Estimating Volatilities and Correlations for a Three-Asset FX Model
Summary
The document offers historical volatility and correlation estimates as starting points for a foreign currency option model with domestic equities, foreign equities, and an exchange rate. Using weekly observations over five years for the DAX, S&P, and EUR measured against the US dollar, the cited answer reports annualized volatilities of 18%, 12%, and 8.8%, respectively. It also reports correlations of −0.23 between EUR and DAX, −0.05 between EUR and S&P, and 0.725 between the two equity indices.
The estimates illustrate that equity indices can move together while currency-to-equity relationships are weaker and may be negative. The answer offers rough generalizations: major currencies may be less volatile than stock indices, and individual stocks may be more volatile than indices. These figures are sample- and instrument-specific, however; they are not universal parameters or a complete calibration procedure. The document does not resolve the question about why a pricing formula approaches zero for some volatility-vector choices, so model assumptions and implementation still need review.
Key ideas
- The cited estimates use weekly data over five years for two equity indices and the EUR exchange rate.
- The reported annualized volatilities are 18% for DAX, 12% for S&P, and 8.8% for EUR.
- The two equity indices have a reported correlation of 0.725, while EUR correlations with them are weaker and negative.
- The figures are historical examples and should not be treated as universal model inputs.
- The document does not diagnose the pricing function’s sensitivity to volatility vectors.
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Full text
# What are some reasonable parameters with three Wiener processes? # What are some reasonable parameters with three Wiener processes? In a foreign currency model, domestic and foreign stocks + exchange rate is modelled via 3 Wiener processes. I am trying to price options in this model, however, I am unsure what some realistic values for the volatility vectors are? I am noticing in particular that the value of my price ranges wildly (mainly it goes to 0) for certain volatility choices. This is because the pricing function contains an exponential to the power of the negative of a dot product of volatility vectors.... so a poor choice of volatilities leads to an incredibly low value due to this exponential factor. ## Answer by nbbo2 (score 1) https://quant.stackexchange.com/a/32655 Using weekly data for the last 5 years for DAX (german equity index), S&P (US equity index) and EUR (value of "german" currency priced in US dollars) we have the following results: EUR fluctuates with an annualized volatility of 8.8% a year, DAX with vol of 18% a year and S&P vol 12% a year. In my experience it is generally true that a currency is less volatile than a major stock index (as a rough guide about half as volatile). I am a little surprised by how low S&P vol has been, my guess would have been 14 or 15%; DAX typically a little more volatile than S&P and this is confirmed. The correlations of the currency with the two stock indexes are as follows: EUR with DAX is -0.23. This is in agreement with a general observation I/many people made over the years: when American investors see a foreign stock market go up, they usually see the currency of that country go down and vice versa (not always true of course). Between EUR and S&P correlation is -0.05, perhaps a little surprising to me, I would have expected it slightly positive. But generally these correlations are between -0.25 and 0.25 in my experience, i.e. not very large. [I left out the correlation between DAX and S&P: it is 0.725. Major world stock markets are always positively correlated]. Of course for a stock (as opposed to a stock index) the volatilities would be much higher (maybe double).
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