Law of the Minimal Price

Transcription

1 Law of the Minimal Price Eckhard Platen School of Finance and Economics and Department of Mathematical Sciences University of Technology, Sydney Lit: Platen, E. & Heath, D.: A Benchmark Approach to Quantitative Finance Springer Finance, 700 pp., 199 illus., Hardcover, ISBN (2006). Platen, E.: A benchmark approach to finance. Mathematical Finance 16(1), (2006). Platen, E.: A benchmark approach to asset management. J. Asset Managem. 6(6), (2006).

2 Law of One Price All replicating portfolios of a payoff have the same price! Debreu (1959), Sharpe (1964), Lintner (1965), Merton (1973a, 1973b), Ross (1976), Harrison & Kreps (1979), Cochrane (2001),... will be, in general, violated under the benchmark approach. c Copyright Eckhard Platen 08 Law Minimal Price 1

3 1 0 ln(savings bond) ln(fair zero coupon bond) ln(savings account) time Figure 1: Logarithms of savings bond, fair zero coupon bond and savings account. c Copyright Eckhard Platen 08 Law Minimal Price 2

4 Two Asset Market Example trading times t i = i h, h > 0 risky asset S 1 (S&P500 accumulation index) asset ratio A 1 t i,h = S1 t i +h S 1 t i (0, ) can reach any finite strictly positive value savings account S 0 (US savings account) asset ratio A 0 t i,h = S0 t i +h S 0 t i > 0 c Copyright Eckhard Platen 08 Law Minimal Price 3

5 time Figure 2: Discounted S&P500. c Copyright Eckhard Platen 08 Law Minimal Price 4

6 8 ln(s&p500 accumulation index) ln(savings account) time Figure 3: ln(s&p500 accumulation index) and ln(savings account). c Copyright Eckhard Platen 08 Law Minimal Price 5

7 portfolio S δ strict positivity of S δ π 0 δ,t i = δ 0 t i S 0 t i S δ t i [0, 1] A δ t i,h = Sδ t i +h S δ t i = π 0 δ,t i A 0 t i,h + ( 1 π 0 δ,t i ) A 1 t i,h find best performing strictly positive portfolio c Copyright Eckhard Platen 08 Law Minimal Price 6

8 expected growth g δ t i,h = E t i (ln ( )) A δ t i,h g δ t i,h π 0 δ,t i = E ti ( A 0 ti,h A1 t i,h A δ t i,h ) second derivative negative = one genuine maximum at π 0 δ,t i (, ) for g δ t i,h π 0 δ,t i = 0 c Copyright Eckhard Platen 08 Law Minimal Price 7

9 Three cases: (i) π 0 δ,t i [0, 1] classical case (ii) π 0 δ,t i < 0 savings account performs poorly, risk premium = index is best performing portfolio (iii) π 0 δ,t i > 1 index performs poorly = savings account is best performing portfolio c Copyright Eckhard Platen 08 Law Minimal Price 8

10 classical case (i): π 0 δ,t i [0, 1] = E ti ( A 1 ti,h A δ t i,h ) = E ti ( A 0 ti,h A δ t i,h ) = 1 = genuine maximum g δ t i,h = gδ t i,h S 0 t i and S 1 t i are constituents of S δ t i S 0 t i S δ ti and S 1 t i S δ ti are martingales = fair = all benchmarked portfolios are fair = Law of One Price holds c Copyright Eckhard Platen 08 Law Minimal Price 9

11 other cases rewrite first order condition g δ t i,h π 0 δ,t i = E ti ( Q ti,h π 0 δ,t i (Q ti,h) π 0 δ,t i Q ti,h ) = 0 with S 0 t+h Q t,h = S 1 t+h S 0 t S 1 t 1 = 1 π 0 δ,t = lim h 0 h E t(q t,h ) ( 1 lim h 0 h E (Qt,h ) 2 t 1+π 0 δ,t Q t,h ) c Copyright Eckhard Platen 08 Law Minimal Price 10

12 time Figure 4: US-benchmarked savings account. c Copyright Eckhard Platen 08 Law Minimal Price 11

13 annualized returns of benchmarked savings account n = 1052 ˆµ = 1 n 1 n + 1 h Q t i,h ˆσ = 1 n i=0 n i=0 1 h (Q t i,h) negative fraction π 0 δ,t = ˆµˆσ = savings account unlikely to be fair c Copyright Eckhard Platen 08 Law Minimal Price 12

