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frolloos
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### hedging shorter dated with longer dated

I probably should know this, but is there a general formula/method for hedging shorter dated European path-independent derivatives with longer dated of the same type and payoff function?

Thanks.

list1
Posts: 1668
Joined: July 22nd, 2015, 2:12 pm

### Re: hedging shorter dated with longer dated

frolloos wrote:
I probably should know this, but is there a general formula/method for hedging shorter dated European path-independent derivatives with longer dated of the same type and payoff function?

Thanks.

It seems that $\ Delta ( t ) = C_{S}^{/} ( t , S ( t ) ; T , K )$ shares of S ( t ) provides the hedged portion of underlyings which hedge the call option. If in general  $T_1 < T_2$ we could not say anything If $T_j$ , j = 1, 2 . If one of the maturities is sufficiently small or large we need to use asymptotic formula for $S ( T_j ; t , S ( t ))$ that defines C ( t , S ( t )) and suppose  that $T_1$ is $\epsilon \,T$ and/ or $T_2$ is $\frac {T}{\epsilon}$

outrun
Posts: 3623
Joined: April 29th, 2016, 1:40 pm

### Re: hedging shorter dated with longer dated

interesting!

Vega hedging? I would start by looking at the covariance matrix of ATM implied for various time till expiration and extract the principal component for the vol dynamics. Ideally your mix of long dated options should highly correlate with the short one.

In commodity markets (and maybe vol is a commodity?) this is often done the other way around. You might have long dated positions like a wind turbine that will generate electricity for the next 10 years but only a liquid electricity market for the next 3 years. One approach there is to use future "stacking", sell electricity futures for the next three years, but sell 7 years extra worth of electricity for year 3. The assumption is that the price of year 4,5,6..,10 will highly correlates (today!) with the price of year 3 and so you can use year 3 as a proxy year 4,5,...10. Points further out in the curve are typically less volatility than point closerby and so you need to correct for that leverage.. and this is where the principal components come in. It's however not risk free, you'll take *big* spread positions and have rollover risk and so you need to trade off risk reduction by hedging vs new basis risk introduced by hedging with proxies. In your case you could perhaps get a 1 month vol by buying 13 times the 1 year vol and selling 12.8 times the 2 year vol (which is scary). You'll probably end up with some risk reduction portfolio optimization problem that has extra constraints coming from risk policies. The stability of the hedge is also important, you would want to prefer a static hedge over a dynamic hedge that requires lots of trading.

Besides the empirical data driven Principal Components approach you can also pick a stock vol model and extract implied vol forward dynamics from the model.

frolloos
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### Re: hedging shorter dated with longer dated

List1, thanks, but I don't understand what you mean.

Thinking about the problem more this is what I came up with (for plain vanilla, which is sufficient), and working in a local vol world for now:

$C(t,K,T) = \int_t^{T'} \delta(S-T) C(t,K,S) dS \quad (T' > T)$
$= C(t,K,T') - \int_t^{T'} \theta(S-T) \frac{\partial C(t,K,S)}{\partial S} dS$
$= C(t,K,T') - \frac{1}{2} \int_T^{T'} \sigma^2_{loc} (K,S) \frac{\partial^2 C(t,K,S)}{\partial K^2} dS$

I'm a bit jet-lagged still after a long flight so maybe made an error. If correct though, then above says the short dated option can be hedged by holding the longer dated option and short a strip of butterflies with maturities S between T and T' (weighted by the local vol function for strike K and corresponding maturities). Not quite sure why this should be so..

Background to this question: When trading OTC derivs the maturity date does not always have to correspond to a listed maturity.

Thanks.

(S is not the spot obviously, but the maturity date)

outrun
Posts: 3623
Joined: April 29th, 2016, 1:40 pm

### Re: hedging shorter dated with longer dated

What risk are you trying to hedge/minimize? Once you have a cost functions that you aim to minimize you can then construct a (static?) portfolio that minimizes it. Eg the butterfly will be an approximation when you have a limited number of strikes and so you should optimize using only tradable products, right?

