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6.1.1 The American Black-Scholes model for dummies

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Since an American option confers its holder the right to buy or sell an underlying share $ S$ any time up to the expiry date, the option price $ V(S,t)$ can never drop below the intrinsic value that has been defined in sect.2.1.3. Indeed, if the price dropped below, a crowd of arbitragers would immediately seize the opportunity and buy a large amount of options only to exercise them immediately for a risk less profit $ \Lambda(S)-V(S,t)$ . The VMARKET applet below illustrates this with an American put option, where the price never drops below the intrinsic value even in the presence of a finite interest rate.

VMARKET applet:  press Start/Stop to calculate the price of an American put option struck for 10 EUR up to 9 months before the expiry, in a market with 40% volatility and a spot rate of 3%. The black (alt. grey) line shows the present (alt. intrinsic) value of the option V(S,t) for a range of underlying prices 0 < S < 20, as the time runs from the expiry date (T=0) back to three quarter of a year (T-t=0.75) before the expiry date.

Virtual market experiments: American options
  1. Compare the payoff from both the American and European exercise styles; for which value of the underlying is the difference largest?
  2. Switch to Call, VSpread and SuperShr to study how the exercise style affects the price of both vanilla and binary options.
  3. Compare the true American payoff with an approximation obtained by taking the larger of the European payoff and the intrinsic value (6.1.1#eq.1).

Since the price necessarily exceeds the intrinsic value, American options never develop the negative time value $ V(S,t)-\Lambda(S)$ that has previously been observed for European options. In fact, the experiments above suggest that a crude estimate for the value of an American option can be obtained simply by choosing whichever is larger, the European Black-Scholes formula or the intrinsic value

$\displaystyle V_\textrm{AMR}(S,t)\approx\max(V_\textrm{EUR}(S,t),\Lambda(S))$ (6.1.1#eq.1)

Discontinuities where the European payoff intersects the intrinsic value are in contradiction with the efficient market hypothesis: indeed, delta-hedging strategies exist from which risk-free profits can be made and arbitragers quickly smooth out the transition. This shows that the approximation (6.1.1#eq.1) is not in general sufficient for the pricing of American options, but it can nevertheless be useful when an explicit formula is needed.

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