Aggregating greeks to portfolio level

Question

I have been asked to calculate/aggregate certain Greeks (delta, gamma, and vega) up to portfolio level for a portfolio consisting of a range of (long and short) equities, convertible bonds, and options -- the request has come from a risk manager in my team. I have little practical experience with risk management, and only an understanding of simple cases from theory, so I have three main questions about this request.

1. I've seen other answers on this site (e.g. here and here) asserting that it only makes sense to aggregate Greeks when the same underlying is concerned -- it isn't obvious to me why that should be the case. Can someone explain or refer me to the relevant theory? It seems to me reasonable to define a delta which characterises the change in price associated with a 1% movement in the underlying, and at an aggregate level would inform you about the sensitivity of the portfolio given every underlying moves 1%. While I may be inexperienced and naïve, I don't think the risk manager would have asked for this calculation if it doesn't make sense either.

2. For the delta, we were advised by our risk analytics system provider to use the delta-adjusted notional value for each position and security type, for which we can simply divide by the total notional market value for the portfolio and sum across positions to achieve a portfolio level delta. This approach assumes the 'contract delta' for the convertibles, and it also gives a delta=1 for equities as we would expect. The contract delta tells you about relative changes and is defined as $$\textrm{Contract delta} = \frac{(\Delta V/V)}{(\Delta \pi/\pi)} = \left(\frac{\% \textrm{ change in CB value}}{1\% \textrm{ change in parity}}\right), $$ where $V$ is the convertible bond value, $\pi = R\cdot S$ is the parity ($R$ is the conversion ratio, $S$ is the underlying price, and we ignore FX rates here).
   
   We were advised to take this approach as, we were told, only the contract delta aggregates meaningfully. I know, however, it is possible to modify the approach to yield the 'parity delta' for the convertible positions in the aggregation by an adjustment involving the conversion premium through the relation: $$\frac{\Delta V}{\Delta \pi} = \textrm{Parity delta} = \textrm{Contract delta}\times (1 + \textrm{Conversion premium %}). $$ I don't see why this would be any less meaningful to aggregate the parity delta than the contract delta -- does anyone with a better understanding see an issue? The reason we might want to do so is because the parity delta seems more relevant for risk management -- the contract delta is generally always smaller and seems like an underestimate, whereas the parity delta reflects the sensitivity in the bond value to movements in the underlying price which is what we want. (I welcome comments on the interpretation of the contract vs parity delta.)

3. I'm not sure how to go about aggregating vega across the portfolio: for the vega we have two forms available to us at a position level in our risk system -- an 'absolute vega' and an 'option vega' (defined as the expected dollar change of the option for a 1% change in the implied volatility of the underlying), with both types available for the options and converts positions. (We can ignore the equities for vega as they will contribute zero.) Is there a way to meaningfully combine these, given the different underlyings involved?

Answer 1

1. In a restricted sense, you can only add Greeks of the same underlying. The Greeks in the Black-Scholes-Merton model are partial derivatives of Option Price against its own variables (time to expiration and underlying price) or parameters (interest rate and volatility).

It really doesn't make much sense to ask "What should be the price of my Apple Call, if General Motors goes up by 1%?", agree?

I can't really say what your risk team is really imagining, but I would guess is one of bellow:

1.1 - They want a vector, not a single number: They want to know the Delta Exposure to each underlying, so you would reply "Our Apple Delta is \$5mm (delta of each position times the notional), Google is \$3mm, GM is \$1mm...."

1.2 - In financial institutions, people sometimes mix different concepts to get a ballpark figures. They might have the CAPM framework in mind. In CAPM model, stocks returns follows a linear regression with the benchmark:

$GM_{ret} = \beta_{GM} SP550_{ret} + \epsilon_{GM}$

The $\beta$ is the stock sensitivity to the SP500 and $\epsilon$ is a random variable with mean 0, uncorrelated with the Index.

They might want to know how the portfolio might behave with a 1% move in the SP500. This way, you should multiply the Greek of each underlying by its respective $\beta$ and notional and sum everything.

Is this number a good approximate what might happen with the portfolio? Well, in normal days probably yes, but in a turmoil...