delta hedging

[Eveline]

Hi David,

Sorry to flood you with questions! This should be the last one, so sorry. Pls take your time... I can always come back to it.

Qn:
A trader buys an at-the-money call option with the intention of delta-hedging it to maturity. Which one of the following is likely to be most profitable over the life of the option?

1) an increase in implied volatility
2) the underlying price steadily rising over the life of the option
3) the underlying price steadily decreasing over the life of the option
4) the underlying price drifting back and forth around the strike price over the life of the option

Ans; 4

Thank you so much!

 

[notjusttp]

Hi Eveline,

A long position in call option always has a positive gamma and when gamma is positive a hedger gains from large changes in the underlying price hence the answer 4 is correct. Trust this clarifies...Amit

 

[Eveline]

Hi Amit,

Thanks a lot for your help! I've typed out the FRM handbook answer (didn't quite understand it, so left it out). Would it be similar to what you're saying?

Answer:
An important fact is that the option is held to maturity.
Changes in the implied volatility would change the value of the option but this has no effect when holding to maturity.
It does not depend on whether the option ends up in-the-money or not.
The profit from the dynamic portfolio will depend on whether the actual volatility differs from the initial volatility. The portfolio will be profitable if the actual volatility is small which implies small moves around the strike price

Again thanks a lot!

 

[notjusttp]

Hi Eveline,

You are welcome. Pls note that this is an option where the intention is to hold till maturity. If you keenly observe the question it speaks about profitable value "over the life" of the call option. Also delta hedging could be achieved by shorting an underlying hence one ways movement is not going to impart any value to the option over the life. So you could also arrive at the answer by eliminating straight away options 2 and 3 and anywhichways 1 is redundant because u hold the option till maturity..hence 4 can be arrived at by elimination also.

The answer from handbook is not exactly implying what i am saying because there is no mention of gamma which is the rate of change of delta..instead its touching upon vega ( since mentions about volatility)..i had got the result about gamma in one of my visits to BT discussion which i m unfortunately not getting now....

Trust this helps...I m sure if David gets sometime to look thru this he can give a more intuitive answer.

Cheers..Amit

 

[David Harper CFA FRM]

@Amit, thanks for the help!

it may help to restate what the delta-hedge is here:
long call option, so...
short underlying share

e.g.,
long X call options +
short (Delta * X) shares

to maintain this *dynamically* the short position must be rebalanced when the underlying stock changes.
(I have to walk thru this myself b/c i tend to forget this part...the long position in X options is constant and the hedge is a changing/dynamic short position in delta * underlying shares;
I have an XLS that mimic Hull's delta hedge here:
http://www.bionicturtle.com/premium/spreadsheet/4.b.5_dynamic_delta_hedge/
...and that is like the "other side of the trade:" short options (constant) and hedge with dynamic (changing) long position in (delta multiplied by) shares)

..okay, so if you are good with: the key to the delta hedge is here a dynamic short position, then consider the behavior of delta:

http://learn.bionicturtle.com/images/forum/0926_deltahedge.png

when stock goes up, delta (the slope of the tangent line) is increasing, and to maintain hedge, we are selling more shares (i.e., increase the short position)
when stock goes down, delta (flatter tangent) is descreasing and we are buying shares to decrease the short position...

oscillation produces a buy low/sell high!

Amit is right about gamma ... with long calls, the position is "long volatility" (vega), the position profits if the actual volatilty (i.e., realized vol produced by the up/down) is greater than the implied volatility (the vol implied at purchase of the long calls). hope that helps, it is *tough* question .... David

 

[rvalive]

I think I understood David's reply but, I didn't understand FRM explanation (based on Eveline's post) about small moves around strike price. Shouldn't large moves around strike make more profit?

Thanks

 

[David Harper CFA FRM]

@rvalive,

Right, agreed, that part of the handbook answer looks wrong to me. (Nice attention to detail!)
I agree with: "Shouldn’t large moves around strike make more profit?"

First, regarding "the profit from the dynamic portfolio will depend on whether the actual volatility differs from the initial volatility" ...
should say: will depend on whether the actual volatility differs from *implied* volatility (i.e., at the time of option purchase) as "initial volatiltiy" is ambigious.

Then: the position is profitable if actual (aka realized volatility) is greater than implied volatlity (or "initial implied volatility" i suppose) ...
so I think the next sentence has it backward:
"The portfolio will be profitable if the actual volatility is small which implies small moves around the strike price"
should be: "will be profitable if actual (realized) volatility is greater than (initial) implied volatility"

David

 

[ajsa]

Hi David,

so if The portfolio “will be profitable if actual (realized) volatility is greater than (initial) implied volatility”, I feel a is more correct than d.. Could you pls clarify? maybe you can explain what "an increase in implied volatility" means?

thanks.

 

[ajsa]

sorry David, I think I know what a means now. increase in implied vol means opt value increase with other variables staying constant, right?

