Martingale System for Stock Investments

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.

Sell in month 8 @ \$5/share = \$10,770
Total invested (7 x \$1260) = 8,820
Gain = \$1,950

DCA Theorem

a = \$’s invested each month
Xk = share price in month k

Theorem. If Xk, k = 0,1....., is a simple random walk conditioned to return to its starting point X0 at time n (and n < 2x0), then EYn > 0.

Proof. Follows from the reflection principle and the harmonic/arithmetic mean inequality.

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.

A Martingale Framework

Fk , k = 0; 1; : : :, a filtration on 
Xk , k = 0; 1; : : :, a positive martingale wrt Fk
Ak , k = 0; 1; : : :, a nonnegative process adapted to Fk

= net gain up to month n from investment stream Ak .

Theorem. Yn, n= 0; 1; : : :, is a mean-zero martingale.

Proof. Note that

Take conditional expectations and note adaptedness of Ak =Xk:

The quadratic variation of X is

Theorem. The quadratic variation of the gain process, Y, is given by where

Proof. From the previous proof and the definition of process S, we have

Since the stock price and gain processes, X and Y, are martingales, we see that

From the definition of quadratic variation, we see that

Summing both sides completes the proof.

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. The fact that the gain process Y is a martingale follows from this theorem by taking

Discounting Trends

Suppose now that X is an arbitrary positive adapted stochastic process.

Put

where

The process R is predictable:

Theorem. The process is a martingale.

Proof.

Following the Trend

Convert all dollar amounts to discounted (time 0) values. The gain process is measured in discounted terms:

where

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.

A Continuous Version

Xt = positive continuous local martingale—the share price
Bt = semi-martingale—total amount invested up to time t

Theorem. The gain Yt at time t is given by

If B is locally of bounded variation, then Y is a continuous local martingale.

Proof

From the definition of S, we see that

Hence the formula for Y.

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.

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:

Suppose that Bt = Xt for all t. Using Ito’s formula we get

Hence,

Since (X, log(X)) is increasing, the gain process Y is a submartingale.

Conclusion

A seemingly harmless assumption, the ability to buy and sell very fast with no transaction cost, leads to a plainly absurd result.

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!)