Pricing Discontinuous Option Payoffs and Managing Numerical Noise
Summary
The document explains that discontinuities in digital, barrier, and knockout payoffs are features of the cash-flow function, rather than separate jump processes that must be modeled in the payoff itself. For valuation, the key market risk is exposure to the volatility skew. A digital option’s value depends on the volatility smile around its strike, including how implied volatility changes with strike, rather than only the volatility quoted exactly at the strike.
The practical difficulty arises in numerical valuation and risk calculations. Monte Carlo and finite-difference methods can produce noisy Greeks around a discontinuity. A common remedy is to smooth the payoff, such as approximating a digital with a call spread. With interacting digitals, smoothing becomes more complex, especially when the approximation must conservatively stay below the original payoff.
The discussion is qualitative: it provides no implementation, worked pricing example, or quantitative comparison of smoothing methods. It points to jump-process literature, but the accepted response focuses on payoff discontinuity and numerical treatment.
Key ideas
- A discontinuous payoff is a cash-flow function feature and does not itself require a separate jump model.
- Digital option valuation is sensitive to the volatility smile and its strike slope.
- Discontinuities can make Monte Carlo and finite-difference Greek estimates noisy.
- Smoothing, such as representing a digital with a call spread, can reduce numerical noise.
- Conservative smoothing becomes harder when several digital features interact.
Tags
Full text
# How to price jumps in payoffs # How to price jumps in payoffs 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. ## Answer by AFK (score 1, accepted) https://quant.stackexchange.com/a/24389 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). ## Answer by Neeraj (score 0) https://quant.stackexchange.com/a/24385 There are sufficient literature that deals with jump process. You may look at these papers: - Financial modelling with jump processes, by Cont and Tankov (1975) - Option pricing when underlying stock returns are discontinuous by Merton (1976)
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.