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Dynamic Delta Hedging

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I know that there are a few of you out there that know about these techniques. Could someone give me a nice introductory example, or a good reference (nothing very useful coming up on google)

Many Thanks

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I know that there are a few of you out there that know about these techniques. Could someone give me a nice introductory example, or a good reference (nothing very useful coming up on google)

Many Thanks

It's under Black Scholes. There's nothing very simple about it, Hull is the standard reference.

or look at:

http://en.wikipedia.org/wiki/Delta_hedging

The dynamic part just means that you have to continually adjust the amounts of securities you hold in the hedge as prices change.

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It's under Black Scholes.

It's related to Black Scholes... though I'd say BS is used to price options - while delta hedging is a strategy investment banks use to manage risk.

Delta is one of the "Greeks" - which, in turn are financial metrics. The "Greeks" arise when someone has written a futures contract or an option, and needs to manage its exposure to the risk that generates. Several assumptions are made about prices in general - including that markets are liquid and market prices transition smoothly... yadda yadda... these are the same sort of assumptions made by Black Scholes.

If we assume that we're hedging the net exposure (by way of options contracts we've written) to - say - IBM shares... and we assume we have a liability if the price of IBM shares go up, but make profit when they go down... we might sit staring at a feed of IBM share prices - and try to time it just-right so that we only own IBM shares when the price of IBM shares is moving against our interests. This means that unwanted risks can be mitigated at a cost that is substantially lower than the fees we charged when we engaged in trade with our customers... leading to profits.

Delta refers to the ratio of the price of the option to the price of the underlying (price of the IBM option to the price of the IBM share - for example) and can be positive and negative. Deltas can be positive or negative - and can be computed for portfolios... when a portfolio has a delta of zero, it is called delta hedged... and will typically generate far smaller profit/loss than a portfolio that is not delta hedged. Delta hedging is usually the first step to fully hedge a portfolio... once this is done, tweaks can be done to attempt to get a zero sum of the other greeks - such as:

Beta (ratio of volatility of price from the option to the underlying)

Gamma (ratio of rate of change of price from the option to the underlying)

Vega (rate of change of price of option with respect to volatility of the underlying)

Rho (rate of change of price of option with respect to interest rates)

Theta (rate of change of price of option over time)

Essentially, it's a game of finding almost equal and opposite bets - preferably as cheaply as possible. If the basis of this strategy was perfect, and someone achieved a perfectly hedged position, they would have eliminated all risk. In practice, this doesn't really happen - not least of all because the assumption that price transitions are smooth is a fantasy... but also because even if two numeric series of price data show perfect historic correlation, any unexpected event might introduce arbitrary divergence between assets and liabilities within a portfolio. The "dynamic" bit, essentially, is frantically trying to keep on top of every change as it happens - with trading-monkeys glued to computer terminals tasked with being faster than each other to avoid exposure to risk as it emerges.

I strongly suspect that this sort of hedging was employed in the context of purely financial assets - both in banks and within the secretive shadow banking system (hedge funds, etc.) It is my opinion that this likely played a significant part in the context of risk management prior to the credit crunch - with the freezing of wholesale credit markets and the dramatic widening of CDS spreads. Essentially, and in short, I think that a belief in the ability to hedge risk (and hence avoid needing to hold costly capital) was instrumental in bringing credit spreads down to minuscule levels on the run-up to 2007... and then, when an unanticipated shock arose, the system was extremely fragile... and the systemic effect was to prevent most/all market participants from hedging their risk as they previously believed they had. This lead to a widespread capital crisis (the only other way to mitigate risk is to have the funds to permit write-downs without reaching insolvency.) This caused banks globally to draw back from accepting any new risk - and to fight for survival... but, amusingly, this withdrawal from lending (which is the only sane strategy for pretty-much every institution individually) causes a systemic credit crisis... whereupon... the value of assets declines (since there is less leverage available to buyers... leading to a deflationary spiral... all be it with every effort of every government and central bank being to embark (for the first time) on epic "counter-cyclical" (i.e. against the prevailing trend) policies. What's playing out now is a global experiment to establish if the central banks are capable of halting a deflationary spiral through (admittedly drastic) monetary and fiscal policies. I don't think they will be entirely successful - though, of course, because there's only one global economy, it will be very difficult to be objective about their success or otherwise... since we won't know what outcome would be found for alternative policies.

