Dynamic Models

Washington University in Saint Louis

- Fixed proportion portfolio strategies
- Log-optimal policy
- Review of binomial option pricing

It should perhaps be obvious that investments in actuality span many short periods. In general, there will be connections across the periods. If there is consumption withdrawal, then there is the question of how much to take out for use this period and how much to leave in the portfolio for future use. Even when we are accumulating to a fixed horizon (for example, building wealth that will be paid out in a lump sum at retirement), there is a question of whether investment proportions should vary depending on time or investment performance. Finally, even given fixed proportions, we want to learn what our choices about asset proportions imply about the distribution of wealth available at the end.

We are going to start with the simplest dynamic model, in which we take investment proportions as fixed. We will take very simple assumptions; volatilities and expected returns are known in advance and do not vary over time. Literally speaking, these are not the most accurate assumptions we could make, but the results we obtain will not mislead us.

If we start with initial wealth *W0* and invest in a
portfolio whose rate of return in each period *t* is
*Rt*, then the wealth after *T* periods is given
by:

T W = W product (1+R ). T 0 t=1 t

In statistics, we are more familiar with dealing with sums
than with products, which we can do by taking logarithms. (We
could use any base for the logarithms, but for convenience we
will use base e. This is convenient because, for *R* not
too large, *log(1+R) ~ R* when the logarithm is in base
*e*.)

T log(W ) = log(W ) + sum log(1+R ). T 0 t=1 t

Now, let us assume the returns in each period are
independent and have the same distribution. Then, if *
Rt* has mean *mu* and variance *V*, *
log(1+Rt)* has a mean of approximately *mu-V/2* and
a variance of *V*. (Take my word or use a Taylor series
expansion.) Therefore, in a sense I am not making precise, *
log(WT)* has an approximate normal distribution with mean
*log(W0) + (mu-V/2)T* and variance *VT*. In fact,
these expressions are exact in the continuous-time model used
in the very successful Black-Scholes formula, so this is likely
to be a very reasonable approximation over short time
intervals. Using historical averages for mean and variance will
tend to overstate average performance and understate
uncertainty, since we are after all studying the US stock
market because it has been more successful than average. Also,
assuming variance is constant may tend to understate
volatility. Nonetheless, these calculations are reasonable
guesses and are much better than, for example, just assuming
the mean return will hold for sure!

What is the probability that the stock market will
outperform T-Bills at various horizons? The previous slide
allows us to give a reasonable answer. Take the riskfree rate
to be *5%*, the mean stock market return to be *
15%*, and the market standard deviation to be *25%*,
all on an annual basis. Then the analysis of the previous slide
tells us that for investing in the market, *log(WT/W0)*
has mean *(15%-.03125)T* and standard deviation *
sqrt(.0625T)*. Using the same analysis says that for
rolling over T-Bills, *log(WT/W0)* has mean *5%T*
and standard deviation *0*. By the properties of a
normal distribution, we have that the probability that the
market will outperform T-Bills over a given period of length
*T* is the same as the probability that a unit normal
random variable is bigger than -(15%-.03125-5%)T/sqrt(.0625T),
which can be obtained from standard tables. This is tabulated
on the following slide.

------------------------------------------------------------------------- # yrs | Treasuries | risky asset | prob(mkt > Trs) | log(WT/W0) | mean(log(WT/W0)) | std(log(WT/W0)) | ------------------------------------------------------------------------- 1/12 | .0042 | .0099 | .0722 | .532 1/2 | .0250 | .0594 | .1768 | .577 1 | .0500 | .1188 | .2500 | .608 3 | .1500 | .3562 | .4330 | .683 5 | .2500 | .5938 | .5590 | .731 10 | .5000 | 1.1875 | .7906 | .808 30 | 1.5000 | 3.5625 | 1.3693 | .934 -------------------------------------------------------------------------

This table is based on historical numbers being good predictors of future returns. Personally, I think those numbers overstate reasonable market returns because we are studying a successful market because it was successful. Even given these optimistic numbers, it is a bad idea to take as a sure thing that the realized returns will be bigger than we would get in Treasuries, even over decades!

The *log-optimal* policy is the strategy that
maximizes the expected log of terminal wealth, and is also the
strategy that maximizes the long-term growth rate of the
portfolio's value. Some people have argued that this is the
best portfolio for anyone with a long horizon, since this
strategy will eventually outperform any other strategy. The
problem with this reasoning is that we might have to wait
hundreds or thousands of years for the superior performance. In
the meantime, the performance could be disastrous.

Among fixed-proportion strategies, the expected log of
terminal wealth is given by *E[log(WT)] = log(W0)+(r +
w1(mu-r) - w1^2 s^2/2)T*, where *r* is the riskfree
rate, *mu* is the mean return on the market, *s*
is the standard deviation of the market, and *T* is the
time horizon. This expected log of terminal wealth is maximized
by choosing a portfolio weight *w1 = (mu-r)/s^2* in the
risky market portfolio. If mu=15%, r=5%, and s=20%, this
implies that .1/.04 = 250% of the portfolio should be invested
in the risky asset, which is a very aggressive policy!

