# Construction of stochastic volatility model from a given local volatility model

The Gyongy's theorem:

Let $$X_t$$ be a stochastic process satisfying $$dX_t = \mu_t dt+\sigma_tdW_t$$ where $$\mu_t, \sigma_t$$ are bounded stochastic process adapted to the filtration $$\mathcal{F}_t$$. Then there exists a stochastic differential equation $$dY_t = b(t,Y_t) +s(t,Y_t)dW_t$$ such that $$Y_t$$ and $$X_t$$ have the same probability distribution for every $$t$$. In addition, the two functions $$b(t,y)$$ and $$s(t,y)$$ satisfy $$b(t,y) =\mathbb{E}(\mu_t|X_t=y)$$ $$s^2(t,y) = \mathbb{E}(\sigma_t^2|X_t=y)$$

This beautiful theorem gives us a method to construct an equivalent local volatility model (i.e. the model of the process $$Y_t$$) from a given stochastic volatility model (i.e. the model of the process $$X_t$$).

I would like to know whether there exists a method to construct an equivalent stochastic volatility model from a given local volatility model. In other words, given the process $$Y_t$$, can we construct the process $$X_t$$ such that $$X_t$$ and $$Y_t$$ have the same probability distribution for every $$t$$?

If there are LVMs that can not be generated from any SVM, I would like to know their characteristics?

PS: we may have $$(\mu_t, \sigma_t) = (b(t,X_t), s(t,X_t))$$, but this result is too trivial. Is there a way to contruct $$(\mu_t, \sigma_t)$$ such that they follow some stochastic differential equations that are not directly related to $$X_t$$? (for example, $$\sigma_t$$ follows a CIR model)

• Great question, have often wondered this myself. The answer I suspect is no, but would love to see somebody show / prove this. Jun 15 at 22:36
• Why is what you describe in PS too trivial? What you are trying to construct instead boils down to matching the distribution of $s(t,X_t)$ with an SDE where another $\sigma_t$ and $X_t$ are only correlated and not related by a possibly nonlinear function. Unlikely in my opinion. There is also a lot of literature about the different smile dynamics of local volatility vs. stochastic vol models. These issues would not exist if your proposal worked. Jun 16 at 10:01
• @KurtG. If what you said is true, the class of LVM must be larger than the class of SVM. We can split the LVM class into two sub-classes: the first sub-class contains LVM that are generated by some SVM (via the Gyongy's theorem), the second sub-class contains the remaing LVM (that cannot be constructed from any SVM).  In this case, my question is "what are the characteristics of these two sub-classes of LVM"? In other words, given a LVM, how do we know whether it is generated by a SVM?
– NN2
Jun 18 at 6:21
• I said mainly the same what you said yourself: for the reverse LVM$\to$SVM the solution we are seeking is $\sigma_t=s(t,X_t)\,.$ It is trivial and imho not too trivial to be true. Jun 18 at 12:34
• @KurtG. no, what I’m seeking is $\sigma_t = \sigma(t, B_t,…)$ which is not directy a function of $X_t$ (whether the solution $\sigma_t=s(t,X_t)$ which depends directly on $X_t$ is trivial or not, it is not important here and so not in the question).  Besides, I modified the question by asking whether there exists any conditions such that the reverse LVM to SVM does ( or does not) have solution.
– NN2
Jun 18 at 13:03