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IAI Actuarial Core Principles · Economic Modelling · Black-Scholes derivative-pricing model

A share pays no dividends and follows geometric Brownian motion with sigma = 0.25 and risk-free rate r = 0.06 continuously compounded. Which PDE must the value V(S,t) of any derivative satisfy?

The PDE is dV/dt + 0.5 sigma squared S squared d2V/dS2 + r S dV/dS - rV = 0, here with sigma 0.25 and r 0.06. The risk-free rate replaces the real-world drift, and the final term is minus rV.

  1. AdV/dt + 0.5(0.25)^2 S^2 d2V/dS2 + 0.06 S dV/dS - 0.06 V = 0Correct
  2. BdV/dt + 0.5(0.25)^2 S^2 d2V/dS2 + 0.06 S dV/dS + 0.06 V = 0
  3. CdV/dt + 0.25 S^2 d2V/dS2 + 0.06 S dV/dS - 0.06 V = 0
  4. DdV/dt + 0.5(0.25)^2 S^2 d2V/dS2 + mu S dV/dS - 0.06 V = 0, mu being the real-world drift
  5. dV/dt + 0.5(0.25)^2 S^2 d2V/dS2 - 0.06 S dV/dS - 0.06 V = 0

Explanation

The Black-Scholes PDE is dV/dt + 0.5 sigma^2 S^2 V_SS + r S V_S - rV = 0. Option 2 has the wrong sign on rV; option 3 uses sigma instead of sigma^2/2 ; option 4 uses real-world drift, which does not appear; option 5 has the wrong sign on the S V_S term.

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