one, i. e. an illiquid asset. The marketability and potential benefits of the latter are greatly doubted, which is associated with great risks, decreased market agent confidence, and falling trade rates in the illiquid asset market (Garleanu, 2008, pp. pp. 535 – 536; Brenner et al., 2002, p. 792; Longstaff, 2009, p. 1123; Muzziolli et al., 2001, pp. 4 – 5). In this respect, scholars also consider the potential and actual effects of illiquidity upon market assets’ prices and the impact of price changes upon the observed liquidity rates dynamics.

Further on, Frey et al. (2002) also consider the two-asset stylized financial market model, but introduce new terminology and alternative equations to help in calculating the asset risks and illiquidity of the market. Thus, Frey et al. (2002, p. 3) refer to the two asset types as bond and stock, with the former being a liquid asset that presents no risk and is marketable and tradable, and the latter being a risky and a rather hard-to-trade asset. Operating with notions of stochastic process S_t, market liquidity p, and volatility σ, the authors derive the stochastic differential equation to calculate the dynamics of asset pricing processes in the illiquid market where at least one of the assets observed is illiquid and present high risks (equation 6; Frey et al., 2002, p. 4):

• dS_t = σS_t – dW_t + pλ (S_t – ) S_t – dα_t+

The similar terminology is employed by Janosi et al. (2002, pp. 7 – 8), who develop the reduced form credit risk model for illiquid markets and express it in the equation that allows considering asset price dynamics and the equilibrium rates at the same time (equation 7):

• B(t, T) = _nΣ_{j = 1} C_tj p (t, t_j)

However, there is literature that supports the somewhat different viewpoint, according to which the asset market always develops according to the trends towards equilibrium of liquidity and prices (Capozza et al., 2003, pp. 4 – 5). Accordingly, Capozza et al. (2001, p. 4) express this equilibrium as follows (equation 8):

• P_t^* = p (X_t),

where P denotes the very equilibrium, while t marks the period of time, during which this equilibrium is observed. At the same time, the dynamics of price changes that is often observed even in equilibrium markets can be seen from equation 9 (Capozza et al., 2003, p. 5):

• ∆ P_t = α∆ P_t-1 + β (P_t-1^* – P_t-1) + γ∆ P_t^*

Schönbucher and Wilmott (2000, p. 235) also pay attention to the topic of market equilibrium but consider it from the point of view of hedging, pricing, and option choices and, similar to Capozza et al. (2001, pp. 4 – 5), they consider the potential for essential changes in pricing policies and trading strategies under the conditions of market illiquidity.

Finally, Chan and Faff (2002) provide one more insight in the topic of asset pricing in illiquid markets using the so-called “Fama and French three-factor asset pricing model” expressed in equation 10 (pp. 7 – 8):

• E (R_i) – R_f = b_i [E (R_m) – R_f] + s_i E (SMB) + h_i (HML) + l_i (IMV)

In the above equation, the three factors that give the name tot he model are returns on the asset, R_i, return on the liquid asset, R_f, and return on market portfolio, R_m (Chan and Faff, 2002, pp. 7 – 8).

Research Gaps

Thus, the above presented review of the relevant literature on the topic of pricing policies observed in the illiquid markets in the contexts of portfolio choices and overall economic conditions reveals that prior scholars have paid much attention to its various aspects. At the same time, the previous research presents considerable gaps, one which the proposed paper will attempt to fill in. In more detail, the previous research, at least on the scope of the articles considered in the presented review, seemingly fails to do two basic things:

1. The previous research on the topic does not provide a specific picture of things; instead it generalizes its findings;
2. The previous research is inconsistent in its approach to the model of studying the asset market and its illiquidity.

Accordingly, the proposed research paper will aim at eliminating these drawbacks of the previous research and filling in the gaps it has left. More specifically, the proposed research will try to provide specific examples to illustrate the theoretical points made. In addition, it is expected that the proposed research will find a single approach to studying the pricing policies in the illiquid markets. Such an approach will allow seeing the actual picture of illiquid asset markets and their implications on real life conditions of asset pricing in various settings. The following data sets and methodological approaches are set to assist the developers of the proposed research with fulfilling the latter’s tasks and answering its questions.

Methodology

Data Requirements

Naturally, the first step in developing the methodological approach to the proposed paper will be the process of data collection. Accordingly, it is necessary to define two major notions:

1. Which data will be needed for the proposed paper?
2. Which data will be retrieved? (Which data will the researcher manage to obtain for the paper?)

