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002_relative_strength_index

Relative Strength Index: Theory and Formalization

Relative Strength Index: Theory and Formalization

I. Historical Context and Motivation

The Relative Strength Index (RSI), introduced by J. Welles Wilder Jr. in 1978, constitutes a normalized momentum oscillator. Its objective is to quantify the speed and magnitude of price variations to identify overbought and oversold conditions.

II. Mathematical Construction

Definition 2.1 (Price Variation)

For a price series {Pₜ}ₜ∈ℕ, we define the variation: ΔPt=Pt−Pt−1\Delta P_t = P_t - P_{t-1}

Definition 2.2 (Gains and Losses)

We decompose the variation into positive and negative components: Gt=max⁡(ΔPt,0)(Gain)G_t = \max(\Delta P_t, 0) \quad \text{(Gain)} Lt=max⁡(−ΔPt,0)(Loss)L_t = \max(-\Delta P_t, 0) \quad \text{(Loss)}

Fundamental property: ∀t, Δ P_t = G_t - L_t and G_t · L_t = 0

Definition 2.3 (EMA of Gains/Losses)

Let n be the period (typically n=14). We calculate: G‾n(t)=EMAn(Gt)=1n∑i=0n−1Gt−i(Wilder method)\overline{G}_n(t) = EMA_n(G_t) = \frac{1}{n}\sum_{i=0}^{n-1} G_{t-i} \quad \text{(Wilder method)} L‾n(t)=EMAn(Lt)=1n∑i=0n−1Lt−i\overline{L}_n(t) = EMA_n(L_t) = \frac{1}{n}\sum_{i=0}^{n-1} L_{t-i}

Definition 2.4 (Relative Strength)

RS(t)=G‾n(t)L‾n(t)RS(t) = \frac{\overline{G}_n(t)}{\overline{L}_n(t)}

Definition 2.5 (RSI)

RSI(t)=100−1001+RS(t)=100⋅G‾n(t)G‾n(t)+L‾n(t)RSI(t) = 100 - \frac{100}{1 + RS(t)} = \frac{100 \cdot \overline{G}_n(t)}{\overline{G}_n(t) + \overline{L}_n(t)}

III. Mathematical Analysis

Theorem 3.1 (RSI Bounds)

∀t ∈ ℕ : RSI(t) ∈ [0, 100]

Proof: By definition, Ḡ_n(t) ≥ 0 and L̄_n(t) ≥ 0.

  • If L̄_n(t) = 0 (pure bullish market): RS → +∞, hence RSI → 100
  • If Ḡ_n(t) = 0 (pure bearish market): RS = 0, hence RSI = 0
  • Otherwise: RS > 0, hence 0 < RSI < 100 ∎

Theorem 3.2 (Probabilistic Interpretation)

The RSI can be interpreted as an estimate of the probability that the next movement is bullish: RSI(t)/100≈P(ΔPt+1>0∣Ft)RSI(t)/100 \approx P(\Delta P_{t+1} > 0 | \mathcal{F}_t)

Proposition 3.3 (Reversal Symmetry)

If we define P'_t = -P_t, then RSI'(t) = 100 - RSI(t)

Proof: Gains become losses and vice versa, thus RS' = 1/RS. RSI′=100−1001+1/RS=100−100⋅RSRS+1=100−RSIRSI' = 100 - \frac{100}{1 + 1/RS} = 100 - \frac{100 \cdot RS}{RS + 1} = 100 - RSI ∎

IV. Critical Zones and Thresholds

4.1 Regime Definitions

| RSI Value | Interpretation | Condition | |-----------|----------------|-----------| | RSI > 70 | Overbought | Ḡ >> L̄ | | RSI < 30 | Oversold | L̄ >> Ḡ | | RSI ≈ 50 | Equilibrium | Ḡ ≈ L̄ |

4.2 Statistical Justification of Thresholds

The choice of 70/30 corresponds approximately to:

  • RSI = 70 ⟹ Ḡ/L̄ = 7/3 ≈ 2.33
  • RSI = 30 ⟹ Ḡ/L̄ = 3/7 ≈ 0.43

These ratios represent significant asymmetry (> 2σ under normal hypothesis).

V. Divergences: Formalization

Definition 5.1 (Bullish Divergence)

A bullish divergence exists if: ∃t1<t2:Pt2<Pt1∧RSI(t2)>RSI(t1)\exists t_1 < t_2 : P_{t_2} < P_{t_1} \land RSI(t_2) > RSI(t_1)

Price makes a lower low while RSI makes a higher low.

Definition 5.2 (Bearish Divergence)

∃t1<t2:Pt2>Pt1∧RSI(t2)<RSI(t1)\exists t_1 < t_2 : P_{t_2} > P_{t_1} \land RSI(t_2) < RSI(t_1)

Physical Interpretation

Divergence signals momentum weakening: the trend "engine" is losing power before price reflects this change.

VI. Extension: Stochastic RSI

Definition 6.1

StochRSI(t)=RSI(t)−min⁡k∈[t−n,t](RSIk)max⁡k∈[t−n,t](RSIk)−min⁡k∈[t−n,t](RSIk)StochRSI(t) = \frac{RSI(t) - \min_{k \in [t-n, t]}(RSI_k)}{\max_{k \in [t-n, t]}(RSI_k) - \min_{k \in [t-n, t]}(RSI_k)}

VII. Exercises

Exercise 1: Prove that if Pₜ follows a symmetric random walk, then E[RSI(t)] = 50.

Exercise 2: Calculate RSI(t) for the sequence P = [100, 101, 99, 102, 100] with n=3.

Exercise 3: Show that RSI is invariant under positive affine transformation of price.

VIII. References

  • Wilder, J.W. (1978). New Concepts in Technical Trading Systems
  • Achelis, S.B. (2001). Technical Analysis from A to Z
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