SERVING THE QUANTITATIVE FINANCE COMMUNITY

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

Hedging an option with a correlated asset

September 25th, 2003, 10:04 am

I can see the attraction of doing what you say. It's quick and easy and would allow you to form your initial pricing by finding the price that gives say 95% probability of no loss.However, this approach is exactly equivalent to assuming a stepped utility function, i.e. with a large negative utility for any loss and a small positive utility for any profit. So all your approach does is to choose an unrealistic utility function for you, without you realising what you're doing. The shape of the utility function that you're implicitly choosing means that you will always undervalue options compared to people using monotonically increasing functions (i.e. the whole of the rest of the market). Therefore you should only use your approach for buying options and never for selling them.Given that you are implicitly using a utility function and that your approach can only be justified for buying options and not for selling, I wonder why you don't just choose a utility function up front and be done?
Last edited by Johnny on September 24th, 2003, 10:00 pm, edited 1 time in total.
 
User avatar
eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

Hedging an option with a correlated asset

September 25th, 2003, 10:48 am

>>However, this approach is exactly equivalent to assuming a stepped utility functionDo you have a proof? I am interested>>you will always undervalue options compared to people using monotonically increasing functions sounds to me that the stepped function you describe is monotonically increasing... did you mean continuously increasing? or strictly increasing? or something else?Should my approach indeed be exactly equivalent to choosing a stepped utility function, I am also interested in a proof that, whichever monotonically increasing (or any appropriate assumption that would match the whole of the rest of the market) utility function you pick, the stepped function undervalues the option.But indeed the methodology I describe would only make sense for OTC transactions, not for market-making. I also think it actually makes more sense for an option issuer than an option buyer as the latter is guaranteed to make losses 95% of the time should he follow a delta-hedging strategy.Since you are apparently very knowledgeable about utility functions, do you know of any way to determine it in the market?Regs,e.
 
User avatar
Johnny
Posts: 2431
Joined: October 18th, 2001, 3:26 pm

Hedging an option with a correlated asset

September 25th, 2003, 11:26 am

You said that utility functions were "very unpractical". All I'm doing is pointing out that you too are assuming a utility function. It's just not a very good one. You asked for proof that your approach assumes a stepped utility function. Sure! This is not rocket science! In your approach you assume that a loss (any loss) is bad, i.e. has negative utility whilst a profit (any profit) is good, i.e. has positive utility. mmm ... that's it. There's your stepped utility function: {negative utility for losses, positive utility for profits}.You must do what you like. I know what works for me; something else may work for you.
 
User avatar
eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

Hedging an option with a correlated asset

September 25th, 2003, 12:03 pm

This is not rocket science indeed... I thought you were saying that BS + VaR(PL delta-hedging) = E(u(PL of something not clear))with u(x) = -alpha for x<0 and beta for x >0but apparently I did not estimate correctly the depth of your answer So you actually trade options on underlying hedged with a proxy using a model that has a utility function? And you actually make money? That's... fascinating!e.
 
User avatar
Johnny
Posts: 2431
Joined: October 18th, 2001, 3:26 pm

Hedging an option with a correlated asset

September 25th, 2003, 1:04 pm

I think we agree on the main issue, which is that you can't approach this just by a simple Black Scholes. You need to adjust for risk somehow, whether by your method of using VaR to aim for a 95% prob of not losing money, or by explicitly stating a utility function. The main point is that we agree that something needs to be done.The place where I have come across this problem most has been in hedging convertibles on stocks which are either thinly traded or which are hard to borrow. Whatever the motivation, you need to find some way of dealing with the additional risk and, crucially, of quantifying the risk so that you can work out if it's a trade you want to do.
 
User avatar
eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

Hedging an option with a correlated asset

September 25th, 2003, 1:31 pm

...yesmy point being that all sensible risk corrections do not reduce to a choice of a specific utility function, until proved otherwisein any case I'm unhappy with either the VaR or utility methods, the former being too simplistic and the latter being too, say, sophisticatede.
 
User avatar
kr
Posts: 1885
Joined: September 27th, 2002, 1:19 pm

Hedging an option with a correlated asset

September 25th, 2003, 2:12 pm

Ok, so hopefully we're now in agreement that any sellside business actually works this way... you say you are ripping the client's face off but mainly you are running a business of taking on risk in exchange for a big premium. Just like any other gambler then, you hope to do this all day long and reduce your variance and manage your deal size, so that in general you are improving your risk through the addition of more uncorrelated events, without being overexposed to any given trade. So you do this all afternoon...and then, what 'VaR' are you using? it's a marginal trade to your overall portfolio, right? what I'm getting at is that you can be a little more at ease over your choice of utility function when the distribution of risk is better-behaved... viewed in a one-off manner, of course you are going to be very risk-averse. But the card sharks are much more comfortable because they've been playing all afternoon. conclusion - the quote you offer to the market is a function of your strategy and your state (i.e. existing book).
 
