# Jim Gatheral's ansatz

In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes $$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$ where $$\hat w_t:=\int_0^t \hat v_s ds$$ is the expected total variance to time $$t$$, $$\hat v_s$$ is the unconditional expectation of the instantaneous variance $$v_s$$ at time $$s$$, $$x_s = \log(S_s/K)$$, and $$T$$ the time of the European option expiry.

Does anyone have a reference to a somewhat rigorous justification for this ansatz?

As shown below in the comments, Quantuple found the answer by Peter Friz, Slade then provided a newer version.