Introduction - The subject of mathematical finance can both be better understood, and related to the way the economy, and especially markets operate, with an appreciation of the economic issues involved. This seminar, as an optional session will explain, from a pure economics and finance viewpoint what markets are, how they operate, and how that relates to some of the criteria upon which the models in the course are either built or dependent. Economics is a broad field, and some background in the subject not merely helps with the stochastic modelling course, but also to see where the methods developed in the course can be applied in areas other than finance. The concepts of utility, markets and prices pervade the whole of economics and finance. They represent the fundamental mechanisms of an economy, whatever the political context, and you can indeed view the world as one single economy, trading and financial system. 1
Values ------ When studying the workings of an economic (or financial) system, literally everything which might have the slightest impact on how that system behaves should be quantified. In economics, and especially finance, quantifying means attaching values, to the agent, or player, involved be it an individual, an institution or a company (a profit seeking business) of some good, or action, its utility. Most often, utility features decreasing marginal rate of growth, that is, the first unit always counts (is worth) more than the next and so on. This explains the "Risk Aversion" of most agents, because a garanteed quantity of goods is preferred to a random outcome with the same expected level. The utility of (known) expected quantities of goods is at least as great as the utility of random amounts of goods coming on average to the same expected quantity. Value therefore how much that agent prepared to pay to get the good, or be paid to part with it. This idea extends to sometimes unexpected concepts : -leisure, -sleep, -rubbish (you have to pay to have it taken away, or spend time getting rid of it otherwise), -Image, or prestige, -opportunity (say being able to eat food now, not later). 2
Opportunities -- Opportunity costs in fact represent a key principle. Not merely possession of something is valuable, but the possibility to use it. Thus, borrowing, whatever, be it some object, money, a book at the library, should involve a compensation to the owner. During the loan, the latter cannot enjoy the benefit of that property. Lending cash to a friend tonight deprives you of the chance to buy a pint. Your loan will get repaid tomorrow, but you should claim some reward for the lost opportunity to spend the cash yourself. In money terms, opportunity costs often take the form of interest, but not necessarily. Readers may pay a subscription to a library, for the priviledge of borrowing (or a University fee covering that). Drivers pay an insurance premium, regardless of claiming or not. At expiry, the agent (borrower) owns nothing. Investment and borrowing allow an agent to adjust his pattern of temporal consumption. Invest (lend assets) to consume them later, or borrow to consume now, paying later (with interest added as the lender effectively sold the opportunity). Because using an opportunity equates to carrying out some economic activity, the level of interest rates controls the financial benefit derived from activity, compared to lending the assets instead. Thus, interest rates, together with the quantity of cash floating in the system dictate the overall economic activity, the economy s Output. 3
Markets (1) - Ancestral - An efficient arena where to trade, that is exchange goods, or reallocate them. Trade is so easily found beneficial to all parties that it needed no economic theorists to develop. Markets allow not just the optimum allocation of existing goods (endowments) but also dissociating production from consumption. All parties (agents) can obtain a bigger basket of consumption goods by trading off what they are best at making and getting what costs them proportionally more to produce themselves. - Agents (or Players) - Buyers (possibly companies). - Sellers (again, may be companies, also possibly governments). - Speculators (after quick profit, and it is their quest which makes markets more efficient). - Prices - Mutual agreement between buyer(s) and seller(s), on amounts changing hands (ownership) when a trade settled. - Need not be monetary! - Market clearing forces marginal (utility) price. - No memory - Current prices reflect all known information, as otherwise, arbitrage opportunities arise. 4
Markets (2) - In efficient markets - Goods of consistent quality and price. - All agents have access to same information. - All agents who wish to trade can (market clears). - Trading incurs no costs (approximately valid when lots of agents, representing large a turnover of trades). - Trading may be in any quantities, at any moment, either buying (getting "long"), or selling ("shorting", even borrowed goods not at the time owned by the agent). This property is called Liquidity. - No arbitrage. Above forces that! 5