14 Extreme Maturity Bond need model with downward trending to reflect reality S0 t S δ t assume short rate deterministic savings bond P (t, T) = S0 t S 0 T select index S 1 as numeraire portfolio S δ c Copyright Eckhard Platen 08 Law Minimal Price 13

15 benchmarked fair zero coupon bond ˆP(t, T) = P(t, T) S δ t = E t ( 1 S δ T ) martingale = real world pricing formula ( 1 ) P(t, T) = S δ t E t S δ T c Copyright Eckhard Platen 08 Law Minimal Price 14

16 discounted numeraire portfolio S δ t = Sδ t St 0 for continuous market generally satisfies SDE d S δ t = α t dt + S δ t α t dw t is time transformed squared Bessel process c Copyright Eckhard Platen 08 Law Minimal Price 15

17 model the drift of discounted index as α t = α η exp{η t} = Minimal Market Model MMM, see Pl. & Heath (2006) net growth rate η with R 2 of 0.88 c Copyright Eckhard Platen 08 Law Minimal Price 16

18 5 4.5 log discounted index trend line time Figure 5: Logarithm of discounted index. c Copyright Eckhard Platen 08 Law Minimal Price 17

19 normalized index Y t = S δ t α t dy t = (1 η Y t ) dt + Y t dw t quadratic variation of Y t d Y t = dw t V t,h = i t l=1 ( Ytl Y tl 1 ) 2 [ Y ] t = t 4 scaling parameter α with R 2 of c Copyright Eckhard Platen 08 Law Minimal Price 18

20 16 14 [sqrt(y)] trend line time Figure 6: Quadratic Variation of Y t. c Copyright Eckhard Platen 08 Law Minimal Price 19

21 fair zero coupon bond (MMM) ( { }) P(t, T) = P 2 η S δ t (t, T) 1 exp α(exp{η T} exp{η t}) initial prices in 1920: P (0, T) = P(0, T) = P(0,T) P (0,T) < = 3.12% c Copyright Eckhard Platen 08 Law Minimal Price 20

22 0.12 benchmarked savings bond benchmarked fair zero coupon bond time Figure 7: Benchmarked savings bond and benchmarked fair zero coupon bond. c Copyright Eckhard Platen 08 Law Minimal Price 21

23 1 0.9 savings bond fair zero coupon bond savings account time Figure 8: Savings bond, fair zero coupon bond and savings account. c Copyright Eckhard Platen 08 Law Minimal Price 22

24 self-financing hedge portfolio hedge ratio δ t = P(t, T) S δ t = P (t, T) exp { 2 η S δ t α(exp{η T} exp{η t}) } 2 η α(exp{η T} exp{η t}) c Copyright Eckhard Platen 08 Law Minimal Price 23

25 1 0 ln(zero coupon bond) Self-financing hedge portfolio time Figure 9: Logarithm of zero coupon bond and self-financing hedge portfolio. c Copyright Eckhard Platen 08 Law Minimal Price 24

26 e-005 6e-005 4e-005 2e e-005-4e time Figure 10: Benchmarked P&L. c Copyright Eckhard Platen 08 Law Minimal Price 25

27 time Figure 11: Fraction invested in the index. c Copyright Eckhard Platen 08 Law Minimal Price 26

28 Financial Market jth primary security account S j t j {0, 1,..., d} savings account S 0 t t 0 c Copyright Eckhard Platen 08 Law Minimal Price 27

29 strategy δ = {δ t = (δ 0 t, δ1 t,..., δd t ), t 0} predictable portfolio S δ t = d j=0 δ j t S j t self-financing ds δ t = d j=0 δ j t ds j t c Copyright Eckhard Platen 08 Law Minimal Price 28

30 Numeraire Portfolio Definition 1 S δ V + x E t numeraire portfolio if S δ t+h S δ t+h S δ t S δ t 1 for all nonnegative S δ and t, h [0, ). 0 S δ best performing portfolio Long (1990), Becherer (2001), Pl. (2002, 2006), Bühlmann & Pl. (2003), Goll & Kallsen (2003), Karatzas & Kardaras (2007) c Copyright Eckhard Platen 08 Law Minimal Price 29

31 Main Assumption of the Benchmark Approach Assumption 2 There exists a numeraire portfolio S δ V + x. c Copyright Eckhard Platen 08 Law Minimal Price 30