The book that taught me about the Greeks (and explains it better than I have) is by Hull:

http://www.amazon.co.uk/Options-Futures-Ot...n/dp/0131499084

I found it a fascinating read... partly because it re-focused my attention on mathematical principles I'd last used at A-level - in a practical context. I don't share Hull's views about - erm - probably - anything... though he explains what are lots of allegedly complex concepts in finance with a helpful clarity - and targets those with little or no financial experience - starting with real basics. I bought the book as I understood that it is/was the core text for a Masters in Finance at Harvard... which, I suspect, means that it gives an insight into the intellectual lingua franca of today's practitioners. To my mind, some of Hull's flippant dismissals are a bit scary... but, if this is what the educated learn to become qualified, it comes as no surprise that they don't ask deeper questions when they are holding down high pressure jobs.

Edited by A.steve

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I found it a fascinating read... partly because it re-focused my attention on mathematical principles I'd last used at A-level - in a practical context. I don't share Hull's views about -erm - probably - anything... though he explains what are lots of allegedly complex concepts in finance with a helpful clarity - and targets those with little or no financial experience - starting with real basics. I bought the book as I understood that it is/was the core text for a Masters in Finance at Harvard... which, I suspect, means that it gives an insight into the intellectuallingua franca of today's practitioners. To my mind, some of Hull's flippant dismissals are a bit scary... but, if this is what the educated learn to become qualified, it comes as no surprise that they don't ask deeper questions when they are holding down high pressure jobs.

Hull's dismissals aren't a statement of his opinion, only a requirement that to the extent that they are true then (in his view) the rest follows... If something dismissed from his theory ends up happening it does not mean his models have been invalidated.

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Delta hedging works only works reasonably well in markets that are trading continuously (i.e without large price jumps up and down) and with sufficient liquidity (i.e allowing instantaneous buyng and selling in any volume that may be required).

Neither of those conditions presently exist in financial markets. Imagine you were trying to delta hedge the RBS share price yesterday for instance. In practice, only futures and options broker/market makers can come close to being able to delta hedge efficiently.

As with all theoretical constructs in financal market they work only in normal market conditions and then only in theory.

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Hull's dismissals aren't a statement of his opinion, only a requirement that to the extent that they are true then (in his view) the rest follows... If something dismissed from his theory ends up happening it does not mean his models have been invalidated.

Which rather supports A.steves conclusions.

If I can paraphrase

If I have a theory about how many angels I can fit on a pinhead, then I can happily ignore the non-existance of angels because within my fantasy world where angels do exist, their 'pinhead area occupation delta' can be precisely calculated.

No wonder we're in such a mess. This is what happens when mathematicians are allowed to run things.

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It's under Black Scholes. There's nothing very simple about it, Hull is the standard reference.

or look at:

http://en.wikipedia.org/wiki/Delta_hedging

The dynamic part just means that you have to continually adjust the amounts of securities you hold in the hedge as prices change.

In a theoretical sense, delta hedging is completely and utterly trivial to anyone with an education that includes high school calculus...typical economics/finance where they make a huge thing out of a first order derivative (in the mathematical sense). The same goes for all the other "greeks". They are obfuscatory terms for first and second order derivatives (in the mathematical sense).

Delta hedging basically says that if you have a derivative (in the financial sense), the value of one unit of which is denoted V, of some underlying thing (share, cash, gold), the value of one unit denoted T, and you own an amount, A, of the derivative then you need -A dV/dT of the underlying thing so that the value of your portfolio of the derivative and the underlying does not change with small changes in the value of the underlying, T.