Markets (3) - Deviations from Optimality in Markets - Costs of Exchange - Communication of Prices, Bids, and Offers for goods, effectively information costs. - Contract negotiations, agreeing prices, but also payment terms, delivery schedules, and warrantees. - Monitoring and enforcing contracts. Agents must get what was paid for. - Policing and punishing fraud. - Transaction Costs - Reduce volume of trade, and thus the Consumer surplus, while also effecting transfers to those collecting the costs, quite often the state, as tax(es). - Proportional costs lead to lesser quantities of goods changing hands. Lump sum costs push agents to only trade large lots of goods at a time. - Inventories - Stocks of goods to cover market closure, or gaps between less frequent trades when Lump Sum transaction costs (taxes). Idle goods entail warehousing costs, and monetary value not earning return. - Liquidity limits - When an agent bids for or offers a big part of a good s market wide endowment, prices will move up or down. This also happens with thinly traded goods. 6
Markets (4) - Money - Convenient standard unit (numeraire) for pricing. - Very concentrated store of value. - Provides flexibility by serving as intermediary stage in trades. Without it, in barter situations, exchange of end goods far more complicated (and thus expensive). - Modern money is fiat money. - Interest is the opportunity cost of money, and therefore controls overall economic activity. - Interest rates exhibit a Term Structure, partly shaped by applying opportunity costing. - Money remains a good, subject to the laws of supply and demand. Seignorage is the right to issue money. Inflation can result, when too much money is made to chase the goods in the economy. Deflation, the inverse phenomenon can lock itself into a spiral. - Because of above, nominal and real interest rates apply as lenders ask compensation against both changes in prices, and (temporarily) lost opportunities. - Loans - Multiply the assets an agent s endowment alone would purchase. Possible gains likewise factored up, but also loss from an unfortunate outcome. Called Leverage in business terms. For share holders, loans raise capital without dilution of ownership. - Risk relates to the possible outcome of economic and financial decisions, while credit ratings express the likelihood a borrower would default on repaying a loan. 7
Markets (5) - "Primary" Assets - Simple unconditional contracts or goods. - Tangible (physical items), or intangible, like services (air tickets), financial assets, or access rights (memberships). - Cash, including foreign. - Bonds - Government (in UK, guilts), - Corporate, - Individual (like car loans, mortgages). - May Pay Interest (coupons) at set times. - Worth Face value at emission and expiry, but otherwise priced by arbitrage. - Shares - Claims on part of a company s assets, and its earnings (return on capital). - Pricing - By book value, - Earnings relationships, say price to earnings or dividend ratio. - Brand and other aspects. - May pay Dividends, at discretion of issuer (company). - Futures - Advance Contracts for delivery, of set amounts, at set time and price. - Called Forwards when (off Market) private deal between parties, and thus not a publicly traded asset. 8
Markets (6) - "Derivative" Assets - Contracts based on "if-then" clauses, thus "contingent" claims, often involving a conditional right to effect some action, usually on some primary asset. - Options - Claims on other assets, depending on price movements of these assets. - "Vanilla" options, only involve price of underlying asset, when exercised or expiring, - "Exotic" options, can involve shape of price movements, or many underlying assets. - Swaps - Exchanges of revenue streams, often Interest Payments, possibly with different modalities, and sometimes in different currencies. - Other - Insurance, bets, lottery tickets. - Uses - Protection of investment (insurance), - Leverage, to multiply effects of base asset price movements, speculation. - Pricing - Principle of fair bet, risk neutrality. 9
Literature ---------- Vast numbers of books on economics, or just finance have been written. In the context of the course, it s appropriate to pick texts going into enough detail, and into the techniques used in modelling markets. The book by Hull on the reading List is excellent, and also gets into some of the more mathematical aspects considered over the course. The accompanying solutions book to the problems, helps when finding it hard to solve real market pricing problems from the generic equations. Options, Futures and Other Derivatives, J. Hull, Prentice Hall, 5th Edition, 2002. J. Hull also wrote a less mathematically and more market orientated title, which comes as a nice complement, very good to get a better understanding on how (actual) markets operate. Introduction to Futures and Options Markets, J. Hull, Prentice Hall, 2nd Edition, 1995, ISBN 0-13-183252-2. In that same line, the Economist (newspaper) published, a set of short briefs well worth a read. It also provides good regular cover, as well as periodic in depth surveys of finance and banking. Economics : Ten Modern Classics, The Economist Newspaper, 1991. For a purely economics and finance based approach, textbook I used myself, and whose money and fiscality focus I much appreciated : Money, Banking and the Economy, T. Mayer, J.S. Duesenberry, R.Z. Aliber, W.W. Norton and Company, 5th Edition, 1993, ISBN 0-393-96300-4. Finally, for a gentle introduction to mathematical finance : An Introduction to Mathematical Finance, S.M. Ross, Cambridge University Press, 1999, ISBN 0-521-77043-2, 10