Pricing of financial instruments. Josechu Fernández Analistas Financieros Internacionales Universidad Autónoma de Madrid

Transcripción

1 Pricing of financial instruments Josechu Fernández Analistas Financieros Internacionales Universidad Autónoma de Madrid

2 Financial acousmatics The sect was divided between the akousmatikoi (hearers) and the mathematikoi (learned) Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

3 L acusmatico Leibnitz: calculating machines, calculus notation (practical approach), Probability of 1 head and 1 tail C and C 2. C and X 3. X and X well, hum! 1/3 Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

4 André Weil 1 1 Trinck signifie beuvez dans L Oracle de la Dive Bouteille de la Princesse Babuc à Panurge pour le guérir de ses angoisses (Rabelais : 5 livre). Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

5 Trend of the day Mathematicians becoming acousmatician: computers, (as laboratory) digital era, (everything is 0 s and 1 s) information (informational science), complex systems Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

6 Akusmatika (well, Matematika) in finance Actuarial science Credit [Banks] Insurance [Insurance companies] Risk management Uncertainty about values and cash flows. Companies must have enough capital to be able to sustain a fair proportion of adverse scenarios. Pricing of financial instruments This aspect is the one that originally attracted the interest for/and of mathematicians. It is the one that demands the most advanced acousmatics: Partial differential equations, Stochastic processes, Numerics, Data mining and Pattern recognition Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

7 Data mining Marketing. Credit scoring. Speculation Systems to detect patterns of market behavior. Neural networks and others. Once the trader or speculator has a view, maths helps to design a procedure that would take the greatest advantage of that view as opposed to the market view. Design a bet, if the view is right..., but if the view is wrong... Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

8 From Pattern recognition, by William Gibson. Fully imagined cultural futures were the luxury of another day, one in which now was of some greater duration. For us, of course, things can change so abruptly, so violently, so profoundly, that futures like our grandparents have insufficient now to stand on. We have no future because our present is too volatile.... We have only risk management. The spinning of the given moment s scenarios. Pattern recognition. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

9 Financial instruments The financial system has generated so-called financial instruments to better serve its purposes. A financial instrument is a contract between two parties, these parties promise to each other certain cash-flows at certain times in the future, and agree to a certain price to be paid by one party to the other at inception. Financial instrument: Uncertain cash flows at future times for a price today. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

10 Stocks and corporate bonds and loans Stock options Treasury Bonds Inflation linked bonds Money market instruments Swaps. Caps. Floors Mutual funds, Pension funds Mortgages, loans,... Insurance policies... Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

11 Old, very old, history. Outsmarting Savoranola: The Medicis and the Pope. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

12 Pricing, what s pricing? Quite conceptually. You have a certain universe of financial instruments, let us denote them by I. These instruments promise certain cash flows (which we should consider as uncertain) at specified dates in the future (the dates themselves could be uncertain). A pricer P is a functional that associates to every instrument in I a real number, its price. The price, by the way, could be a negative number. instrument I I price: P(I) R cash flows at future dates equivalent value today We should think of I as a large vector space. A portfolio being the same as a linear combination of financial instruments, itself a financial instrument. Of course, a pricer has to fulfill certain properties. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

13 1. Actuarial pricing This is the origin. You have a large number N of similar individuals exposed to the same specific risk. As a consequence of this risk they may incur in certain damage. Years of experience tells us that the potential damage (moneywise) for individual i during a single year could be represented by a random variable X i, whose distribution is known and common to all of those similar individuals. Denote this common distribution by X. The total potential loss is then N i=1 X i The potential individuals damages are independent (although at this point this is not relevant). Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

14 Mathematics or... Even the stupidest man by some instinct of nature per se and by no previous instruction (this is truly amazing) knows for sure that the more observations...that are taken, the less the danger will be of straying from the mark. Ars Conjectandi and/or in a letter to Leibnitz. Jacob Bernoulli tell us that quite surely the average is quite close to 1 N N i=1 E (X) X i Suddenly, the random nature of the events has disappeared, if you take the aggregate then it becomes a sure thing. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