User avatar
eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

Hedging an option with a correlated asset

September 25th, 2003, 2:42 pm

krI m not sure that I have understood all you've said, but one thing is clear: there is no such thing as 'client rip-off'. There are bids and offers, period.Also, a financial institution is not an insurance company and its role is not to mitigate losses. So, when issuing an illiquid security, there is absolutely no reason to be less risk-averse than a normal investor would be.That said, if a trade significantly reduces your overall risk, you will of course be more aggressive in your pricing. But this is about risk reduction, not reduction of risk aversion.e.
 
User avatar
kr
Posts: 1885
Joined: September 27th, 2002, 1:19 pm

Hedging an option with a correlated asset

September 25th, 2003, 3:20 pm

ok, so perhaps gentlemen use different language in public than they use behind closed doors... now that I am buyside I am a little more sensitive to what constitutes effort, when I'm paying for it. Just yesterday, we "offered" to deal with certain parties ourselves rather than let the counterparty do it, and this managed to get the phone ringing again and prices have come in half a "buck". Maybe there is a polite term for what happened, but my style tends to directness rather than politeness.A 'normal investor' gets into a position because they believe the vol will, ultimately and on average, go their way. Sellside has classically started from the position that the fee will remain on the books, provided that unforeseen events do not loom too large. That is, the dealer books were managed more from an accounting perspective than an investing one. I would say that this shows less risk-aversion from a sellside p.o.v., and the reason that this lower sensitivity persists is because of firmwide averaging, especially when the events are expected to be uncorrelated, and uncorrelated over time as well. Still, you've got it - if the trade reduces your risk, you might even give the client a small break on the fees and lock in a good thing rather than let it slip away to your competitor. As a concrete example, before CDS this was rarely the case for corp bond desks, because you had to run an inventory to keep the bar stocked for your clients. I knew a certain high-yield trading desk that managed to lose a really large amount of money this way not all that long ago. Taleb's 'Fooled' tells the same story about EM people as well.
 
User avatar
ClosetChartist
Posts: 190
Joined: July 17th, 2003, 4:41 pm

Hedging an option with a correlated asset

September 25th, 2003, 3:36 pm

Ladies and Gentlemen! What's all of this talk about risk preferences? Step right this way place your bets. Today we're offering:50:1 against Black-Scholes prices3:2 against LIBOR Market Model prices5:2 against Markovian Volatility processes20:3 against Gaussian innovationsWho's a taker? Who wants to win BIG! If you don't play, we can't pay!
 
User avatar
adannenberg
Posts: 284
Joined: July 14th, 2002, 3:00 am

Hedging an option with a correlated asset

September 25th, 2003, 3:47 pm

eiriamjh, what do you mean "one thing is clear: there is no such thing as 'client rip-off'. There are bids and offers, period." This is just naive or dumb. Every exotics trader, unless reined in by relationship managers, bosses or the law, will try to take out as much margin as he can (neglecting golden goose care and feeding, etc.). If the client is unsophisticated and can't figure out by model or by telephone what a given gizmo is really worth, then he may sell it to the exotics trader for much less than the trader would actually be willing to pay. This is obviously predatory on the part of the trader, i.e. a rip-off, but no one's constructed clear, quantitative guidelines for how much margin is too much. Does the trader have a fiduciary responsibility to the client, as an agent of the bank (say)? Or is it a free market and "let the buyer beware"? Are the rules different for different market participants? ...Bankers Trust got spanked hard many years ago, but there are still no clear rules of engagement, as far as I know. And without them, the traders job is to maximize his p/l (over a suitable time horizon). Just because the rules allow a rip-off doesn't change the fact that it's a rip-off...
Last edited by adannenberg on September 24th, 2003, 10:00 pm, edited 1 time in total.
 
User avatar
eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

Hedging an option with a correlated asset

September 25th, 2003, 4:13 pm

adannenbergI guess I'm just being a bit extreme and provocativemy point was to say that, for securities where the BS "fair value" does not apply (i.e. when there is no risk-free replication strategy), the price you should charge is really a risk premium, and since there are many ways of assessing risk, there is no telling whether an offer is a 'rip-off'. In the end it is a matter of market participants willing or not to take risks at certain prices.this doesn't mean that financial institutions have no responsibility in deciding whether a certain trade is suitable for a client - especially when a client would be exposed to unlimited downsideif there were to be a regulation to limit excessive charges for OTC transactions, I guess a possible idea could be to oblige the institution to show a rewind price on each transaction-but this idea will not bring you many friends e.
 