 

[David Harper CFA FRM]

Hi asja,

Implied volatility only has meaning in the context of the option pricing model: when the trader buys the option, the price paid reflects the implied volatility (i.e., a forward-looking estimate of volatility). What does the hedge then consist of? A sequence of purchase/sales in the underlyng stock (not in more options!)
(this is why i think it may be fruitful to study the XLS example of dynamic hedge:
http://www.bionicturtle.com/premium/spreadsheet/4.b.5_dynamic_delta_hedge/
the initial trade is long 100,000 options, then the options are done
...each week therefafter, outright shares, not options, are sold or purchased to maintain delta-neutral hedge)

so, the answer "increase in implied volatility" is incorrect b/c that implies future call option prices ... it is worth a meditation to see why this answer is not correct.. implied vol is meaningless without the call option prices ... an increase in implied volatility requires future call option prices...will an increase in realized (actual) volatility increase future implied volatiltiies? Yes, probably, but it is beyond the question as the hedger is not buying any more options ... so (d) is correct because the determinant here is the realized volatility of the underlying...i hope that helps

(the final bow on the package is to see Hull's point about the delta hedge simulation in 4.b.5:
if the realized volatility--i.e., the oscillation of the underlying--ends up matching the intial, implied volatility--i.e., as reflected in the option call premium,
then the cost of the option will equal the profits on the delta hedge! ... if the implied volatilty volatility predicts correctly, no free lunch)

David

 

[gzxjj]

Hi David,

I have just discovered this wonderful place to learn financial maths.

I do not quite understand this question: in my text book, it states that "hedge" can only reduce risk and maintain the value of portfolio, but here we say that the trader can earn money by hedging. Could you tell me why, please?

Thanks.

gzxjj

 

[James89]

Hi David,

Why is it that when the realized (actual) volatility turns out to be higher than initial implied volatility, then our strategy is profitable (assuming that we bought the call option in the first place)? If we believe that the call was underpriced because the implied volatility quoted on the market is lower than our estimated future volatility (maybe calculated using historical volatility), then we buy the call and hedge the position by selling the stock. However, as one month has elapsed, how do we know whether that call was truly underpriced? Is it by comparing the actual realized volatility over the past month with the implied volatility when we bought the call? But how do we know exactly the profit we have made due to the call being underpriced? Do we also need to look at the call prices or stock prices at regular intervals during the month to determine whether we were correct in saying that the call was underpriced?

Thank you.

James.

 

[David Harper CFA FRM]

Hi James,

The original option premium is priced, by definition, to assume that today's (i.e., at option purchase) implied volatility will turn out to be a good predictor of tomorrow's realized volatility. So, yes, if realized volatility (e.g., at expiration in the case of Euro) turns out to be greater than implied volatility, we can say the option was originally under-priced (cheap). In theory, this would be exposed even before maturity with an increase in the M2M option value. This is a sense in which options are, firstly, volatility instruments and buying an option is a long volatility trade.

But the above refers to the dynamic delta hedge, which is best illustrated by Hull's Table 18.2 and 18.3;
i.e., long a call option hedged by short (delta) fractional share
(this may help with further detail on the impact of gamma in the dynamic delta hedge)

If an option did not have any gamma, then this dynamic hedge would be profitless as option losses would be perfectly hedged by share gains. However, the gamma (curvature) is favorable on both the up/downside to the long option holder, such that re-balancing creates forward profits over the life of the hedge. (importantly, omitting transaction costs). Again, best understood with Hull's illustration. But the moral of his story is:

• If the long call, in the dynamic hedge has a cost of X, then if the implied volatility = subsequently realized volatility, then the cumulative profit engendered by gamma will equal FV(X); i.e., the option was perfectly priced if the "rebalancing payouts" created by gamma are equal to the option premium, such that the PV(payoffs) - premium = 0.

And, in fact, this equality is the key premise in the BSM option pricing model itself. I hope that helps,

 

[James89]

Hi David,

Thanks for your reply. So do you mean that only by dynamic hedging will our exposure to positive gamma be profitable when realized volatility is larger than implied volatility? The reason is because I tried identifying an underpriced put (based on volatility), weekly delta hedging and then ending up with a loss after 3 weeks? Is that because hedging is not done continuously in this case?

 

[David Harper CFA FRM]

Hi Jarnes, yes the hedge frequency is a practical difference. The breakeven equivalence assumes continuous rebalancing and without transaction costs, so it is "idealized." However, I *think* another practical difference is the BSM assumption of constant volatility: the initial option purchase and each delta re-hedge--i.e., where delta = N(d1) and the d1 assumes constant volatility--assumes constant volatility, so I *think* it's not only that the realized volatility is higher per se but that it time-varies. Thanks,

 

[James89]

Hi David

So in this case, does it mean that in order to benefit from long volatility, we should get exposed to positive gamma and should not gamma hedge so that we can lock in any profit when delta changes?

 

[James89]

If stock price changes by a relatively large amount (as opposed to small changes in the case of delta) , but we do delta-hedge and rebalance our portfolio continuously, then does it matter if we still use delta-hedging rather than gamma-hedging?

 

[David Harper CFA FRM]

Hi James,

Although the dynamic delta hedge (i.e., long call + short delta share continuous rebalanced to maintain neutral delta) is long volatility, in my opinion, to go "long volatility" is a broader concept; for example, simply long a (naked) call or put is "long volatility" as this is the essence of the option. (and, of course, vega is a more direct Greek. We can position to the portfolio simply for positive vega for direct volatility exposure!).

In regard to your second question, I think one of Hull's Q&A might be helpful:

"Hull 18.5 What is meant by the gamma of an option position? What are the risks in the situation where the gamma of a position is large and negative and the delta is zero?

Answer: The gamma of an option position is the rate of change of the delta of the position with respect to the asset price. For example, a gamma of 0.1 would indicate that when the asset price increases by a certain small amount delta increases by 0.1 of this amount. When the gamma of an option writer’s position is large and negative and the delta is zero, the option writer will lose significant amounts of money if there is a large movement (either an increase or a decrease) in the asset price."

I like this Q&A because it illustrates gamma as a key indicator of the option's directional risk: the option buyer pays the seller, to some degree, for the benefit of positive gamma (the curvature that implies the buyer benefits under a large move in either direction).