SERVING THE QUANTITATIVE FINANCE COMMUNITY

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Mink
Topic Author
Posts: 16
Joined: May 14th, 2003, 11:08 pm

### Hedging an option with a correlated asset

Hi all,I posted a question to the student forum but got no answers, so perhaps it is technical enough to post here? The question:Suppose you have an option on Z with payoff P(Z). However, you can't trade Z; you can only trade H, which is correlated to Z. Let's say you know the joint probability distribution PDF(Z,H). What would be everyone's best guess re: numerical methods to value this puppy?Thanks everyone!

liefje
Posts: 86
Joined: September 26th, 2002, 2:11 pm

### Hedging an option with a correlated asset

Unless correlation is equal to one, otherwise you can't....How will you hedge the missing information???

Posts: 284
Joined: July 14th, 2002, 3:00 am

### Hedging an option with a correlated asset

As a first pass, get a handle on the problem via simulation: To proceed semi-blindly, treat a call on Z as having a delta of "Q"*delta_BS and do minimum variance hedging w/ H (hedge ratio = -correln * sig_Z/sig_H if memory serves). Run lots of simln paths and look at the distribution of your final, hedged p/l. Vary "Q" so as to minimize the spread in this distribn. Use some reasonable utility function to equate in expectation the distribution of future outcomes to a given premium today. Then PAY LESS / CHARGE MORE. Don't believe any closed form solution that's radically different from the above unless you know why the above is wrong(!).

ClosetChartist
Posts: 190
Joined: July 17th, 2003, 4:41 pm

### Hedging an option with a correlated asset

Perhaps we shouldn't take Mink's question quite so literally. You can value the option using a partially correlated instrument and you can't completely immunize the risk by hedging with a partially correlated asset. But you CAN partially hedge the option. Depending upon the asset model you have chosen, you can redefine: dZ = a(Z,H)*dH + random error Loosely speaking, you have converted your single asset option into a basket option written on H and Z-H. You can use this setup to hedge the H-portion of your option. This approach can be a bit delicate, however!- Identifying and parameterizing a good {Z H} model can be difficult.- If the correlation is "weak" you may dig a deep hole even though the math looks nice.- If the option form is "interesting" you may dig a much deeper hole.

Aaron
Posts: 6433
Joined: July 23rd, 2001, 3:46 pm

### Hedging an option with a correlated asset

This happens all the time. You write an option on a stock price, but all you can trade is the stock-plus-dividends. You write an derivative on an untradeable reference rate.In these cases the correlation is near one, so you hedge as if you can trade the underlying and make adjustments.If the correlation is farther from one, you might take ClosetChartist's solution, especially if you have reason to believe that Z - Beta*H has low volatility or can be diversified away. But I'd also consider other approaches. Is Z totally untradeable or just illiquid? Is there a larger structure? For example, suppose you had to hedge individual equity options using a broad index. That would be highly risky for an individual option, but for a basket of options that matched the index it would be riskless. For 20 randomly selected option of roughtly equal volatility, it would be reasonably low risk.

eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

### Hedging an option with a correlated asset

I agree with the simulation-based methodology, however I would probably compute VaR for the final loss and add it to the regular option premium.I would be interested to know if any reliable theoretical results have been derived.e.

Martingale
Posts: 511
Joined: November 15th, 2001, 7:54 pm

### Hedging an option with a correlated asset

Here is a paper maybe of some interest.http://www.ma.ic.ac.uk/~mdavis/docs/basisrisk.pdf

eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

### Hedging an option with a correlated asset

thank you martingale, I had a look at that paperit is indeed addressing interesting issues, however I doubt that an approach based on utility functions can be of any practical usee.

Johnny
Posts: 2431
Joined: October 18th, 2001, 3:26 pm

### Hedging an option with a correlated asset

There's only one place in finance where the "risk preference" term cancels out and that's where markets are complete. By construction this doesn't apply in this case. So any approach that doesn't take risk preferences into account (whether by utility function or some other way) is a guaranteed route to the poor house. Take care!

pobazee
Posts: 246
Joined: September 23rd, 2003, 5:32 pm

### Hedging an option with a correlated asset

Johnny:You’re correct, it is amazing how people despise utility function, and proudly say their model consistently values an asset of any type. The fact of the matter is if a model is arbitrage free (which is a market consistency requirement), then it implies, in finite dimensional case, a continuity of preference (investor prefer more wealth to less) which can be supported in equilibrium. While in practical modeling situation, we are always bent on dispensing with utility assumption, the fact remain that the vestige is in the black box. A choice among available investment is cast through the “invisible hand” of utility function.

Martingale
Posts: 511
Joined: November 15th, 2001, 7:54 pm

### Hedging an option with a correlated asset

Well, I was bumped into this while talking with someone on a interview (I am the interviewee), Guess what I got. I should have said, damn it, go ask Johnny :

Johnny
Posts: 2431
Joined: October 18th, 2001, 3:26 pm

### Hedging an option with a correlated asset

Martingale, if you didn't want the job you should have just told the interviewer straight out ...

eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

### Hedging an option with a correlated asset

QuoteOriginally posted by: JohnnyThere's only one place in finance where the "risk preference" term cancels out and that's where markets are complete. By construction this doesn't apply in this case. So any approach that doesn't take risk preferences into account (whether by utility function or some other way) is a guaranteed route to the poor house. Take care!I am not saying that the concept of risk preference is uninteresting. All I am saying is that utility functions are very unpractical, and that any paper based on this concept has a guaranteed route to the practioner's trash can.Furthermore, in the particular case of the paper by Mark Davis, the analysis applies only to payoffs with lower bounds, which disqualifies short calls, and although I did not look at the math in detail, I suspect that this assymetry entails different theoretical values for the option buyer and the option issuer, which would definitely be another annoying aspect.In any case I am not criticizing fundamental research. I just happen to be more interested in implementable research. The success of the Black-Scholes framework is that it relies on an implementable replication strategy.Cheers,e.

Johnny
Posts: 2431
Joined: October 18th, 2001, 3:26 pm

### Hedging an option with a correlated asset

I'm interested ... how do you price trades when you can't eliminate the risk, as in this case?

eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

### Hedging an option with a correlated asset

As suggested before, a practical approach could be to mark the option using BS, simulate your losses and take a reserve such as VaR to make sure you end up with a positive PL in the end e.g. 95% of the time. Nothing impressive from a theoretical point of view, but at least you don't have to make utility function assumptions. Obviously this can only work with a decently high correlation level, otherwise you are exposed to drift differential issues.e.