Paul has written about static hedges, https://www.wilmott.com/wp-content/uplo ... ilmott.pdf

list1
Posts: 1668
Joined: July 22nd, 2015, 2:12 pm

### Re: hedging shorter dated with longer dated

frolloos wrote:
List1, thanks, but I don't understand what you mean.

Thinking about the problem more this is what I came up with (for plain vanilla, which is sufficient), and working in a local vol world for now:

$C(t,K,T) = \int_t^{T'} \delta(S-T) C(t,K,S) dS \quad (T' > T)$
$= C(t,K,T') - \int_t^{T'} \theta(S-T) \frac{\partial C(t,K,S)}{\partial S} dS$
$= C(t,K,T') - \frac{1}{2} \int_T^{T'} \sigma^2_{loc} (K,S) \frac{\partial^2 C(t,K,S)}{\partial K^2} dS$

I'm a bit jet-lagged still after a long flight so maybe made an error. If correct though, then above says the short dated option can be hedged by holding the longer dated option and short a strip of butterflies with maturities S between T and T' (weighted by the local vol function for strike K and corresponding maturities). Not quite sure why this should be so..

Background to this question: When trading OTC derivs the maturity date does not always have to correspond to a listed maturity.

Thanks.

(S is not the spot obviously, but the maturity date)

Dealing hedging with loc vol is a little bit new and I think we should verify our points. Here is some preliminary thoughts which could be incorrect too  and you probably better comprehend the subject.
Locvol concept interprets BS call option C ( t , S ; K , T ) as a function of variables K , T when t and S are fixed parameters. One can try to hedge option by using inverse time loc vol stoch. process  k ( t ; T , K ). Here variable t is fixed and parameters K , T are considered as variables. In this case
C ( t , S ; K , T )  =  E max { S - k ( t ; T , K ) , 0 }
Though it is possible to present delta hedging by using k ( t ; T , K ) it does not make real sense as far as this underlying does not a tradable asset. It exists only on the paper. Now we should specify hedging concept in LV setting. In BS setting we hedge C ( t , S ; K , T ) for infinitesimal changes t when S, T , K are fixed. In LV the variable is T hence do you think that we should establish hedge of the C ( t , S ; K , T ) for a small change t from T to T - dT by using real security S?

Of course it is not answer on your specific question " is there a general formula/method for hedging shorter dated European path-independent derivatives with longer dated of the same type and payoff function?" On the other hand why we should apply local volatility concept to answer?

frolloos
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### Re: hedging shorter dated with longer dated

outrun wrote:
What risk are you trying to hedge/minimize? Once you have a cost functions that you aim to minimize you can then construct a (static?) portfolio that minimizes it. Eg the butterfly will be an approximation when you have a limited number of strikes and so you should optimize using only tradable products, right?

Paul has written about static hedges, https://www.wilmott.com/wp-content/uplo ... ilmott.pdf

Hi Outrun, thanks for your reply! I am trying to hedge variance swaps with maturity date in between two listed maturities.

I think Carr has written a paper on how to hedge longer dated options with shorter dated options and then delta hedging after the shorter dated options expire. For variance swaps this strategy wouldn't work I think.

My main question right now is whether the formula I came up with is correct or not (I  think it could be right as it is a consequence of Dupire eqt). If it is, then the next step would be to do some optimization as indeed inifnitely tight butterflies are not traded, and strikes are spaced discretely apart. And also there is model dependency (local vol), but then at least I know how to use longer dated maturities (keeping in mind the practical limitations and assumptions used).

