Sunday, July 26, 2009

Stochastic Finance on Stock Returns

The stochastic calculus based interpretation of stock returns offers a glimpse of institutional perspective and indirectly reveals how buying and holding leads to a negative expectancy. While numbers and symbols may look intimidating initially, they tell a very simple story.

Variable definitions

The excerpt comes from Stochastic Calculus for Finance Vol1.

S0: Stock price at time step 0

S1: Stock price at time step 1

r: risk free interest rate (usually off treasury debt)

S1(H): Stock price at time step 1 if “heads” were to occur off a random coin toss

S1(T): Stock price at time step 1 if “tail” was to occur off a random coin toss

p: probability of H occurring

q: probability of T occurring

u: multiple applied to S if H occurs (e.g. S1(H) = 2S0)

d: multiple applied to S if T occurs (e.g. S1(T) = 0.5S0)

Note: p + q = 1, so they are collectively exhaustive

Formula 1.1.8 displays “expected” probabilities for future price moves.

Slower than actual rate of inflation

The expected returns adjust simply with r. The “risk free” (questionably) debt instruments usually offer the lowest of interest returns compared to institutional, private loans. As way more money becomes created via institutional and private borrowing, the actual rate of inflation is naturally greater than whatever return off government debt.

Credit risk ignorance makes the model impractical

The binomial pricing model assumes all listed companies to operate with infinite lifespan. Is this belief viable for actual money management? I’d say not. By the way this also invalidates general portfolio theory, sorry Markowitz!

While some revealed, many issues remain to be addressed with quantitative financial theories. Until academic ideas become aligned with empirical evidence, traders must search out personal, unique ways to remedy quantitative model weaknesses to become/remain profitable.