Stochastic Oscillator: Probability and Momentum Analysis

I. Introduction

The Stochastic Oscillator, developed by George Lane in the 1950s, measures the position of closing price relative to the high-low range over a given period. It is based on the observation that in uptrends, prices tend to close near the high.

II. Mathematical Construction

Definition 2.1 (Fast Stochastic %K)

$\%K_n(t) = \frac{C_t - L_n(t)}{H_n(t) - L_n(t)} \times 100$

Where:

• C_t = Current closing price
• L_n(t) = Lowest low over last n periods
• H_n(t) = Highest high over last n periods

Definition 2.2 (Slow Stochastic %D)

$\%D_m(t) = SMA_m(\%K_n(t))$

Typically m = 3 (3-period simple moving average of %K).

Definition 2.3 (Full Stochastic)

• %K (Slow) = SMA_m of Fast %K
• %D = SMA_m of Slow %K

Standard parameters: (14, 3, 3)

III. Theoretical Properties

Theorem 3.1 (Bounded Domain)

∀t: 0 ≤ %K_n(t) ≤ 100

Proof: By definition, L_n(t) ≤ C_t ≤ H_n(t). Therefore: 0 ≤ C_t - L_n(t) ≤ H_n(t) - L_n(t) Dividing: 0 ≤ (C_t - L_n(t))/(H_n(t) - L_n(t)) ≤ 1 ∎

Theorem 3.2 (Extreme Values)

• %K = 100 ⟺ C_t = H_n(t) (close at highest high)
• %K = 0 ⟺ C_t = L_n(t) (close at lowest low)

Proposition 3.3 (Probabilistic Interpretation)

Under uniform distribution of closes within the range: $E[\%K] = 50$

The stochastic oscillator measures the percentile rank of the closing price.

Theorem 3.4 (Trend Behavior)

In a perfect uptrend with C_t = H_t ∀t: $\%K_n(t) = 100 - \frac{H_n(t) - C_t}{H_n(t) - L_n(t)} \times 100 = 100$

Conversely, in a perfect downtrend with C_t = L_t ∀t: $\%K_n(t) = 0$

IV. Signal Generation

4.1 Overbought/Oversold Zones

| Zone | %K Range | Interpretation | |------|----------|----------------| | Overbought | > 80 | Momentum exhaustion (sell zone) | | Neutral | 20-80 | Normal trading range | | Oversold | < 20 | Momentum exhaustion (buy zone) |

4.2 Crossover Signals

Bullish Signal: $\%K(t^-) < \%D(t^-) \land \%K(t^+) > \%D(t^+) \land \%K(t) < 20$

Bearish Signal: $\%K(t^-) > \%D(t^-) \land \%K(t^+) < \%D(t^+) \land \%K(t) > 80$

4.3 Divergences

Bullish Divergence: $Price_{t_2} < Price_{t_1} \land \%K_{t_2} > \%K_{t_1}$

Bearish Divergence: $Price_{t_2} > Price_{t_1} \land \%K_{t_2} < \%K_{t_1}$

V. Comparison with RSI

5.1 Structural Differences

| Aspect | Stochastic | RSI | |--------|-----------|-----| | Measures | Price position in range | Gain/loss ratio | | Input | High, Low, Close | Close only | | Sensitivity | Higher | Lower | | Noise | More | Less | | Best for | Ranging markets | All conditions |

5.2 Mathematical Relationship

Both are bounded oscillators but measure different phenomena:

• Stochastic: Where is price within its range?
• RSI: What is the relative strength of gains vs losses?

VI. Advanced Variants

6.1 Williams %R

$\%R = \frac{H_n - C}{H_n - L_n} \times (-100) = \%K - 100$

Williams %R is simply an inverted stochastic.

6.2 Stochastic RSI

$StochRSI = \frac{RSI - RSI_{min}}{RSI_{max} - RSI_{min}}$

Combines both indicators' strengths.

6.3 Double Smoothed Stochastic

$DSS = SMA_m(SMA_k(\%K))$

Reduces noise further with double averaging.

VII. Parameter Optimization

7.1 Standard Parameters

• Lane's original: (14, 3, 3)
• Short-term: (5, 3, 3)
• Long-term: (21, 5, 5)

7.2 Optimization Criterion

Minimize false signals while maintaining responsiveness: $\min_{n,m} \sum_{i} |Signal_i - TrueReversal_i|$

VIII. Exercises

Exercise 1: Calculate %K(5) for the data: Closes: [100, 102, 99, 103, 101] Highs: [101, 103, 100, 104, 102] Lows: [99, 100, 98, 101, 99]

Exercise 2: Prove that %K is invariant under positive linear transformation of prices.

Exercise 3: Show that Fast %K is always more volatile than Slow %K.

Exercise 4: Calculate %D given %K = [30, 45, 60, 75, 90] with m=3.

IX. References

• Lane, G. (1984). Lane's Stochastics
• Murphy, J.J. (1999). Technical Analysis of the Financial Markets