User avatar
Aaron
Posts: 6433
Joined: July 23rd, 2001, 3:46 pm

Hedging an option with a correlated asset

September 26th, 2003, 6:06 pm

I take an intermediate position. There are bids and offers and anyone who trades in the institutional market has to take care of herself. Being stupid (or "unsophisticated") doesn't make your broker or counterparty evil for making too much money. There are lots of good jobs for stupid people, you should get one of those rather than whining about how much money you lost your employers. Your counterparties are doing everyone a favor by speeding your exit. Someone is going to take your money, it might as well be them.However, there are common market practices that should be labeled fraudulent. If you provide your customer with a model or analysis, it should be accurate, not tailored to make money for you. If you offer your customer an opinion, it should be truthful, sincere and not couched to mislead. You don't have to tell your customers how much money you make from their trades, but you shouldn't lie about it either.There is no essential difference between financial intermediaries and professional gamblers. Neither one gambles. For some games you take formal risk, but diversify it away (say, roulette in a casino) with limits and other protections to ensure diversification is sufficient. When you cannot diversify, say for a sports book, you balance the book so you are hedged. Where you cannot diversify or hedge, you lay off.
 
User avatar
trc
Posts: 95
Joined: April 4th, 2002, 2:28 pm

Hedging an option with a correlated asset

September 28th, 2003, 6:20 am

Lets denote the option payoff by H and the assets by S and X respectively and assume the option expires at time T.I am assuming that you mean the payoff H=h(S) is a function of the entire path t->S(t) since obviously path dependent option payoffs are not deterministic functions of the asset price S(T) at expiration only.Now let F_T denote the sigma field generated by the process t->S(t) up to time T.A random variable H is a deterministic function H=h(path(S)) if and only if H is F_T-measurable.So if the second asset price process X generates the same sigma field F_T as S does thenH=g(path(X))is another deterministic function of the path t->X(t). What you might not like is that whilethe first representation H=h(path(S)) might be how the option is defined, the second representationH=g(path(X)) is typically not known. Moreover if the option is path independent "as an option on S"H=h(S(T))(depends only on S at expiry) the same might not be true as an option on X.For example often you can represent S(T) as a stochastic integralS(T)=integral_0^T v(s)dX(s)and then H=h(S(T))=h( integral_0^T v(s)dX(s) )(path independent as an option on S but path dependent as an option on X).Purely mathematically there really is no unique "underlying".There are only options H (F_T-measurable random variables) and assets S.The option H can be replicated by trading in S if we have a representationH=c+integral_0^T u(t)dS(t)which is a very special explicit way of writing H as a deterministic function of thepath t->S(t). The valuation is by the martingale pricing formulac_t = E_t[ H ]with no explicit connection to the assets S. The assets S only determine the equivalent martingalemeasure (simplistically speaking since there might be more than one).Quite possibly this same option can be replicated by trading in Xalso regardless of how S and X are correlated. The correlation between S and X is not what is important here.Easy example: single asset marketdS(t)=sigma(t)S(t)dW(t).with W a Brownian motion on a the probability space (O,F_T,P). If the volatility process is bounded and bounded away from zero0<e<=|sigma(t)|<=K (*)then every P-integrable random variable H can be represented asH=E(H)+integral_0^T u(t)dS(t)(replicated by trading in S). Note that the condition is P-integrability not any explicit connectionto the process S. Consequently we can replace S with any other process X satisfyingdX(t)=nu(t)X(t)dW(t)with volatility nu(t) satisfying (*) and H will be replicable by trading in X.I understand where the confusion comes from.In the frequent case where the option payoff H=h(S(T)) is path independent we can hope that the price process c_t has the formc_t=C(t,S(t))(itself only dependent on the value S(t) and not the whole path of S up to time t).In this case we can then consider the deterministic function C(t,s) and the DELTASdC(t,s)/ds evaluated at s=S(t)and try delta hedging. In general the deltas are not defined so there is no delta hedging.A reasonable procedure of finding a hedge in general is to minimize the hedge variance between hedge trades.There is an easy solution to finding the optimal hedge weigths for each hedge trade.(download the book from http://martingale.berlios.de, chapter 5, p117). The equations defining the variance minimizing hedge weights (equation 5.13, p129 for the single asset case, equation 5.40, p150 for the basket case) just take H as a random variable regardlesswether H=H(S(T)) or not and compute the hedge weights with respect to the assets S.You simply replace S with X and get the hedge weights with respect to X.In the martingale probability minimizing the variance over each hedge trade also minimizes the total hedge variance.In the market probability the situation is considerably more complicated.There may not be a variance minimizing hedge at all.
Last edited by trc on September 27th, 2003, 10:00 pm, edited 1 time in total.
ABOUT WILMOTT

PW by JB

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...


Twitter LinkedIn Instagram

JOBS BOARD

JOBS BOARD

Looking for a quant job, risk, algo trading,...? Browse jobs here...


GZIP: On