On the basis of the proposed paper topic, it becomes obvious that the following sets of data will be necessary:

• Theoretical considerations, i. e. equations and regression models that will help in calculating specific research results;
• Stock market index data;
• Such macroeconomic sets of data as interest rates, inflation rates, and their dynamics;
• Historical data regarding the market liquidity rates dynamics;
• Asset pricing data for specific markets under analysis.

Basically, all the above listed data can be retrieved for the purposes of the proposed research with various degree of effort taken. However, if all these data sets are obtained and properly analyzed, the researcher will be able to answer the research questions and consider the topic to the fullest extent possible. Stock market and macroeconomic data are crucially important for the background of the proposed research, while asset pricing data sets will allow tracing the liquidity rate changes in the analyzed markets (Longstaff, 2009, pp. 1120 – 1121). Finally, historical records regarding market liquidity and illiquidity switching rates, if combined with all other data required, will allow the researcher to implement the further discussed research methodology and especially its basic element, i. e. Markov switching approach (Janosi et al., 2002, p. 5; Garleanu, 2008, pp. 536, 539).

Methodologies to Be Used

Accordingly, the above presented discussion of data required for the proposed paper allows formulating the methodological strategy to be used in the paper. Basically, this strategy will consists of four major elements, including the further review of the relevant literature on the research topic; the descriptive analysis of the retrieved data; the estimation of regression models considered by previous scholars; and the analysis of all research data time series with the help of the Markov switching model, one of the best means to trace the dynamics of economic phenomena like liquidity rates or asset price and regime changes (Garleanu, 2008, pp. 536, 539).

In more detail, the first step in the methodology of the proposed paper will be the further literature review. Consideration of secondary sources of data is expected to provide the sufficient background for further paper development. The next step will be the descriptive analysis of the collected data. This research method is reported to provide a deeper insight into the research topic (Chan and Faff, 2002, pp. 15 – 16) and can be considered as a preliminary data processing before the regression models can be estimated.

Finally, the process of regression model estimation and the analysis of regression time series data with the help of the Markov switching model can be viewed as a single, two-fold method of data collection and analysis. Using the following equations (11 and 12), the researcher is expected to cope with the calculations and come closer to making conclusions regarding the research topic (Garleanu, 2008, p. 540):

• P = 1 / r K₁ (θ_k, Σ_j μ_jkpj), where θ_k = 1 / yσ_D² (m_D / r – P) – ση / σ_D Σ_j μ_jkpj

The briefly discussed methodology is expected to bring the perfect results and help the researcher to study the proposed paper topic in detail. Needless to say, such methodology has its limitations as the chosen methods may prove inapplicable to certain specific situations. If such cases arise, the researcher will modify the methodology in respect of the challenges faced.

Conclusion

So, the whole above presented discussion reveals that the topic of pricing policies observed in the illiquid markets in the contexts of portfolio choices and overall economic conditions is rather relevant in the light of the modern economic conditions observed globally. Naturally, such a significant topic has received considerable attention from scholars, who examined it in various aspects. At the same time, previous research literature on the topic lacks specificity and consistency to some extent, and the proposed paper will try to fill in this research gaps using the discussed data sets and methodological strategies for this purpose.

Reference

Brenner, M. et al. (2001) The Price of Options Illiquidity. The Journal of Finance, 56(2), 789 – 805.

Capozza, D. et al. (2003) An Anatomy of Price Dynamics in Illiquid Markets: Analysis and Evidence from Local Housing Markets. SSRN, 19(8), 1 – 38.

Chan, H. and Faff, R. (2002) Asset Pricing and the Illiquidity Premium. Financial Review, 2, 1 – 34.

Frey, R. et al. (2002) Risk Management for Derivatives in Illiquid Markets: A Simulation Study. Mathematics Subject Classification, 2(13), 1 – 20.

Garleanu, N. (2008) Portfolio choice and pricing in illiquid markets. Journal of Economic Theory ,144, 532–564.

Janosi, T. et al. (2002) Estimating Expected Losses and Liquidity Discounts Implicit in Debt Prices. Journal of Risk, 6(20), 1 – 39.

Longstaff, F. (2009) Portfolio Claustrophobia: Asset Pricing in Markets with Illiquid Assets. American Economic Review, 99(4), 1119 – 1144.

Muzziolli, S. et al. (2001) Implied Trees in Illiquid Markets: a Choquet Pricing Approach. International Journal of Intelligent Economics, 6, 1 – 19.

Schönbucher, P. and Wilmott, P. (2000) The Feedback Effect of Hedging in Illiquid Markets. SIAM Journal on Applied Mathematics, 61(1), 232 – 272.