With issues of discreteness, transaction costs etc. the practical implementation of this is more taxing...and there is nothing to stop you hedging with things other than the underlying etc. but it is all based on a trivial observation that anyone doing A level maths should understand immediately.

The simplest example of when a normal(ish) person would want to use such an observation would be sizing a spreadbet to hedge the value of their gold portfolio or perhaps locking in a favourable rate exchange rate when they know they will be emigrating in 6 months time or similar.

Edited by D'oh

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Which rather supports A.steves conclusions.

If I can paraphrase

If I have a theory about how many angels I can fit on a pinhead, then I can happily ignore the non-existance of angels because within my fantasy world where angels do exist, their 'pinhead area occupation delta' can be precisely calculated.

No wonder we're in such a mess. This is what happens when mathematicians are allowed to run things.

If my theory assumes the non-existence of angels and angels do turn up it doesn't invalidate my theory given the express statement that it relies on the absence of angels.

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If my theory assumes the non-existence of angels and angels do turn up it doesn't invalidate my theory given the express statement that it relies on the absence of angels.

:-) I was careful not to get into unnecessary theological discussion.

I think Hull's discussion of the abstract concepts to be sound - but, I'm less convinced by the way he connects theory and practice. A naive reader could easily assume that it is a defacto truth that hedging works and the economic implications are understood and benign. I can definitely imagine a career minded student drawing the conclusion that it "just works" - and using the techniques without further thought. I can also imagine such a person gaining rapid career advancement.

My objection isn't so much with what Hull says - but, rather, with what he doesn't say. He utterly neglects to discuss the systemic effects of the practices he describes. It is self-evident that the techniques would likely be useful if there was one small-time practitioner... but, in teaching the technique, it becomes essential to ask how it influences systems if use is widespread. I think it worthwhile pondering that trading in securitized debt, for example (where risks would likely have been aggressively hedged) was traded principally by professionals - most of whom would have been familiar with the techniques Hull describes...

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:-) I was careful not to get into unnecessary theological discussion.

I think Hull's discussion of the abstract concepts to be sound - but, I'm less convinced by the way he connects theory and practice. A naive reader could easily assume that it is adefacto truth that hedging works and the economic implications are understood and benign. I can definitely imagine a career minded student drawing the conclusion that it "just works" - and using the techniques without further thought. I can also imagine such a person gaining rapid career advancement.

My objection isn't so much with what Hull says - but, rather, with what he doesn't say. He utterly neglects to discuss the systemic effects of the practices he describes. It is self-evident that the techniques would likely be useful if there was one small-time practitioner... but, in teaching the technique, it becomes essential to ask how it influences systems if use is widespread. I think it worthwhile pondering that trading insecuritized debt, for example (where risks would likely have been aggressively hedged) was traded principally by professionals - most of whom would have been familiar with the techniques Hull describes...

If there are systematic effects and he was aware of them, it is not surprising that he would choose not to voice his concerns given the likely social vilification he could have expected to receive.

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Neither of those conditions presently exist in financial markets. Imagine you were trying to delta hedge the RBS share price yesterday for instance. In practice, only futures and options broker/market makers can come close to being able to delta hedge efficiently.

yes the options market makers have a huge number of tools in their arsenal to be able to delta hedge appropriately.

  • massive liquidity advantages (e.g by running a big book they can mange delta relativly easily)

  • high markups (e.g wide spreads make hedging relativly cheap)

  • cartel access in some cases to provide liquidity

  • automatic options book delta protection (e.g orders get pulled if a certain delta limit is hit )

  • options markets may be suspended if an underlying market is not available for delta hedging.

In return they have to buy the market making rights in an auction and have an obligation to provide a certain quality of market (size and spread)

Edited by jonpo

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Hi

Thanks for all the excellent responses; you've helped clarify most of my problems with this.

I study theoretical physics, so the mathematics of (mathematical) derivatives is the simple part, the application in the unfamiliar world of finance (with its own nomenclature), is the harder part.

