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Does the volatility of a Currency Pair depend on the currency in which the premium is paid? For example- will the Volatility of USDJPY change if the premium is paid in USD instead of JPY. Is there any mathematical formulation for this?

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It depends, on what you mean by returns. For simple returns: no, for log returns yes. To recap, simple returns are given by $$R_\textrm{simple} = \frac{P_{t+1}}{P_t}-1$$ and log returns are given by $$R_\textrm{log} = \log \left(\frac{P_{t+1}}{P_t}\right).$$ The rate of change is given by $$R = \frac{P_{t+1}}{P_t}.$$

A percentage increase in one currency of a pair, implies a decrease in the other of the same size, so $$R^\textrm{USDJPY} = \frac{P_{t+1}}{P_t} = x$$ implies $$R^\textrm{JPYUSD}\frac{P'_{t+1}}{P'_t} = \frac{1}{x}$$ where $P'_t$ is the reverse rate.

In words, if EURUSD is trading at 1.20 today and at 1.212 tomorrow the return from a USD perspective is $1.212 / 1.20 - 1 = 1\%$ as today the USD holder was holding 120 cents of USD and tomorrow he would be holding 1.212 cents of USD. On the other hand, from a EUR perspective the loss is $1.20 / 1.212 - 1 = -0.99\%$.

We can now do a simple experiment to get a feeling of the volatility for these types of returns in R:

> # Simple returns
> set.seed(1)
> returns <- rnorm(10, 1, 0.01) # One added back to R_simple
> returns
 [1] 0.9937355 1.0018364 0.9916437 1.0159528 1.0032951 0.9917953 1.0048743
 [8] 1.0073832 1.0057578 0.9969461
> sd(returns - 1)
[1] 0.00780586
> sd(1/returns - 1)
[1] 0.007769419

Clearly, the volatility of simple returns is not the same. Using the same sample suggests that the volatility of the log returns is equal:

> sd(log(returns))
[1] 0.0077874
> sd(log(1/returns))
[1] 0.0077874

This can be shown to always hold with $x$ defined as above. The log returns for $P_t$ and $P'_t$ are then given by $\log(x)$ and \begin{align} \log(1/x) &=\log{1} - \log{x} \\ &= -\log{x} \end{align}

The standard deviation of sample is equal to standard deviation of the mirrored around its mean.

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  • $\begingroup$ I think in your last code block, you meant to type sd(log(1/returns)) for the second command -- though that does not affect the answer. $\endgroup$ – kurtosis Jul 31 '20 at 21:56
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    $\begingroup$ Thanks, fixed it! $\endgroup$ – Bob Jansen Aug 1 '20 at 5:39
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If you're modelling the FX rate as a geometric brownian motion and asking whether the volatility depends on whether you model the rate or the inverse rate, then the answer is no - and we can demonstrate it using Ito's lemma

Assuming the rate $X$ obeys \begin{align} {\frac {dX} X} = rdt + \sigma dW \end{align}

for some rate $r$ and volatility $\sigma$, lemma says that for a function $f(X,t)$

\begin{align} df = \Bigl( {\frac {\partial f} {\partial t}} + r X {\frac {\partial f} {\partial X}} + {\frac {\sigma^2 X^2} 2} {\frac {\partial^2 f} {\partial X^2}} \Bigr) dt + \sigma {\frac {\partial f} {\partial X}} dW \end{align}

Substituting in $f(X,t) = {\frac 1 X}$, we get

\begin{align} d{\frac 1 X} &= \Bigl( rX {\frac {-1} {X^2}} + {\frac {\sigma^2 X^2 } 2} {\frac {2} {X^3}} \Bigr) dt - \sigma X {\frac {1} {X^2}} dW\\ &= - {\frac 1 X} \Bigl( \bigr(r - \sigma^2 \bigl) dt + \sigma dW\Bigr) \end{align}

So the inverse process ${\frac 1 X}$ also follows a geometric brownian motion, with a drift of $-r + \sigma^2$ and a volatility of $\sigma$ (ie. the same volatility as $X$)

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  • $\begingroup$ You are missing a final measure change here to bring everything into alignment - as the sde you have written is still under the domestic numeraire and not the foreign numeraire. A 1 year outright forward on USDEUR is just the inverse of 1 year outright on EURUSD. There is no convexity adjustment required. $\endgroup$ – river_rat Sep 16 '20 at 22:51
  • $\begingroup$ In domestic currency, for GBM the inverse contract is volatility-dependent. This shouldn't be so surprising - we need a vol correction term to make sure the expectations match, otherwise large-vol pairs would have inflated forwards due to the exponential. $\endgroup$ – StackG Sep 16 '20 at 23:53
  • $\begingroup$ Sure, as long as we all can agree that there is no fx volatility information available in fx forward prices. A USD investor looking at a EUR/USD fwd would see a drift of libor-euribor for example while a eur investor looking at a USD/EUR fwd would see a drift of euribor-libor. $\endgroup$ – river_rat Sep 17 '20 at 10:38
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    $\begingroup$ Agreed - the misleading post where I implied the opposite has been deleted. $\endgroup$ – StackG Sep 17 '20 at 10:45
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As a general rule of thumb, the price of a thing should not depend intrinsically in the units of value in question. Since quoting something in X per 1 unit of a base currency or 1/X per 1 unit of counter currency happen to be questions revolving around units, the price of an option should not depend on that choice. The entire area of measure change and looking at different numeraires is effectively this simple fact taken to its logical conclusion.

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