VOLATILITY, EQUILIBRIUM AND INCOMPLETE MARKETS

9.3 VOLATILITY, EQUILIBRIUM AND INCOMPLETE MARKETS

Volatility, and in particular stochastic volatility, is an increasingly important issue dealt with by financial managers. The Financial Times for example, reported in 1997 (although it could be any year):

The New York Stock Exchange (NYSE) has been swinging far more wildly than in previous years. The events of 1997 that struck stock markets throughout Asia and subsequently in Europe and in the US are additional proof that volatility (stochastic) is becoming a determinant factor of stock values. This has an important effect on investments and investing behaviour. Some investors, for example, are ‘tiptoeing’ away from the stock market and sitting on cash rather than stocks. Others are lulled by the swings in stock values and as a result are becoming less sensitive to these variations (which may be a costly strategy to follow if the stock market were to decline significantly). This year for example, there were daily price drops of more than 3 %,

a phenomenon which in years past would have attracted a great deal of attention and warnings. Past experience has also indicated that when the volatility increases, it may signal a downturn on the stock market (although, it has also signalled upturns on the stock market – but less often). In any case, a growth of volatility makes investors rethink their strategy and thereby, to change their portfolio holdings.

Stochastic volatility is often used as a proof that markets are incomplete (since the former implies the latter). In other words, it implies an underlying departure from conventional approaches to economics and finance that invalidates risk- neutral pricing. Incompleteness, thus, reflects our inability to explain uniquely prices’ formation. Ever since the Second World War change has been plentiful, providing an opportunity to explain why volatility may have grown or changed. Some factors contributing to an appreciable change in economics and finance theories that seek to explain the behaviour of financial markets include among others:

r The demise of Bretton Woods. r The liberalization of the financial sector worldwide. r Globalization through the growth of multinational firms, cross-boundary cap-

ital flows etc. r The growth of derivatives and related products that have enriched financial

theories and financial markets but at the same time have allowed the use of financial products on an unprecedented scale.

Explicitly, economic theory has changed! Classical equilibrium precepts, coined by the Arrow–Debreu–Mackenzie studies have diverted attention to dis- equilibrium theories, information asymmetries, organization and the effects of contracts on economic behaviour. Economic and finance theories have recog- nized these changes that led to new approaches – both theoretical and practical and underlie to a large extent fundamental finance. The assumption of the ratio- nal expectations hypothesis that markets clear (i.e. decision and expectations are compatible both in current and derivatives markets in the present and the future), the assumption that decision makers are homogeneous, self-interested, rational

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

and informed, with common knowledge of the market statistics came in some cases to be doubtful. As a result, the study of incompleteness and situations in- volving bounded rationality, information asymmetry, utility maximizing decision makers etc. have also become important elements to reckon with in devising a mechanism for the valuation and pricing of assets.

Although financial economics has greatly contributed to finance practice, both through its approach to valuation by risk-neutral pricing and in a better under- standing of financial market mechanisms, there are some problems to be reck- oned with. First, financial theory is based on assumptions that are not always right. In this case, we ought to develop other theories to compensate for the- oretical imperfections. For these reasons, making sure that financial theory as- sumptions are validated is essential for making money using ‘complete market models’.

9.3.1 Incomplete markets

Markets are incomplete when any random cash flow cannot be generated by some portfolio strategy. The market is then deemed ‘not rich enough’. Technically, this means that the number of assets that make up a portfolio is smaller than the number of market risk sources plus one, or:

number of assets ≤ Number of risk sources + 1

When this is not the case, we cannot replicate, for example, an option’s implied cash flow and thus, are unable to value the option uniquely. For this, as well as other reasons, incompleteness, implying non-uniqueness in pricing, is particularly important. Non-uniqueness can arise for many reasons, however, including for example issues:

r Due to pricing, rationality and psychology. r Due to information asymmetries and networking. r Due to transaction costs. r Due to stochastic volatility.

If markets are not complete or close to it, financial markets have problems to value assets and investments. Some cases are well studied, however (transaction costs for certain types of assets, some problems associated with stochastic volatility), where one uses additional sources of information to replicate a derived finan- cial asset. Financial markets may be perceived as too risky, perhaps ‘chaotic’, and therefore profits may be too volatile; the risk premium would then be too high and investment horizons smaller, thereby reducing investments. Finally, contingent claims may have an infinite number of prices (or equivalently an infinite number of martingale measures). As a result, valuation becomes forcibly utility-based, which is ‘subjective’ rather than based on the market mechanism. In these circum- stances, the SDF (stochastic discount factor) framework presented in Chapter 3 is

277 particularly useful, providing an empirical approach to pricing (risk-discounting)

VOLATILITY , EQUILIBRIUM AND INCOMPLETE MARKETS

financial assets.

Example: Sources of incompleteness

(1) Incompleteness can arise in many circumstances. Below, a few are summa- rized briefly: r Because of lack of liquidity (leading to market-makers’ and bid/ask spreads – for which trading micro-models are constructed). r Because of excessive friction defined in terms of: taxes; indivisibility of assets; varying rates for lending and borrowing, such as no short sales and various portfolio constraints.

r Because of transaction costs leading to ‘friction’ in market transactions. r Because of insiders trading introducing a risk originating in information

asymmetries and leading thereby to assets mis-pricing. (2) Arbitrage: The existence of arbitrage opportunities implies nonviable mar- kets rendering the unique determination of contingent claim prices impos- sible. If there is arbitrage, there will be trade only out of equilibrium and thus the fundamental theory of finance will not be again applicable and risk-neutral pricing cannot be applied.

(3) Network and information asymmetries: Networks of hedge funds, commu- nicating with each other and often coordinated explicitly and implicitly and herding into speculative activities can lead to market inefficiencies, thus contradicting a basic hypothesis in finance assuming that agents are price takers. In networks, the information exchange provides a potential for in- formation asymmetries or at least delays in information. In this sense, the existence of networks in their broadest and weakest form may also cause market incompleteness.

(4) Pricing and classical contract theory: Transaction costs, informational asymmetries in the Arrow–Debreu paradigm, lead to significant amend- ments of classical analysis. For example, analysis of competition in the presence of moral hazard and adverse selection lead to stressing substantial differences between trade on ‘contracts’ and trade on contingent commodi- ties. The profit associated with the sale of one unit of a (contingent) good depends then only on its price. Further, the profitability of the sale of one contract may also depend on the identity of the buyer. Identity matters, either because the buyer has bought other contracts (the exclusivity problem) or because profitability of the sales depends on the buyer’s characteristics, also known as the screening problem. These issues relate to financial intermedia- tion too, where special attention must be given to the effects of informational asymmetries to better understand prices and how they differ from the ‘social values of commodities’.

(5) Psychology and rationality: The Financial Times has pointed out that some investment funds seek to capitalize on human frailties to make money. For example: Are financial managers human? Are they always rational, mim- icking Star Trek’s Mr Spock? Are they devoid of emotions and irrationality?

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

Psychological decision-making processes integrated in economic rationales have raised serious concerns regarding the rationality axioms of DM pro- cesses, as was discussed in Chapters 2 and 3. There are, of course, many challenges to reckon with in understanding human behaviour. Some of these include:

r Thought processes based on decision-making approaches focusing on the big picture versus compartmentalization.

r The effects of under- and over-confidence on decision making. r The application of heuristics of various sorts applied to trading.

These psychological aspects underpin an important trend in finance called ‘Be- havioral Finance’ and at the same they provide and presume important sources of incompleteness, stimulating research to bridge observed and normative economic behaviour.