I got An Introduction to Global Financial Markets by Stephen Valdez from amazon for Christmas. Its a very poor book on the subject, very long on unnecessary detail, and unfortunately short on pedagogical explanations.

The example he gives is a convertible, which he describes as an option to buy shares in a particular company in the future for a set price. He then says that the position can be dynamically delta hedged by buying or selling the current stock in the company.

Is the idea that the person who buys the convertible thinks that it is currently undervalued relative to the price of the stock, and by shorting the stock by the appropriate amount (dynamically changing it depending on the stock price movements), can lock in the difference between the current stock price and the current convertible price. So even if the stock price falls in the future, necessarily taking the convertible price with it, then there is still profit if the relative difference has moved in favour of the convertible.

Once again thanks for all the help.

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The example he gives is a convertible, which he describes as an option to buy shares in a particular company in the future for a set price. He then says that the position can be dynamically delta hedged by buying or selling the current stock in the company.

Is the idea that the person who buys the convertible thinks that it is currently undervalued relative to the price of the stock, and by shorting the stock by the appropriate amount (dynamically changing it depending on the stock price movements), can lock in the difference between the current stock price and the current convertible price. So even if the stock price falls in the future, necessarily taking the convertible price with it, then there is still profit if the relative difference has moved in favour of the convertible.

Yes, that could work... but if the person who buys the convertible isn't a full-time trader, they might just as well enter a futures contract (cheaper than a put option - which could also be used) to sell the stock at the maturity date of the convertible bond for around the current market price. This would also lock-in the profit - but with the advantage that there would be no need to actively trade to maintain the hedge... leaving that task to professionals who can manage this risk (using delta hedging or otherwise) more cost effectively owning to economies of scale.

The good bit about this significantly more straightforward strategy (assuming suitable contracts can be purchased) is to eliminate the need to hold liquid capital to fund the delta hedging... as well as the effort required to maintain the hedge and it eliminates the risk of a portfolio diverging from its objectives - owing to trading delays and/or rapid movement in stock prices.

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Hi

Thanks for all the excellent responses; you've helped clarify most of my problems with this.

I study theoretical physics, so the mathematics of (mathematical) derivatives is the simple part, the application in the unfamiliar world of finance (with its own nomenclature), is the harder part.

I got An Introduction to Global Financial Markets by Stephen Valdez from amazon for Christmas. Its a very poor book on the subject, very long on unnecessary detail, and unfortunately short on pedagogical explanations.

The example he gives is a convertible, which he describes as an option to buy shares in a particular company in the future for a set price. He then says that the position can be dynamically delta hedged by buying or selling the current stock in the company.

Is the idea that the person who buys the convertible thinks that it is currently undervalued relative to the price of the stock, and by shorting the stock by the appropriate amount (dynamically changing it depending on the stock price movements), can lock in the difference between the current stock price and the current convertible price. So even if the stock price falls in the future, necessarily taking the convertible price with it, then there is still profit if the relative difference has moved in favour of the convertible.

Once again thanks for all the help.

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if delta is hedged then you have still vega and gamma risk....

delta neutral trades are also called 'vol trades' the reasons should be obvious...

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If an options holder (call, put, convertible) expects to profit from dynamically hedging the underlying, they are just essentially profiting from the long gamma... hedging the moves up and down (you get longer deltas on a rally therefore sell futures and you get shorter shorter on the way down therefore buy futures).

In this view, you dont really care which way futures move as long as they move everyday. This is the way options are priced, it is argued that even if an out of the money option expires worthless, the option holder may still have profitted if they hedged the gamma.

Your question is why someone may buy a convertible/call- well it could just be a directional punt (unhedged), pay premium for P&L on the rally with a limited downside loss (no more than the premium) or as I say vol punt; i.e. vol is cheap and it is likely to move a lot, or this vol will trade higher, I will buy it now, hedge it and hopefully someone will pay more for it.

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