list1
Posts: 1668
Joined: July 22nd, 2015, 2:12 pm

### Re: hedging shorter dated with longer dated

I think that an approach to answer for original question can be presented in next form.
$C ( t , S ( t ) ; T_1 , K ) \, \sim \, C_{S}^{/} ( t , S ( t ) ; T_1 , K ) \, S ( t )$
$C ( t , S ( t ) ; T_2 , K ) \, \sim \, C_{S}^{/} ( t , S ( t ) ; T_2 , K ) \, S ( t )$
where sign $\sim$ means 'hedged by' and $T_1 < T_2$. Then date-t hedge of $C ( t , S ( t ) ; T_1 , K )$ can be realized by the portion of

$\frac{C_{S}^{/} ( t , S ( t ) ; T_1 , K ) }{ C_{S}^{/} ( t , S ( t ) ; T_ 2, K ) }$

of stocks which is equivalent to the same portion of the calls with $T_2$ maturity. This number of calls should be adjusted at the end of the day.
Intuitively the use of LV model where T is considered as variable suggests that the hedge might be exist by using underline of the LV model. Nevertheless underling of the LV pricing model is a heuristic function k ( t , T , K ) which should present the same result as BS model as far as regardless of BS or LV our point on call we deal with the same function C ( t , S ; T , K ).

outrun
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Joined: April 29th, 2016, 1:40 pm

### Re: hedging shorter dated with longer dated

The equations look good to me (fwiw).

Would be nice to try and get a relations that is as much model free as possible. Focusing on the realized variance like you do (instead of implied) makes that better to manage. At some point you will however have to take care of conditional variance and then it becomes model specific?

list1
Posts: 1668
Joined: July 22nd, 2015, 2:12 pm

### Re: hedging shorter dated with longer dated

outrun wrote:
The equations look good to me (fwiw).

Would be nice to try and get a relations that is as much model free as possible. Focusing on the realized variance like you do (instead of implied) makes that better to manage. At some point you will however have to take care of conditional variance and then it becomes model specific?

equation is probably good but dS does not notation for the stock differential. S is  maturity time and this term is confusing

outrun
Posts: 3623
Joined: April 29th, 2016, 1:40 pm

### Re: hedging shorter dated with longer dated

list1 wrote:
outrun wrote:
The equations look good to me (fwiw).

Would be nice to try and get a relations that is as much model free as possible. Focusing on the realized variance like you do (instead of implied) makes that better to manage. At some point you will however have to take care of conditional variance and then it becomes model specific?

equation is probably good but dS does not notation for the stock differential. S is  maturity time and this term is confusing

I agree. My wife's name starts with an S and I though he was differentiating towards her.

DavidJN
Posts: 1685
Joined: July 14th, 2002, 3:00 am

### Re: hedging shorter dated with longer dated

I don't have time to look them up at the moment but there are published papers looking at the reverse of what you are looking at, that is to say they investigate the hedging of longer-dated options using rolled over portfolios shorter-dated options. Needless to say one is accepting vol term structure risk in any such strategy.

frolloos
Topic Author
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Location: Netherlands

### Re: hedging shorter dated with longer dated

outrun wrote:
list1 wrote:
equation is probably good but dS does not notation for the stock differential. S is  maturity time and this term is confusing

I agree. My wife's name starts with an S and I though he was differentiating towards her.

frolloos
Topic Author
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Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: hedging shorter dated with longer dated

David, thanks, I think I know which paper you mean. Reason I'm looking at the reverse is to avoid precisely the term structure risk (or rather forward surface risk). But in return getting model risk in the form of local vol function.

Trying what Outrun suggested (seeking something more model free). But if there is an easier solution to the problem will be glad to hear about it.

list1
Posts: 1668
Joined: July 22nd, 2015, 2:12 pm

### Re: hedging shorter dated with longer dated

frolloos wrote:
outrun wrote:
list1 wrote:
equation is probably good but dS does not notation for the stock differential. S is  maturity time and this term is confusing

I agree. My wife's name starts with an S and I though he was differentiating towards her.

froloos, how do you interpret the integral term as an asset or its approximation sum of assets?