32 Supermartingale Property benchmarked value Ŝ δ t = Sδ t S δ t Corollary 3 For nonnegative S δ Ŝ δ t E t ) (Ŝδ s 0 t s < nonnegative Ŝ δ supermartingale c Copyright Eckhard Platen 08 Law Minimal Price 31

33 Definition 4 Price is fair if, when benchmarked, forms martingale. Ŝ δ t = E t ) (Ŝδ s for 0 t s <. c Copyright Eckhard Platen 08 Law Minimal Price 32

34 Lemma 5 The minimal nonnegative supermartingale that reaches a given benchmarked contingent claim is a martingale. see Pl. & Heath (2006) c Copyright Eckhard Platen 08 Law Minimal Price 33

35 Law of the Minimal Price Theorem 6 If a fair portfolio replicates a nonnegative payoff, then this represents the minimal replicating portfolio. least expensive minimal hedge economically correct price in a competitive market c Copyright Eckhard Platen 08 Law Minimal Price 34

36 0.12 benchmarked savings bond benchmarked fair zero coupon bond time Figure 12: Benchmarked savings bond and benchmarked fair zero coupon bond. c Copyright Eckhard Platen 08 Law Minimal Price 35

37 claim H T E 0 ( HT S δ T ) < Corollary 7 Minimal price for replicable H T is given by real world pricing formula ( HT ) S δ H t = S δ t E t S δ T. c Copyright Eckhard Platen 08 Law Minimal Price 36

38 normalized benchmarked savings account Λ T = Ŝ0 T Ŝ = Λ 0 E 0 (Λ T ) real world pricing formula = = similar for any numeraire S δ H 0 = E 0 ( Λ T S 0 0 S 0 T S δ H 0 H T ) ) S E 0 (Λ 0 0 T H ST 0 T E 0 (Λ T ) c Copyright Eckhard Platen 08 Law Minimal Price 37

39 time Figure 13: Candidate Radon-Nikodym derivative of hypothetical risk neutral measure of real market. c Copyright Eckhard Platen 08 Law Minimal Price 38

40 special case when savings account is fair: = Λ T = dq dp forms martingale; E 0(Λ T ) = 1; equivalent risk neutral probability measure Q exists; Bayes formula = risk neutral pricing formula S δ H 0 = E Q 0 ( S 0 0 S 0 T H T ) Harrison & Kreps (1979), Ingersoll (1987), Constatinides (1992), Duffie (2001), Cochrane (2001),... otherwise risk neutral price real world price c Copyright Eckhard Platen 08 Law Minimal Price 39

41 long term growth rate g δ = lim sup t ( ) 1 S δ t ln t S0 δ Theorem 8 For S δ V + x g δ g δ. pathwise best in the long run Karatzas & Shreve (1998), Pl. (2004), Karatzas & Kardaras (2007) c Copyright Eckhard Platen 08 Law Minimal Price 40

42 Pl. (2004) Definition 9 if Nonnegative portfolio S δ outperforms systematically S δ (i) S0 δ = S δ 0 ; ) (ii) P (S δt S δt ) (iii) P (S δt > S δt = 1 > 0. relative arbitrage Fernholz & Karatzas (2005) c Copyright Eckhard Platen 08 Law Minimal Price 41

43 1 0 ln(fair zero coupon bond) ln(savings account) time Figure 14: Logarithms of fair zero coupon bond and savings account. c Copyright Eckhard Platen 08 Law Minimal Price 42

44 Theorem 10 Numeraire portfolio cannot be outperformed systematically. c Copyright Eckhard Platen 08 Law Minimal Price 43

45 portfolio ratio A δ t,h = Sδ t+h S δ t expected growth g δ t,h = E t ( ( )) ln A δ t,h t, h 0 c Copyright Eckhard Platen 08 Law Minimal Price 44

46 derivative of expected growth S δ V + x S δ S δ ε nonnegative perturbed S δ A δ ε t,h = ε Aδ t,h + (1 ε) Aδ t,h for t, h 0, ε > 0 g δ ε t,h ε = lim ε=0 ε ε 0 1 ( ) g δ ε t,h gδ t,h c Copyright Eckhard Platen 08 Law Minimal Price 45

47 Definition 11 S δ growth optimal if g δ ε t,h ε 0 ε=0 for all t, h 0 and nonnegative S δ. alternative definition to expected log-utility Kelly (1956) Hakansson (1971) Merton (1973a) Roll (1973) Markowitz (1976) Theorem 12 The numeraire portfolio is growth optimal. c Copyright Eckhard Platen 08 Law Minimal Price 46

