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I specifically want to know how to model a jump condition while valuing a derivative.Example :- the jumps which are observed in digital product payoffs, or barriers and knockouts.

Although a mathematical explanation for this would be fairly easy to provide , I'd be more interested in practical aspect of this condition.

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  • $\begingroup$ I don't think there is a general way to handle such payoffs, in particular, for path-dependent, such as barrier or autocallable options. For a simple European style digital, a geared call, or put, spread can be used. $\endgroup$ – Gordon Feb 18 '16 at 18:15
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There is nothing to model in the payoff. A payoff is a collection of cash flows. A cash flow is a function of market observables. Your function just happens to be discontinuous.

From a risk point of view this means that you are exposed to the volatility skew. So any model used for valuation should be calibrated to the volatility smile (you cannot simply value a digital striked at K by looking at the implied volatility at K, you also need the derivative at K of the implied vol wrt to the strike).

Where real work on the payoff needs to be done is when you try to apply standard numerical methods to value it. For example when using Monte Carlo and finite difference to compute Greeks, the discontinuity will generate a lot of noise. The most common solution is to smooth the payoff: replace it with a continuous function. Typically you replace a digital by a call spread but things get a lot more complicated when you have several types of digitals interacting and you want to your smoothing to be conservative (i.e. subreplicate your payoff).

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There are sufficient literature that deals with jump process. You may look at these papers:

  1. Financial modelling with jump processes, by Cont and Tankov (1975)

  2. Option pricing when underlying stock returns are discontinuous by Merton (1976)

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    $\begingroup$ The question is about jumps in derivative payoff, path-dependent dis-continuities. The papers you sited are about jumps in the underlying process, a different topic. $\endgroup$ – onlyvix.blogspot.com Feb 18 '16 at 18:49

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