How do you price an option on multiple things>

Question

Suppose, for simplicity, I want to cover the U.S. stock market by buying ETFs for the Russell 1000 and Russell 2000. But I want to overweight small cap, so the Russell 3000 won't do. Also, let's assume none of these ETFs has options on it, as I am more interested in the delta hedging practice.

Let's also assume I want to protect my whole portfolio against a drop of more than 10%. And I have enough money to be able to go to an investment bank and get a custom-tailored swap to make this happen.

Now the obvious and easy thing to do is for them to create two delta hedged portfolios, using the ETFs to get an exact hedge. But that seems like its overly costly to me, because I was only concerned about the total portfolio dropping more than 10%.

If they did two portfolios, and the R1000 is down 4% (and is the bigger holding), while the R2000 is down 14%, I'm going to get a payoff on the R2000 option. But my total portfolio was down only maybe 6%, so, in essence, I paid too much for the protection I wanted: I only care if the total portfolio drops more than 10%.

Can that be delta-hedged with one portfolio, and if so, how would it be done? And would it indeed be cheaper?

Answer 1

1. Multiple "things" = Multi-Asset option. Based on your description it seems to me that what you need is a 90% basket put option. Call 3-4 banks and ask them to price it, pick the best price. It's far from trivial to price a basket option.

2. Delta-hedging does not have the same effect as an option. Rather, it neutralizes moves in the spot prices, but leaves volatility and correlation open, because Black-Scholes assumptions are always going to be violated in practice. I am referring to Black-Scholes because I am not sure whether you are already thinking of (L)SV modelling.

See for example this thread for intro to basket options pricing.