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

A European call option is written on a share that pays a continuous dividend yield q. Under the Black-Scholes framework, which adjustment gives the correct price of the call?

The correct adjustment is to replace S0 by S0·e^(-qT) in the standard Black-Scholes formula. The holder of the option does not receive dividends, so the share price is reduced by the discounted dividend stream, which equivalently lowers the risk-neutral drift to r-q.

  1. AReplace the strike price K by K·e^(-qT) in the standard formula
  2. BReplace the initial share price S0 by S0·e^(-qT) in the standard formulaCorrect
  3. CReplace the risk-free rate r by r+q in the discounting of the strike only
  4. DReplace the volatility σ by σ-q in the standard formula
  5. Add the present value of the dividends to S0 in the standard formula

Explanation

With a continuous dividend yield q, the share price is reduced by the present value of dividends, so S0 is replaced by S0·e^(-qT). Equivalently the drift in the risk-neutral world becomes r-q. Adding dividends to S0 would go the wrong way, and the strike is not adjusted for dividends.

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