Valuation of Renminbi options using a flucutation-dissipation model

The issue of the valuation of the Renminbi is a very strongly discussed topic. There has been a lot of

How to hedge
You can hedge renminbi options using currency forwards rather than the underlying.

The basic model
Problem, how do you get volatility for Black Scholes. Answer: use fluctuation-dissipation theorem.

The dynamics of the RMB is

$$ \frac{1}{p_0}\frac{dp}{dt} = -\Gamma p + f(t) $$

where $$p$$ is the difference between the price of the renminbi equilibrium price which is derived from currency forwards and the current price, $$\Gamma$$ represents a constant which accounts for the People's Bank of China intervention that prevents the RMB from reaching its equlibrium value, and f(t) is a stochastic process. p_0 is the average level of the RMB which we are setting constant.

Intuitively our model corresponds to an ant being tied to a piece of elastic. The ant moves in random motions, however because of the RMB peg, there is a force that opposes the motion of the the ant.

Following the treatment in Deserno (2004), we have the Green function for the homogenous equation

$$ p_g(t) = \mathbb{I}e^{-\Gamma t}\Theta(t) $$

Convolving this Green function with the stochastic force we get

$$ \begin{align} p(t) & = | p_g \star f|(t)\\ & = \int_{-\infty}^{\infty}dt^{\prime}\Theta(t^{\prime})e^{-\Gamma t}\mathbb{I}f(t-t^{\prime})\\ & = \int_{0}^{\infty}dt^{\prime}e^{-\Gamma t}f(t-t^{\prime}) \end{align} $$

Given the noise factor

$$ \begin{align} \langle f(t) \rangle & = 0\\ \langle f(t_1) f(t_2) \rangle & = \sigma^2(t_1 - t_2) \end{align} $$

Calculating the expection value of p we find

$$ \begin{align} \langle p^2 \rangle & = \langle \int_0^\infty dt_1 e^{-\Gamma t_1} f(t-t_1) \int_0^\infty dt_2 e^{-\Gamma t_2} f(t-t_2)\rangle\\ & = \int_0^{\infty}dt_1 \int_0^{\infty} dt_2 e^{-\Gamma(t_1+t_2)} C(t_1 - t_2). \end{align} $$

This gives you

$$ p^2 = \frac{\sigma^2}{\Gamma} $$

Analysis
Here should be a chart of the expected value of the RMB has described by RMB futures and the implied volatility of options. They should be identical. Cite Zhang's masters thesis here since he has one chart that shows the correlation before 7/22.

Public policy implications
This model suggests that it would be unwise for the People's Bank of China to suddenly stop intervention in the currency markets. PBC intervention acts as a frictional force which dampened currency volatility. If this damping force were removed, the volatility of the market would increase cause widespread instability.