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Estimation of
the European
Equity Model

European Bond
and Currency Markets
in Anticipation of
Monetary Union


Volatile Markets
BARRA Models

Part One:
The Case for
the Market Neutral

BARRA announces new
managing director of
research

The BARRA Brainteaser
for Winter 1999

Solution for the Fall 1998
Brainteaser

The BARRA Brainteaser from Fall 1998

by Eugene Reznik

On the CBOE exchange, stock prices move by exactly one tick from one trade to the next, with a 50% probability of moving either up or down. On average, after how many trades will a stock on the CBOE exchange have traded at N different levels after opening? How is the answer affected if the probabilities of the up and down moves are p and q respectively?

Brain Teaser Solution

by Eugene Reznik and Nick Baturin

Let (N) denote the expected number of trades it takes for the stock to trade at N distinct levels. The stock always opens at some level and so (1) = 0. Since it always takes one trade to get to a new level (2) = 1. We can try to find a recursive solution of the form (N) = F((N-1)).

Let the state (N,k) denote a situation in which the stock has traded at N distinct price levels (numbered 1 through N from lowest to highest) and is currently at level number k. Also, let M(N,k) be the time it takes for the stock to reach a new high or low starting from the state (N,k).

Now M(N,k) satisfies the following difference equation:

M(N,k)=1+pM(N,k+1)+qM(N,k-1) (1)

with boundary conditions

M(N,0)=M(N,N+1)=0 (2)

The solution to equation (1) with boundary conditions (2) is:

In the special case where p = q = 1/2, equation (1) with boundary conditions (2) is solved by M(N,k) = k(N+1-k). Thus,
M(N,1) = M(N,N) = N. This implies that
(N) = (N-1) + (N-1) = (N-1)N/2 and we have the answer to the first part of the problem. Note that in the general case, the formula is not symmetric, i.e.,
M(N,1) M(N,N). The solutions to (1) and (2) above are derived from Feller¹.

In the general case, we can say that:

where H(N) is the probability of being in the state (N,N) conditioned on having just covered a new level. In order to calculate H(N) let’s define h(N,k) as the probability of the stock price reaching a new high from the state (N,k) before reaching a new low.

Like M(N,k), h(N,k) satisfies a simple difference equation:

h(N,k) = p h(N, k+1) + q h(N, k-1) (3)

with boundary conditions

h(N,0) = 0; h(N,N+1) = 1 (4)

We can solve (3) and (4) to obtain

To find H(N), note that H(2) = h(1,1) = p. Note also that H(N) can be expressed in terms of H(N-1), h(N-1,1) and h(N-1,N) as:

H(N) = H(N-1) h(N-1,N-1) + (1-H(N-1)) h(N-1,1)

This completes the solution as we now have all the necessary pieces to recursively calculate
(N). FIGURE 1 contains a plot of as a function of N and p. We can see that
(N) = N(N-1)/2 in the special case where p=1/2 and that (N) = N-1 in the case where
p = 1. 

Figure 1: Number of trades required to cover N different levels as a function of N and probability of an up move p.

¹ W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 1970.