15 The conclusion is: if each of the persons involved puts down an amount of E (X), we will collect N E (X) which quite surely and very approximately equals N i=1 X i The amount E (X) is the pure prime. It is the fair price of the contract that pays the random amount X i to participant i. Dealing with the quite surely and quite approximately qualifications above requires mathematics, statistics and business knowledge: actuaries. The pricing rule comes from the fact that the law of large numbers (the repetition of a large number of independent instances) allows to replace uncertainty future cash flows by a certain amount of money today. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

16 Retail credit The context of retail credit is very similar to that of insurance: large number of small credits. From the point of view of the bank or the saving and loans, the interesting event now is default. The law of large numbers is the driving principle again. Lets us depict this with a simple example. Suppose you (the bank) have made 1000 loans of 1000 euros. These loans constitute a homogeneous block (they are quite similar in all respects). Scoring. These are clients with similar financial statements, similar age, similar.... Comparison with your data bases of clients and defaults occurring in the path will tell you that the probability of default of these individuals is, say, 1 %. This process of assigning a probability of default is termed (credit) scoring. It uses techniques like discriminant analysis, neural networks, classification tress and other data mining procedures. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

17 Pagan justos por pecadores. That s life! The bank wants to make a return of, say, 5 % on the loans. It charges an interest of 5 % plus a spread, x %. Large numbers: { 990, will pay back 1000 (1 + 5 % + x %) 10, will pay back...0 Total received, quite certainly, (1 + 5 % + x %) and total desired, (1 + 5 %) Equating both totals reads 10 (1 + 5 %) = 990 x % 990 (random) individuals pay extra to cover for 10 (random) who default. Of course, WE all are always going to be part of the 990. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

18 2. Opinions and subjective probabilities Decisions, pricing, etc., are based on subjective assignation of probabilities of potential events to come. Of course, this assignation is not based purely on crystal ball, but based on substantial analysis, fundamental knowledge, careful scrutiny of the past, econometrics and projections, and, well,... some crystal ball... for the final touch. Comparisons to form opinions and reach decisions based on this approach makes perfect good sense. Prices obtained by calculating expectations based on these probabilities, when there is only one instance of the experiment, is ludicrous, but some how unavoidable. Fundamental analysis no real models economic reasoning (bayesian experience) cause effect, with natural models which are not, most of the time, spelled out Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

19 3. Marked to market pricing. Relative pricing. Fair value Hugh Grant. The Englishman Who Went Up a Hill But Came Down a Mountain Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

20 Anson and Garrad have explained they must go and calculate the height. Thomas Twp Too: And how d you know later? Reginald Anson: Well, w-we ve made, um, we ve made measurements with those two hills, and w-we already know the height of Newton Beacon and Whitchurch Hill... Thomas Twp: But how were they measured? Reginald Anson: The same way, by comparing them with other hills. Thomas Twp Too: But who measured the first hill? Rev. Robert Jones: whispering God. God, my boy. God. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

21 Come back to the conceptual introduction. We have the whole universe I of financial instruments. But besides this there are a number of instruments M 1,M 2,...,M n which constitute a market. These instruments are traded regularly, with enough liquidity, and their prices are representative opinions. The point here is that the prices that are quoted of these basic instruments are real prices in the sense that at any given time you can buy or sell them at the quoted (bid or ask) price. Let us denote the prices of these instruments at a given time (valuation date) by: π 1,π 2,...,π n These instruments constitute THE MARKET. All other financial instruments are OTC (over the counter). You can buy or sell them, but have to find a seller or a buyer, and the price is not a public affair. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

22 Basic instruments whose price is transparent: stocks Treasury bonds interest rate swaps, vanilla options caps, floor, swaptions corporate bonds credit options inflation linked bonds an ever growing list Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

23 We wish to develop a technique of pricing according to the market, i.e. the pricing tool P has to interpolate: P(M j )=π j Portfolios of these basic instruments should be considered as market instruments. To avoid arbitrage we should have: P ( ) α j M j = α j π j j Thus the pricer should be linear. The coefficients could be positive and negative. short selling. The pricer is given in the subspace M of I generated by the instruments M 1,M 2,... and should be extended to the whole universe I. j Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

24 Dr. Hahn and Dr. Banach, would you please keep quiet in the back row.... Have you ever heard about arbitrage? Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

25 Whypricetomarket? If all OTC instruments are marked to market (priced accordingly to the market) you get a fair perception of the real value of the portfolio. A good indication of the price you could get if you were to transform the whole portfolio into cash. In this volatile world we live in, it is better accounting than the price you pay ab initio. It gives a good reference point to start trading. One byproduct of the pricing to market is that it reveals the probability distribution that the market is assuming (in prices). Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

