Stock Market Analyse

1. Use the log return instead of simple return.

1.1 Defination of log return:

$r_t = ln (\frac{P_t}{P_{t-1}}) = ln(P_t) - ln(P_{t-1})$

• $P_t$ is the price at time t

1.2 Mathematical derivation of log return:

1. Assumption: Exponential Growth Dynamics
   We assume the price P(t) grows continuously at a rate proportional to its current value:$\frac{d P(t)}{dt} = r \cdot P(t)$

• The instantaneous growth rate ($\frac{dP(t)}{dt}$) depends on the current price P(t)
• r is the propotionality constant (e.g., 5% -> r = 0.05)

2. Solve the Differential Equation
   Separate variables and integrate:

   $\int \frac{1}{P(t)}dP = \int r dt$
   $\ln P(t) = rt + C$
   Exponentiate both sides to solve for P(t):
   $P(t) = e^{rt+C} = e^C \cdot e^{rt}$
   set t = 0: $P(0) = e^C$, so:
   $P(t) = P(0) \cdot e^{rt}$
   (Here we get the continuous-time exponential growth formula.)

3. Discrete-Time Version (Adjacent Periods)
   For two adjacent time points t-1 and t:

   $P_t = P_{t-1} \cdot e^{r_t}$

• $r_t$ is the log return from t-1 to t.

Solve for $r_t$:

$r_t = \ln (frac{P_t}{P_{t-1}})$

(This is the defination of log return.)

1.3 Why Use Log Return

1. Time Additivity (Multi-Period Returns)
Log returns are additive over time, For example, the total return over k periods is the sum of individual log returns:

$r_{t\rightarrow t+k} = \sum \limits_{i=1} ^k r_{t+i}$

Simple returns require geometric compounding(multiplication), which is less intuitive for calculations.

2. Symmetry (Equal Treatment of Gains/Losses)
Log returns treat gains and losses symmetrically. Simple returns are asymmetric (e.g., a 50% drop requires a 100% gain to recover)

3. Normality Assumption(Better for Statistical Models)

• Financial Models:
  Many models (e.g., Black-Scholes, Brownian motion) assume returns are normally distributed. Log returns are closer to normality( especially for short horizons), while simple returns can be skewed.
• Continuous Compounding:
  Log returns reflect continuously compounded growth, aligning with theoretical finance (where time periods are infinitesimally small)

4. Numerical Stability (Avoids Extreme Values)

• Guaranteed Positive Prices:
  Since $\ln (P_t)$ is only defined for $P_t > 0$, log returns naturally avoid negative prices(e.g., in derivatives pricing).
• Reduces Outlier Impact:
  Log transforms compress large price swings, making models less sensitive to extreme values.