Hi all, one of the inputs required to correctly price any worst-of options (commonly seen in equity structure products) is the pairwise correlation between the assets.It might seem trivial but I'm having a hard time deciding what is the appropriate way to arrive at a correlation number for this purpose - non of the books I came across on exotic options and structured products really explains this, so any advice is greatly appreciated.

Let's use a simple example of a worst-of put option on AAPL and MSFT, expiry = 3 months, strike = ATM.

2 options I can think of:

1. simply measure the historical return correlation between AAPL and MSFT as of today, using a weekly frequency (to smooth out non-trading days and timing mismatch etc which is not a huge concern here but relevant for cross-market pairs) for a period T. Should T have to match 3 months? or is there merit in using something generic like 1y?

2. still measuring historical correlations, using weekly frequency, but this time take a longer historical data range and apply a rolling window of T. So let's say with 5 years of historical weekly return data, and a rolling window of 1y, I will have 4 years worth of historical correlation data. I can then take some kind of sample statistic from there? say median.

option 1 feels too noisy to me and is likely to suffer from a certain amount of lag - if I use a 1y period to price a 3m option the historical measure could be too slow moving to be robust?

option 2 seems to offer more information value and allows the trader to easily shift his bias based on the current regime - especially since correlation is bound by a range and mean-reverting similar to volatility, but the computation load is much higher especially for a industrial-scale implementation.

Both options are backward-looking and contain zero information about current market conditions. Given that there are readily available broad-market level implied correlation measures (such as those calculated by CBOE), is there a sound way to blend the forward-looking measures into the historical measures?

Both options are backward-looking and contain zero information about current market conditions. Given that there are readily available broad-market level implied correlation measures (such as those calculated by CBOE), is there a sound way to blend the forward-looking measures into the historical measures?

On this last question, say you have some historical correlation at each time t, call it [$]\rho^H_t[$] and the CBOE has their forward-looking average implied correlation [$]\rho^C_t[$]. Then, a predictor [$]\hat{\rho}[$] for the future period [$](t,t+T)[$] that blends both pieces of information is

(*) [$] \hat{\rho}(t,t+T) = a + b \, \rho^H_t + c \, \rho^C_t[$],

where (a,b,c) could be easily estimated from a regression.

Just thinking out loud, I would want to also incorporate into (*) the earnings release days that are expected in [$](t,t+T)[$] for the two stocks.

Also (*) might need some constraints or logs or something to keep it legal.

I like that question. I can’t quite muster the energy to come up with a good answer, but it challenges my world view that fixed income is a lot more complex than equities.

Both options are backward-looking and contain zero information about current market conditions. Given that there are readily available broad-market level implied correlation measures (such as those calculated by CBOE), is there a sound way to blend the forward-looking measures into the historical measures?

On this last question, say you have some historical correlation at each time t, call it [$]\rho^H_t[$] and the CBOE has their forward-looking average implied correlation [$]\rho^C_t[$]. Then, a predictor [$]\hat{\rho}[$] for the future period [$](t,t+T)[$] that blends both pieces of information is

(*) [$] \hat{\rho}(t,t+T) = a + b \, \rho^H_t + c \, \rho^C_t[$],

where (a,b,c) could be easily estimated from a regression.

Just thinking out loud, I would want to also incorporate into (*) the earnings release days that are expected in [$](t,t+T)[$] for the two stocks.

Thank you so much for the suggestion. I like the idea of regression and building in impacts of earnings days.

Would you have any thoughts on the choice of period T? What about if the option to be priced is subject to more frequent knock out / knock in observations?

You're welcome. T is just your option maturity -- 3 months in your example. But you could fit (*) for T - t = 1 day, 2 day, ..., 90 day, ... 365, say --  say, adding dummy variables for earnings release days. So if T - t = n days, the left-hand-side of the regression for (*) would be the historical (log) return cross-correlation [$]corr(x,y)[$] for returns across n days: [$]x_t = \log S_{t+n}/S_t[$].  The regression fits would yield fitted parameters for each horizon [$](a^{(n)}, b^{(n)},c^{(n)}, \vec{\delta}^{(n)})[$], where the delta's are the various dummies.