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4.2.1 Binary options

[ SLIDE super-share - cash-or-nothing || VIDEO modem - LAN - DSL ]

Binary or digital options are a straight forward extension of a plain vanilla contract with a more general terminal payoff $ \Lambda(S)$ : as a consequence, the solution methods and the payoff enjoy many of the same features that have already been discussed for the put / call option. The VMARKET applet below shows the evolution of a super-share option for an increasing time to the expiry date.

VMARKET applet:  press Start/Stop to study the payoff dynamics V(S,t) for a super-share option. Change the SpotRate, Dividend and Volatility parameters and review the dependencies previously discussed in the case of vanilla instruments.

Virtual market experiments: exotic binary options
  1. Set Volatility=0 and keep a finite SpotRate, Dividend to show how large oscillations appear in the proximity of the sharp edges. These oscillations lead to negative option prices and have no financial justification: they are an artifact of the numerical solution and should be avoided.
  2. Modify the StrikePrice, Shape0 and Shape1 parameters defining the center, the height and the width of the box function to calculate the present value of a cash-or-nothing put option that pays EUR 1 if the underlying presently trading for 10 EUR rises to EUR 12 in six months time. Assume a 3% spot rate, 40% volatility and no dividend payment.
  3. Switch from SuperShr to VSpread and use the shaping parameters to reproduce the same payoff starting from a vertical spread option. Note that the payoff from a vertical spread can be replicated with the payoff from two plain vanilla options (exercise 2.05).

Apart from stretching the validity of the numerical method, the sharp edges in the terminal payoff do seriously question whether it is practically possible to perform the delta-hedging in order to eliminate the uncertainty close to the expiry date. Indeed, the option value jumps every time the underlying moves across an edge, so that a large number of underlying shares have to be bought / sold to keep the portfolio value deterministic. Transaction costs play an increasingly important role and have to be taken into account (exercise 3.05).

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