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

A non-dividend share has S0 = 100, and the risk-free rate is 4% continuously compounded. A forward-type derivative pays S_T^2 at time T = 1 with sigma = 0.20. Under the risk-neutral measure, ln S_T is normal with mean ln S0 + (r - 0.5 sigma^2)T and variance sigma^2 T. What is the derivative's value today, to the nearest rupee?

The value is S0 squared times exp(rT + sigma squared T), because E_Q of S_T squared is S0 squared times exp(2rT + sigma squared T) and discounting removes one rT. This gives 10000 x exp(0.08), about Rs 10,833.

  1. ARs 10,000
  2. BRs 10,408Correct
  3. CRs 10,833
  4. DRs 10,202
  5. Rs 10,624

Explanation

E_Q[S_T^2] = S0^2 exp(2(r-0.5 sigma^2)T + 2 sigma^2 T) = S0^2 exp(2rT + sigma^2 T) = 10000 exp(0.08+0.04) = 10000 exp(0.12). Discounting by e^-0.04 gives 10000 exp(0.08)=10000 x 1.0833 = 10,833. So the correct value is Rs 10,833, option 3, not 10,408.

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