-------------------------------------------------------------------------- # yrs | Treasuries | log optimal (l.o.) portfolio | prob(l.o. > Trs) | log(WT/W0) | mean(log(WT/W0)) | std(log(WT/W0)) | -------------------------------------------------------------------------- 1/12 | .0042 | .0146 | .1443 | .529 1/2 | .0250 | .0875 | .3536 | .570 1 | .0500 | .1750 | .5000 | .599 3 | .1500 | .5250 | .8660 | .667 5 | .2500 | .8750 | 1.1180 | .711 10 | .5000 | 1.7500 | 1.5811 | .785 30 | 1.5000 | 5.2500 | 2.7386 | .915 --------------------------------------------------------------------------

There is little assurance over reasonable horizons that the log-optimal strategy will outperform Treasuries, let alone other portfolio strategies taking some advantage of the higher average returns to stocks.

When the mean and variance of stock returns are constant (as
we have been assuming), there is a very interesting property of
fixed-proportion strategies that is a good approximation in
practice. Let *MT* be the value at time *T* of a
policy of staying *100%* invested in an indexed
portfolio of stocks, and let *WT* be the value of being
invested *k%* in stocks, each given the same initial
investment. Then *WT* is approximately proportional to
*MT^k*, or more precisely

k 2 2 . MT (k-k )s WT = W0 (--) exp(((1-k)r + --------)T), M0 2

where *s^2* is the market's variance and *r*
is the riskfree rate. This formula is exact in the ideal limit
of continuous trading, as in the Black-Scholes model.

**Why option pricing?** Option pricing theory
is ideal for analyzing or devising dynamic trading strategies,
since option pricing theory gives us both the value and the
trading strategy for various payoff rules. Many trading
strategies such as portfolio insurance and related strategies
were actually motivated by option pricing models. In general,
using option pricing models allows us to customize our exposure
to risk by following an investment strategy that pays off an
arbitrary function of the market value at the end.

The option pricing model of Black and Scholes revolutionized a literature previously characterized by clever but unreliable rules of thumb. The Black-Scholes model uses continuous-time stochastic process methods that interfere with understanding the simple intuition underlying these models. We will use instead the binomial option pricing model of Cox, Ross, and Rubinstein, which captures all of the economics of the continuous time model but is simple to understand and use. For option pricing problems not appropriately handled by Black-Scholes, some variant of the binomial model is the usual choice of practitioners since it is relatively easy to program, fast, and flexible.

Cox, John C., Stephen A. Ross, and Mark Rubinstein (1979)
``Option Pricing: A Simplified Approach'' *Journal of
Financial Economics* **7**, 229--263

Black, Fischer, and Myron Scholes (1973) ``The Pricing of
Options and Corporate Liabilities'' *Journal of Political
Economy* **81**, 637--654

Riskless bond:

2 3 1 --- r --- r --- r

Stock (u > r > d):

|- uuuS |- uuS -| |- uS -| |- uudS S -| |- udS -| |- dS -| |- uddS |- ddS -| |- dddS

Derivative security (option or whatever):

|- V1 |- ? -| |- ? -| |- V2 ? -| |- ? -| |- ? -| |- V3 |- ? -| |- V4

*What is the price of the derivative security?*

Riskless bond (interest rate is 0):

100 --- 100

Stock:

|- 100 50 -| |- 25

Derivative security (on-the-money call option):

|- 50 ? -| |- 0

To duplicate the call option with alpha_S shares of stock and alpha_B bonds:

50 = 100 alpha_S + 100 alpha_B

0 = 25 alpha_S + 100 alpha_B

Therefore alpha_S = 2/3, alpha_B = -1/6, and the duplicating portfolio is worth 50 alpha_S + 100 alpha_B = 100/6 = 16 2/3. By absence of arbitrage, this must also be the price of the call option.

Compute the duplicating portfolio and the price of the general derivative security below. Assume u > r > d > 0.

Riskless bond:

1 --- r

Stock:

|- uS S -| |- dS

Derivative security:

|- V_u ? -| |- V_d

In general, exactly the same valuation procedure is used many times, taking as given the value at maturity and solving back one period of time until the beginning. This valuation can be viewed in terms of state prices p_u and p_d or risk-neutral probabilities pi*_u and pi*_d, which give the same answer (which is the only one consistent with no arbitrage):

1 * * Value = p V + p V = -(pi V + pi V ) u u d d r u u d d

where

r-d p = ------, u r(u-d)

u-r p = ------, d r(u-d)

* r-d pi = ---, u u-d

and

* u-r pi = ---. d u-d

In the binomial model (with parameters u, d, and r), show that the stock and the bond have the same one-period expected return computed using the artificial probabilities.

Consider the binomial model with u=2, d=1/2, and r=1. What are the risk-neutral probabilities? Assuming the stock price is initially 100, what is the price of a call option with a 90 strike price maturing in two periods? Do the valuation two ways, using the risk neutral probabilities to solve backwards through the tree, and directly using the two-period risk neutral probabilities.

Most texts seem to have unreasonably complicated expressions for u, d, and r in binomial models. From the theory, we know that a good choice is

u = 1 + r * Delta t + sigma * sqrt{Delta t} d = 1 + r * Delta t - sigma * sqrt{Delta t}

with pi*_u=pi*_d=1/2 and Delta t the time increment. This has the two essential features: it equates expected stock and bond returns, and it has the right standard deviation. In addition, it has a continuous stock price (like Black-Scholes) as a limit.

One alternative is to choose the positive solution of ud=1 and u-d=2 sigma sqrt{Delta t} (good for option price as a function of time) or a recombining trinomial (good for including some dependence of variance on the stock price).