26 Arbitrage The main requirement is that pricer should not create arbitrage. An arbitrage opportunity is a trading strategy which requires no initial investment of money but no matter what happens will end up producing positive (or at least non negative) cash flows, surely. Markets are supposed to be so efficient that they do not allow arbitrage. In fact, there are arbitrageurs who are eager to detect arbitrage opportunities and take advantage. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

27 Examples. The price of a call should be less than the price of the underlying stock. Prices of zero coupon bonds should be decreasing with maturity. Call-Put =Stock - Discounted strike Forward on Euribor should be striked at implicit rate. Price of a bond paying annually Eur1Y should be par. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

28 Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

29 So our pricer should not produce arbitrage. Notice: Plenty of rules, plenty of potential strategies. Arbitrage could be: Static If two instruments X and Y, are going to pay in all scenarios the same amounts, then P(X) =P(Y ) Dynamic If trading dynamically with a self-financing strategy based (initially) on a portfolio of M 1,M 2,...,M n (say with weights λ 1,λ 2,...,λ n ) we are able to produce exactly the same cash-flows as a certain instrument X then P(X) = i λ i π i Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

30 Glossary trading dynamically self-financing strategy exactly the same cash-flows (bounds) Technically, those instruments that can be replicated as above could (and do) become part themselves of replicating strategies. Mathematically, this is not a problem, but in practice it places a certain burden of uncertainty in the technology. Let us denote by A those instruments which can be replicated. The pricer P should extend to A without creating arbitrage. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

31 Introducing scenarios. Taming by naming Imagine that we can codify as Ω={ω 1,ω 2,...,ω n } the collection of scenarios that can occur between now and the... future. Our financial instruments have become vectors (columns) in R n, or, if you prefer, as functions from Ω into R. Lets us point out that it is not coincidence that the number of scenarios is the same as the number of basic instruments of reference. This instance would allow us to present in a simple manner the fundamental ideas. M 1 M 2... M n ω 1 M 1 (ω 1 ) M 2 (ω 1 )... M n (ω 1 ) ω 2 M 1 (ω 2 ) M 2 (ω 2 )... M n (ω 2 ) ω n M 1 (ω n ) M 2 (ω n )... M n (ω n ) Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

32 An appropriate change of basis. The Arrow-Debreu securities: AD j. These are given by AD j (ω i )=δ ij AD 1 AD 2... AD n ω ω ω n Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

33 We have X = j X(ω j )AD j No arbitrage tell us: P(X) = j X(ω j )P(AD j ) But no arbitrage tell us also: P(AD j ) > 0 j P(AD j)= 1 1+r =discount=price of the security that pays 1 in any case. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

34 Enters probability Define Observe q j = P(AD j ) (1 + r) q j =1, q j > 0 j Drums! We have a probability Q in ω. And P(X) = 1 1+r } {{ } discount average payoff { }} { E Q (X) Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

35 Bah!, big deal! 1. But, notice. Surely we can write (changing back basis) each AD j = i u i j M i and therefore (pricing rules, no arbitrage) P(AD j )= i u i j P(M i ) and so q j = (1+r) } {{ } market given u i j i π i }{{} market given What! The pricing probability Q is given by prices quoted in the market. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

36 2. The return R X on any instrument X is given by: R X (ω j )= X(ω j) P(X) P(X) and it turns out that E Q (R X )=r = return of money What! To price you should use a probability that makes investing in any instrument equivalent (on average) to have money in the bank. What!, again The odyssey of Black Scholes. What did they do, actually? Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

37 Two recommendations at this point: Library corner Against the Gods. Peter Bernstein Fortune s formula. William Poundstone Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

38 Enters time Instruments can be traded continuously (quite frequently, at least) and have cash flows at certain points in time, not just a single future time in the future. If X t denotes price/value of instrument X at time t, then the pricing probability should verify: 1 ) X t = (1 + r) (s t)e Q (X s F t (Conditional expectations. F t = all information about prices known at time t. Filtration.) Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

