A Martingale System Theorem for Stock Investments

Transcription

1 A Martingale System Theorem for Stock Investments Robert J. Vanderbei April 26, 1999

2 DIMACS New Market Models Workshop 1 Beginning Middle End Controversial Remarks Outline

3 DIMACS New Market Models Workshop 2 The Beginning

4 DIMACS New Market Models Workshop 3 Dollar Cost Averaging (DCA) Invest a fixed dollar amount each month. End up buying relatively more when price is low than when high. Therefore, make money even if market is flat. Example. Invest $1260 each month month $ / share shares Sell in month $5/share = $10,770 Total invested (7 x $1260) = 8,820 Gain = $1,950

5 DIMACS New Market Models Workshop 4 a = $ s invested each month X k = share price in month k DCA Theorem n 1 Y n = X n k=0 a X k an = net gain up to month n Theorem. If X k, k = 0, 1,..., is a simple random walk conditioned to return to its starting point x 0 at time n (and n < 2x 0 ), then EY n > 0. Proof. Follows from the reflection principle and the harmonic/arithmetic mean inequality.

6 DIMACS New Market Models Workshop 5 Comment Conditioning the process to return to its starting point at the end is unrealistic. If we knew that, then we should buy only when the price is lower than the initial price. Such a strategy would dominate DCA.

7 DIMACS New Market Models Workshop 6 The Middle

8 DIMACS New Market Models Workshop 7 A Martingale Framework Fk, k = 0, 1,..., a filtration on (, F, P) X k, k = 0, 1,..., a positive martingale wrt Fk A k, k = 0, 1,..., a nonnegative process adapted to Fk Y n = n 1 k=0 A k ( Xn X k 1 ) = net gain up to month n from investment stream A k. Theorem. Y n, n = 0, 1,..., is a mean-zero martingale. Proof. Note that Y n+1 Y n = n k=0 A k X k (X n+1 X n ) Take conditional expectations and note adaptedness of A k / X k : E [ ] n Y n+1 Y n Fn = k=0 A k E [ ] X n+1 X n Fn = 0 X k

9 DIMACS New Market Models Workshop 8 The quadratic variation of X is X k = Quadratic Variation k 1 j=0 ( [ ] ) E X 2 j+1 j X 2 j F Theorem. The quadratic variation of the gain process, Y, is given by Y n = n 1 j=0 ( X j+1 X j ) S 2 j, where S j = j k=0 A k X k.

10 DIMACS New Market Models Workshop 9 Quadratic Variation Proof Proof. From the previous proof and the definition of process S, we have Y n+1 Y n = (X n+1 X n ) S n. Since the stock price and gain processes, X and Y, are martingales, we see that [ ] [ ] E Yn+1 2 n Yn 2 = E (Y n+1 Y n ) 2 Fn [ F ] = E (X n+1 X n ) 2 Fn Sn 2 ( [ ] ) = E Xn+1 2 n Xn 2 Sn 2 F From the definition of quadratic variation, we see that Summing both sides completes the proof. Y n+1 Y n = ( X n+1 X n ) S 2 n.

11 DIMACS New Market Models Workshop 10 A Stochastic Calculus Approach A fundamental result in stochastic calculus is that the stochastic integral of a predictable process against a martingale is again a martingale. The discrete time analogue is as follows: Theorem. If X is a martingale and B is an adapted process, then is a martingale. Z n = n 1 k=0 B k (X k+1 X k ) The fact that the gain process Y is a martingale follows from this theorem by taking B k = k j=0 A j X j.

12 DIMACS New Market Models Workshop 11 Discounting Trends Suppose now that X is an arbitrary positive adapted stochastic process. Put X k = R k X k, where The process R is predictable: R k = k 1 j=0 X j E [ ]. X j+1 j F R k Fk 1, k = 0, 1,... Theorem. The process X is a martingale. Proof. [ ] E X k+1 Fk = R k+1 E [ ] X k+1 Fk = R k X k = X k.

13 DIMACS New Market Models Workshop 12 Following the Trend Convert all dollar amounts to discounted (time 0) values. The gain process is measured in discounted terms: where Y n = R n X n S n = = n k=0 n k=0 n R k A k k=0 ( ) A k R n X n R k A k X k ) Ã k ( X n X k 1, Ã k = R k A k represents the initial value of the input at time k. Our earlier theorem now says that DCA simply tracks the trend in the stock price.

14 DIMACS New Market Models Workshop 13 The End

15 DIMACS New Market Models Workshop 14 A Continuous Version X t = positive continuous local martingale the share price B t = semi-martingale total amount invested up to time t S t = t db u 0 X u = number of shares owned at time t Y t = X t S t B t = net gain at time t Theorem. The gain Y t at time t is given by Y t = t 0 S u dx u + X, S t. If B is locally of bounded variation, then Y is a continuous local martingale.

16 DIMACS New Market Models Workshop 15 Proof Proof. Start with the stochastic integration by parts formula: X t S t = t From the definition of S, we see that t Hence the formula for Y. 0 S u dx u + 0 t 0 X u ds u = B t. X u ds u + X, S t. If B locally has bounded variation, the S does too. Hence, the cross variation X, S t vanishes. Since S is continuous, it is predictable and so the remaining stochastic integral is a martingale.

17 DIMACS New Market Models Workshop 16 Remarks 1. If shares are purchased but not sold, then B is increasing and hence is locally of bounded variation. 2. If B has unbounded variation, then X, S is not expected to vanish. In this case, the gain process can be a submartingale and that s a good thing. See example on next slide.

18 DIMACS New Market Models Workshop 17 Suppose that B t = X t for all t. Using Ito s formula we get An Example S t = t 0 db u X u = log(x t ) log(x 0 ) t 0 d X u X 2 u. Hence, X, S t = X, log(x) t. Since X, log(x) is increasing, the gain process Y is a submartingale.

19 DIMACS New Market Models Workshop 18 Conclusion A seemingly harmless assumption, the ability to buy and sell very fast with no transaction cost, leads to a plainly absurd result.

20 DIMACS New Market Models Workshop 19 Final Remarks

21 DIMACS New Market Models Workshop 20 Other Erroneous Results The Black-Scholes option pricing formula assumes the ability to trade fast without transaction costs. The result is a formula that depends on volatility but not on drift. Consider historical data for companies A (on the left) and B (on the right): They both have the same volatility. Would you pay the same price for an option on these two companies? (Be honest!)

22 DIMACS New Market Models Workshop 21 An Alternative