Static hedging

hlpsg · Dec 10, 2009

http://ideas.repec.org/p/wpa/wuwpfi/0409016.html

Wonder if anyone has read the paper "Static Hedging of Standard Options" by Peter Carr and Wu Liuren? Still making my way through the paper and was wondering if anyone has an idea of how it is setup and can give an example?

From what I can gather, it involves trading your primary portfolio in a back month, and the hedge is in the front month. You take off the whole trade when the front month hedge expires.

On how it works, this is what I can figure out but I could possibly be wrong.

You look at the gamma curve at expiration of the front month options, and for any point in that curve that is negative, you use the front months to hedge that gamma to neutral or positive gamma at all price points that have a negative gamma at expiration (of the front month).

However, when I tried doing this, it always ends up into a short theta trade.

Is there anyway to make this work for a long theta, short gamma trade? It has to correct, otherwise there'll be no need for this paper in the first place (long gamma positions don't really need to be hedged for big price moves).

hlpsg · Dec 11, 2009

Went to do some research yesterday and it seems static hedging is mentioned in some other books like Wilmott as well. Only there it's addressing exotic (barrier) options so I'm not sure if this paper is actually addressing vanilla options or...?

heech · Dec 12, 2009

The article clearly specifies they're dealing with vanilla European options.

I'm not really understanding it however, and would appreciate some hand-holding. Specifically, it states on page 2:

<i>"As the target optionâs future gamma does not vary with the passage of time or the change in the underlying price, the weights in the portfolio of shorter-term options are static over the life of these options."</i>

Why would the target option's future gamma *not* vary with passage of time or change in underlying price?

heech · Dec 12, 2009

Quote from heech:

Why would the target option's future gamma *not* vary with passage of time or change in underlying price?

Oh, page 8 states the theorem in more detail. I understand now. Each hedging option weighted based on gamma, assuming price is at its strike K.

Original post:

You look at the gamma curve at expiration of the front month options, and for any point in that curve that is negative, you use the front months to hedge that gamma to neutral or positive gamma at all price points that have a negative gamma at expiration (of the front month).

Not sure what you mean, re: negative gamma curve. Why would there be a negative gamma curve with a single option position?

The idea is replication, not a "hedge". You can replace a longer term option with a static set of shorter term options, weighted based on gamma at that strike. So the result for the portfolio should look just like that of a single option, in terms of gamma/theta. If you're short option being replicated, then the portfolio is also short gamma/long theta. If you're long option being replicated, then long gamma/short theta.

If you want to be short gamma/long theta, then sell the front-month options.

There's no arbitrage here, no reason to trade both short-term and long-term options... not unless you can sell the longer-term option for more than fair value, and you're just replicating the option until you can buy it back for "fair" value.

Did you look at the comparison to just basic daily delta-hedging with the underlying future? This is no better than that approach, except when large random jumps make delta-hedging more error-prone.

hlpsg · Dec 12, 2009

Quote from heech:

The article clearly specifies they're dealing with vanilla European options.

I'm not really understanding it however, and would appreciate some hand-holding. Specifically, it states on page 2:

<i>"As the target optionâs future gamma does not vary with the passage of time or the change in the underlying price, the weights in the portfolio of shorter-term options are static over the life of these options."</i>

Why would the target option's future gamma *not* vary with passage of time or change in underlying price?

I think it's because it's referring to the target (i.e. portfolio) option's FUTURE gamma. Not it's instantaneous gamma but the gamma when the front month hedges expire I think.

Hence the weights of shorter-term options will also be static because you calculate those weights based on the future gamma (at expiration of the hedging options) of the target option.

heech · Dec 12, 2009

Quote from hlpsg:

Hence the weights of shorter-term options will also be static because you calculate those weights based on the future gamma (at expiration of the hedging options) of the target option.

I just didn't understand how gamma could be independent of the underlying price. And what they're saying is, independent of the underlying price at time t = 0, by assuming price = K at expiration. And with enough options at different strikes, you have all possibilities covered.

I don't see how a retail trader could use this info. Maybe Buffett is using it for his 15 year puts!

hlpsg · Dec 12, 2009

Not sure what you mean, re: negative gamma curve. Why would there be a negative gamma curve with a single option position?

There is a negative gamma curve even with a single option because I make the assumption that they are selling options in the target backmonth portfolio, not buying them. If you were buying options, there won't be a need for static hedging, because if you look towards the end of the paper (p51 onwards), they do a comparative study using Monte Carlo simulation of dynamic vs. static hedging under the Merton Jump-Diffusion Model.

I could be totally wrong on my assumptions however.

The idea is replication, not a "hedge". You can replace a longer term option with a static set of shorter term options, weighted based on gamma at that strike. So the result for the portfolio should look just like that of a single option, in terms of gamma/theta. If you're short option being replicated, then the portfolio is also short gamma/long theta. If you're long option being replicated, then long gamma/short theta.

If you want to be short gamma/long theta, then sell the front-month options.

Oh...I thought they were proposing a method of hedging the back month options by using a variety of front month options at different strikes? No?

What would be the practical usage of replicating a back month option with front months, wrt hedging? In my mind, it'll just incur more transaction costs, and give no practical benefits? Could you point out which page talks about this replication?

There's no arbitrage here, no reason to trade both short-term and long-term options... not unless you can sell the longer-term option for more than fair value, and you're just replicating the option until you can buy it back for "fair" value.

Did you look at the comparison to just basic daily delta-hedging with the underlying future? This is no better than that approach, except when large random jumps make delta-hedging more error-prone.

Yes I was under the idea that this paper was proposing a method to hedge a portfolio in a superior way to dynamic hedging.

It is superior because of a couple of reasons stated in the paper:
1) Hedging costs (commissions, slippage). Static hedging is one time, incurring one commission and one slippage, whereas dynamic hedging incurs far more.

2) More importantly, hedging robustness - dynamic hedging is obviously not robust because of market gaps, liquidity holes and other practical issues.

hlpsg · Dec 12, 2009

Quote from heech:

I just didn't understand how gamma could be independent of the underlying price. And what they're saying is, independent of the underlying price at time t = 0, by assuming price = K at expiration. And with enough options at different strikes, you have all possibilities covered.

I don't see how a retail trader could use this info. Maybe Buffett is using it for his 15 year puts!

which page is this? perhaps we can look at it together and shed some light on it.

heech · Dec 12, 2009

hlpsp,

The paper talks at length about delta-hedging. If you've looked at how Black-Scholes is derived (or at least one of the more common ways, and the only one that I truly understand)... it's by assuming an arbitrage-less environment, and by comparing an option to the cost of an equivalent "delta-hedged" portfolio.

Ie, on one hand, you have the option. On the other hand, you have an identical delta-hedged porfolio consisting of the underlying + cash. You assume the cost of the option is equivalent to the cost of synthetically replicating the option... and then calculate the latter to get the former.

So, the idea in this paper, is that this is a static-hedging alternative to the dynamically delta-hedged portfolio. But the idea is the same, this is a replication of the longer-term option.

As far as why is this useful? It talks about a scenario in the paper where longer-term options might be less liquid, therefore more expensive because of ask/bid spread if nothing else. So, instead of buying (or selling) the longer-term option, you use the more liquid near-term option. You can then roll forward after expiration, and you eventually end up saving money versus the full longer-term option. That's why I mentioned the Warren Buffet 15-year puts.

heech · Dec 12, 2009

Quote from hlpsg:

which page is this? perhaps we can look at it together and shed some light on it.

I believe this is on page 8, the section where they're discussing the proof of their theorem, and the context of the proof.