39 In other terms: discounted (to present time) prices, i.e must be martingales under Q 1 (1 + r) t X t and viceversa, if discounted prices are martingale there is no arbitrage So, to price instruments (actually only those in A, but never mind), all you have to do is to find a Q so that discounted prices of (actually only) the basic instrument are martingales. In the basic stylized situation we started with, there was only one Q, but that was special. We just need one. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

40 In practice, things proceed as follows. Q, what Q? 1. First, you are given I. There many different I s. All markets (interest rates, stocks, currencies,...) are interconnected, but you may consider some portions of interest as reasonable independent. 2. There are a number of instruments M 1,M 2,...,M n which you may consider basic. There maybe many other transparent but less relevant instruments. 3. Then you propose a model to generate scenarios: usually this is a stochastic process, with some free parameters. 4. Finally you perform some fine tuning (well, bricolage!, actually) so that the M i s appropriately discounted become martingales. 5. Then, finally price the instruments in A by averaging and discounting. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

41 Black-Scholes You have stock and money as basic instruments. Discounted money is constant, so always a martingale. For the stock, postulate (Bachelier, Mandelbrojt, Samuelson,...) ds t = S t (rdt + σdw t ) where W t is brownian process under a certain probability Q. 250 sample of evolutions; r=10%, sigma=40% 200 level of stock time in years Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

42 The deterministic part (and Ito) makes S t e rt a martingale under Q And that is all, no arbitrage possible (within the model) by trading with stock and money. You may price basic (vanilla) stock options, but these are already priced by the market. To be consistent with the market, you choose σ so that this vanilla instruments are correctly priced. And you may price all sorts of derivatives on the stock. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

43 Ito tells that where N is a standard normal variable. A call C with maturity T and strike K pays The price today is: Fortune s formula S T = S 0 e (r σ2 /2)T e σ T N C T =máx ((S T K); 0) ) C 0 = e rt E Q (máx (S T K; 0)) = e rt E Q (máx (S 0 e (r σ2/2)t e σ T N K;0) Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

44 After some algebra this expectation turns out to be: C 0 = Black-Scholes formula = S 0 Φ(d + ) Ke rt Φ(d ) where Φ is the distribution function of the standard normal variable and ln d ± = ( S0 Ke rt σ T ) ± 1 2 σ2 T Probably, the most quoted mathematical formula in history. C 0 = S 0 Φ ( ln ( S0 Ke rt σ T ) σ2 T ) Ke rt Φ ( ln ( S0 Ke rt σ T ) 1 2 σ2 T ) Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

45 Actuarial/Fundamental plus arbitrage gives Black-Scholes Assume that we all believe in geometric brownian motion. As actuaries or as fundamental analists we postulate: ds t = S t (µdt + σdw t ) We know that call-put must be S 0 Ke rt. This is the no-arbitrage bit. But we price (with µ) as call put = e rt ( E((S T K) + (K S T ) +) = e rt (E(S T K)) = e rt (S 0 e µt K) and it should equal S 0 Ke rt. Therefore: µ = r No matter what you think (µ) you should be risk neutral (µ = r) for pricing. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

46 Models for all seasons. Model risk Model risk. Model are interpolation techniques. If the prices obtained are use for hedging, there is risk (uncertainty) involved. And capital needed to cover that potential downturns Rama Cont Uses and needs. Pricing is used for different purpooses and requires different calibrations and tunings to market instruments: Back office Traders Risk Managers Regulators Ethics Model arbitrage. Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

47 Lotteries. Coins People bet on the outcome of a match. Only two possible outcomes of the match, A or B wins. Every bet (ticket) costs 1 euro. There are a million bets. Somebody offers you a contract that pays 4 if A wins and 3 if B wins. We have the following information a) Historically, A have won 60 % of the time (average payoff=1,2) b) 55 % of the bets for A (average payoff=0.85) c) Insider information God is going to decide the winner of the match by tossing a fair coin (average payoff=0,5) Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de

48 People is willing to pay.6 for the contract which pays 1 if heads, and pays 0 otherwise. People is willing to pay.4 for the contract which pays 1 if tails, and pays 0 otherwise. The coin is perfect, unbiased, every test confirms that probability of heads and tails, or throw twice many times and only allow CX or XC. Price (fair price, market price,... ) of the contract that pays 1 if heads, 0 if tails?...2 Escuela de Finanzas Aplicadas Todos los derechos reservados. Fecha de elaboración: 21 de septiembre de