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Please help me understand similarity and differences between local volatility and Stochastic Volatility both intuitively and mathematically.

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I'll try my best to explain them

Both of them aim to match the implied volatility surface as shown by the empirical data.

Local volatility process is a function of Stock and time without any stochastic term (not moving randomly). It changes with with different inputs of stock and time. It matches the implied volatility surface with short term maturity very well, but not well with option with long maturity.

Stochastic volatility is simply volatility that varies stochastically (it moves randomly). It can have mean reverting properties. It matches long term implied volatility surface well, but unable to replicate short term implied volatility skew or smile

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  • $\begingroup$ Thank you for your response. I have few follow up queries. Would we great if these get clarified. 1.Is local volatility a subset of Stochastic Volatility. 2. Do both local and stochastic vol have mean reverting properties or just Stochastic vol? 3. Is there any mathematical relation connecting the two types of volatility? $\endgroup$ – Ussu Oct 6 '19 at 10:48
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    $\begingroup$ I'll answer them according to the numbers you have set. 1. No, they are both different thing. In fact, there are such things called local stochastic volatility model. 2. I am almost surely that the answer is no. Only stochastic volatility has it. 3. Local stochastic volatility model and it's popular :) $\endgroup$ – Crushh Oct 6 '19 at 11:09
  • $\begingroup$ Thank you so much for clarifying things. Please let me know if i infer it correctly: 1. Here we are taking about volatility of the underlier and not the volatility of the option premium. 2. Volatility is a function of spot, time, interest rate , and various other factors. If all the factors that affect volatility are considered it is called Stochastic volatility and if only spot and time is considered it is called local volatility? $\endgroup$ – Ussu Oct 6 '19 at 11:34
  • $\begingroup$ I dont understand ur first question. For the second question: Local volatility can be actually function of spot tkme interest and other factors just without "stochastic" term. Stochastic volatility normally do not take in any other financial factors except volatility itself $\endgroup$ – Crushh Oct 6 '19 at 11:41
  • $\begingroup$ Thank you for your help. I have got a high level idea but am still not very clear and able to distinguish between the two. Any reading or video explanation would be really helpful. If you have any please feel free to share them. Thanks. $\endgroup$ – Ussu Oct 6 '19 at 20:47

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