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Derivatives Pricing and Computational Finance / Foundations of Computational Option Pricing

Derivation: From Risk-Neutral Expectation to the Black-Scholes Formula

L1 jumps from the risk-neutral integral to the Black–Scholes formula in one line. This subnote performs the jump in slow motion: starting from

C(t,S0)=e−rτ EQ[(ST−K)+],τ:=T−t,C(t, S_0) = e^{-r\tau}\, \mathbb{E}^{\mathbb{Q}}\big[ (S_T - K)^+ \big], \qquad \tau := T - t,
we evaluate the expectation explicitly and watch d1d_1 and d2d_2 emerge from completing the square in a Gaussian integral. The technique — split the payoff, complete the square, recognise normal CDFs — recurs throughout computational finance (e.g. in Fourier pricing, Note 3), so it is worth doing carefully once.

In this lesson

  1. Step 0 — Setup
  2. Step 1 — Remove the positive part
  3. Step 2 — The probability term I2I_2
  4. Step 3 — The stock term I1I_1: complete the square
  5. Step 4 — Assemble
  6. Step 5 — The put, for free

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