First of all, I inform you that I am not a financial mathematician and have vague knowledge about an incomplete market.

Stochastic volatility models are incomplete so derivatives cannot be replicated by a single risky asset. But by introducing additional risky assets, we can extend the incomplete model into a complete model. For example, the Heston model is complete by adding a variance derivative.

  1. At this point, my first question arises. As far as I understand, the requirement of additional hedging instrument is the dependence of risk randomnesses. So, By additionally trading a single call written on the same underlying asset, it is able to make the Heston model complete since the option value definitely depends on both random sources. Am I right?

  2. Let's back to a jump model. In contrast with stochastic volatility models, I have seen that a jump-diffusion model is complete only when a continuum of calls and puts is tradable. Why?? For example, Merton's jump-diffusion has three random components: i) Brownian motion for a diffusion part ii) Poisson process for counting jumps iii) Normal distribution for the size of random jumps. My intuitive answer is that the number of jumps can be arbitrarily large and each jump generates identical but independent different random sources. If my intuition is right, we can hedge a derivative within a fixed error under a jump-diffusion model by adding a finite number of calls and puts?

  • $\begingroup$ Where have you read about the continuum of calls and puts for jump-diffusion models? $\endgroup$ – Daneel Olivaw Nov 13 '18 at 10:02
  • $\begingroup$ @DaneelOlivaw link Please, look for the answer by AFK. He/She said "You would need to add an infinite number of derivatives to make the market complete which is absurd. ". And I interpreted an infinite number of derivatives as the continuum of puts and calls. $\endgroup$ – user155214 Nov 13 '18 at 10:24

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