48 Strong Arbitrage market participants can only exploit arbitrage limited liability = nonnegative total wealth of each market participant Definition 13 A nonnegative S δ is a strong arbitrage if S δ 0 = 0 and P ( S δ t > 0) > 0. Pl. (2002)-mathematical arguments Loewenstein & Willard (2000)-economic arguments c Copyright Eckhard Platen 08 Law Minimal Price 47

49 Theorem 14 There is no strong arbitrage. c Copyright Eckhard Platen 08 Law Minimal Price 48

50 = there is no pricing based on strong arbitrage Delbaen & Schachermayer (1998) free lunches with vanishing risk (FLVR) may exist Loewenstein & Willard (2000) free snacks & cheap thrills may exist c Copyright Eckhard Platen 08 Law Minimal Price 49

51 time Figure 15: P(t, T) minus savings account. c Copyright Eckhard Platen 08 Law Minimal Price 50

52 time Figure 16: P(t, T) minus savings account. c Copyright Eckhard Platen 08 Law Minimal Price 51

53 Under BA candidate risk neutral Q may not be equivalent to P since its Radon-Nikodym derivative may be a strict supermartingale. Under BA existence of equivalent risk neutral probability measure is more a mathematical convenience than an economic necessity. c Copyright Eckhard Platen 08 Law Minimal Price 52

54 References Becherer, D. (2001). The numeraire portfolio for unbounded semimartingales. Finance Stoch. 5, Bühlmann, H. & E. Platen (2003). A discrete time benchmark approach for insurance and finance. ASTIN Bulletin 33(2), Cochrane, J. H. (2001). Asset Pricing. Princeton University Press. Constatinides, G. M. (1992). A theory of the nominal structure of interest rates. Rev. Financial Studies 5, Debreu, G. (1959). Theory of Value. Wiley, New York. Delbaen, F. & W. Schachermayer (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, Duffie, D. (2001). Dynamic Asset Pricing Theory (3rd ed.). Princeton, University Press. Fernholz, E. R. & I. Karatzas (2005). Relative arbitrage in volatility-stabilized markets. Annals of Finance 1(2), Goll, T. & J. Kallsen (2003). A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13(2), Hakansson, N. H. (1971). Capital growth and the mean-variance approach to portfolio selection. J. Financial and Quantitative Analysis 6(1), Harrison, J. M. & D. M. Kreps (1979). Martingale and arbitrage in multiperiod securities markets. J. Economic Theory 20, Ingersoll, J. E. (1987). Theory of Financial Decision Making. Studies in Financial Economics. Rowman and Littlefield. Karatzas, I. & C. Kardaras (2007). The numeraire portfolio in semimartingale financial models. Finance Stoch. 11(4), Karatzas, I. & S. E. Shreve (1998). Methods of Mathematical Finance, Volume 39 of Appl. Math. Springer. Kelly, J. R. (1956). A new interpretation of information rate. Bell Syst. Techn. J. 35, Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econom. Statist. 47, Loewenstein, M. & G. A. Willard (2000). Local martingales, arbitrage, and viability: Free snacks and cheap thrills. Econometric Theory 16(1), Long, J. B. (1990). The numeraire portfolio. J. Financial Economics 26, Markowitz, H. (1976). Investment for the long run: New evidence for an old rule. J. Finance XXXI(5), Merton, R. C. (1973a). An intertemporal capital asset pricing model. Econometrica 41, c Copyright Eckhard Platen 08 Law Minimal Price 53

55 Merton, R. C. (1973b). Theory of rational option pricing. Bell J. Econ. Management Sci. 4, Platen, E. (2002). Arbitrage in continuous complete markets. Adv. in Appl. Probab. 34(3), Platen, E. (2004). A benchmark framework for risk management. In Stochastic Processes and Applications to Mathematical Finance, pp Proceedings of the Ritsumeikan Intern. Symposium: World Scientific. Platen, E. (2006). A benchmark approach to finance. Math. Finance 16(1), Platen, E. & D. Heath (2006). A Benchmark Approach to Quantitative Finance. Springer Finance. Springer. Roll, R. (1973). Evidence on the Growth-Optimum model. J. Finance 28(3), Ross, S. A. (1976). The arbitrage theory of capital asset pricing. J. Economic Theory 13, Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. J. Finance 19, c Copyright Eckhard Platen 08 Law Minimal Price 54