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Risk-Neutral Valuation Intuition

Risk-neutral valuation
P0=EQ[e−rTXT]P_0 = E^{\mathbb{Q}}\left[e^{-rT}X_T\right]

Price is the discounted expected payoff under a set of probabilities that are deliberately not forecasts.

This lesson is where those probabilities come from.

Build it from one step

Stock at 100, going to 120 or 90. Risk-free rate 0. Price a call struck at 100, worth 20 or 0.

Replicate. Find Δ\Delta shares and BB cash matching both states:

120Δ+B=20,90Δ+B=0120\Delta + B = 20, \qquad 90\Delta + B = 0

Giving Δ=23\Delta = \frac{2}{3}, B=−60B = -60, so

C=23(100)−60=6.67C = \tfrac{2}{3}(100) - 60 = 6.67

Now rewrite it. Define qq such that the stock's expected value under qq equals its current price:

120q+90(1−q)=100  ⟹  q=13120q + 90(1-q) = 100 \implies q = \tfrac{1}{3}

Then

EQ[payoff]=13(20)+23(0)=6.67E^{\mathbb{Q}}[\text{payoff}] = \tfrac{1}{3}(20) + \tfrac{2}{3}(0) = 6.67

The same answer. The replication price can always be written as an expectation under these particular probabilities.

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