In this working paper a nonparametric hedge ratio is derived for general stochastic volatility models. Enabling the hedging of volswaps with varswaps only.

Any questions and comments welcome.

https://arxiv.org/abs/2001.02404

In this working paper a nonparametric hedge ratio is derived for general stochastic volatility models. Enabling the hedging of volswaps with varswaps only.

Any questions and comments welcome.

https://arxiv.org/abs/2001.02404

Any questions and comments welcome.

https://arxiv.org/abs/2001.02404

I thought the trick with the H-process was clever.

In the final formula (5.15) I suppose [$]\Sigma\_[$] is defined at (4.7). If so, you might explicitly say that, since (as you said) there was a notational abuse. Related, I think you need to show that all is sensible with the proposed hedging of a seasoned contract as [$]t \rightarrow T[$].

Finally, a numerical example of the effectiveness of the proposed hedging formula (5.15) might be interesting and useful. If done via a model, I'd suggest using a vol-of-vol [$]\xi=1[$] (annualized units) in some GBM-type: [$]d \sigma = a(\sigma) \, dt + \xi \, \sigma \, dW[$]. This SDE form might be more realistic than the ubiquitous Heston '93, and, if it's going to be a Monte Carlo exercise anyway, non-affine type shouldn't matter.

In the final formula (5.15) I suppose [$]\Sigma\_[$] is defined at (4.7). If so, you might explicitly say that, since (as you said) there was a notational abuse. Related, I think you need to show that all is sensible with the proposed hedging of a seasoned contract as [$]t \rightarrow T[$].

Finally, a numerical example of the effectiveness of the proposed hedging formula (5.15) might be interesting and useful. If done via a model, I'd suggest using a vol-of-vol [$]\xi=1[$] (annualized units) in some GBM-type: [$]d \sigma = a(\sigma) \, dt + \xi \, \sigma \, dW[$]. This SDE form might be more realistic than the ubiquitous Heston '93, and, if it's going to be a Monte Carlo exercise anyway, non-affine type shouldn't matter.

Thank you for taking the time to read it and for the comments, Alan. Much appreciated.

Yes, although more cumbersome than Heston I should probably use lognormal or 3/2 model for the numerical part (not my forte, but needs to be done). If you or anyone else have suggestions on how best to carry out the MC (a particular MC scheme) I'd be happy to hear and learn about it.

In (5.15), [$] \Sigma_- [$] is as you point out as in (4.17). Hedging should remain sensible as [$] t \rightarrow T [$] since by construction the volswap price will then approach the realised volatility and in the formula for the adjusted options it is the product [$] c\sqrt{T-t} [$] that determines the value and that will remain finite as well, but I will mention that in the updated version.

Thanks again.

Yes, although more cumbersome than Heston I should probably use lognormal or 3/2 model for the numerical part (not my forte, but needs to be done). If you or anyone else have suggestions on how best to carry out the MC (a particular MC scheme) I'd be happy to hear and learn about it.

In (5.15), [$] \Sigma_- [$] is as you point out as in (4.17). Hedging should remain sensible as [$] t \rightarrow T [$] since by construction the volswap price will then approach the realised volatility and in the formula for the adjusted options it is the product [$] c\sqrt{T-t} [$] that determines the value and that will remain finite as well, but I will mention that in the updated version.

Thanks again.

You're welcome.

Re MC. For the model form I posted, likely you'll choose some mean-reverting [$]a(\sigma)[$], so the process: (a) lies on [$](0,\infty)[$], (b) never reaches a boundary in finite time, and (c) has a stationary distribution. Given that, I would transform it to [$]d y = \tilde{a}(y) \, dt + \xi \, dW[$] with [$]y_t = \log \sigma_t[$]. This latter SDE has a straightforward MC: [$]y_{t+1} = y_t + \tilde{a}(y_t) \Delta t + \xi Z_t \sqrt{\Delta t}[$], with [$]Z_t[$] a standard normal draw at each step. Just update (the discrete form of) any needed integrals [$]\int_0^t v(y_s) \, ds[$] at each MC step.

Re MC. For the model form I posted, likely you'll choose some mean-reverting [$]a(\sigma)[$], so the process: (a) lies on [$](0,\infty)[$], (b) never reaches a boundary in finite time, and (c) has a stationary distribution. Given that, I would transform it to [$]d y = \tilde{a}(y) \, dt + \xi \, dW[$] with [$]y_t = \log \sigma_t[$]. This latter SDE has a straightforward MC: [$]y_{t+1} = y_t + \tilde{a}(y_t) \Delta t + \xi Z_t \sqrt{\Delta t}[$], with [$]Z_t[$] a standard normal draw at each step. Just update (the discrete form of) any needed integrals [$]\int_0^t v(y_s) \, ds[$] at each MC step.

Yes, SABR with mean reversion is I think what you are suggesting. I will try that, thank you.

Btw, I wasn't aware of this until only recently, but SABR with positive correlation results in loss of martingality. Not really an issue since I am mainly interested in equity, but could be an issue for FX.

https://www.researchgate.net/publicatio ... Volatility

Btw, I wasn't aware of this until only recently, but SABR with positive correlation results in loss of martingality. Not really an issue since I am mainly interested in equity, but could be an issue for FX.

https://www.researchgate.net/publicatio ... Volatility

Yeah, if you really need to do positive correlation under this type of process, just stick with put options, which have a non-ambiguous value. (See the discussion around Table 8.7 in my Vol II book).

I think actually the section on hedging needs revision because I overlooked something subtle (as usual). The end conclusion though that you need 1/(2[$] \Sigma_- [$]) of varswaps to hedge volswaps I believe remains true.

The subtlety is as follows: given two functions [$] F(x) [$] (the volswap) and [$] G^2(x) [$] (the varswap), with [$] x[$] the instantaneous vol, then it is true that

[$] \frac{\partial (G^2-F^2)}{\partial x} \approx 0 [$]

which means that to first order the convexity adjustment does not change with level of instantaneous vol. However,

[$] \frac{\partial G^2}{\partial x} \not\approx 2 F \frac{\partial F}{\partial x} [$]

The subtlety is as follows: given two functions [$] F(x) [$] (the volswap) and [$] G^2(x) [$] (the varswap), with [$] x[$] the instantaneous vol, then it is true that

[$] \frac{\partial (G^2-F^2)}{\partial x} \approx 0 [$]

which means that to first order the convexity adjustment does not change with level of instantaneous vol. However,

[$] \frac{\partial G^2}{\partial x} \not\approx 2 F \frac{\partial F}{\partial x} [$]

Update:

I derived a more accurate formula for the hedge ratio, and I will update the arXiv paper accordingly. See the formula below and also a histogram of the hedge p/l in volatility points. I.e. if final value of volatility swap is 20% and the hedge is 20.2%, then p/l is 0.2%.

I ran 500 simulations of the Heston model of daily hedging of a 1 year volatility swap until maturity with variance swap using my formula below for the hedge ratio.

Formula:

[$]

d \mathcal{V}(t) = \frac{1}{ \mathcal{V}(t) + V^2(t) / \mathcal{V}(t)} d V^2(t)

[$]

where [$] \mathcal{V}(t) [$] is the (seasoned) volswap price at time [$] t [$] and [$] V^2(t) [$] is the (seasoned) varswap price at time [$] t [$]. Implication of my paper is that this is robust for practically any SV model.

I am quite happy with this actually. Approximations can be elegant too

The histogram of the PNL:

[img]file:///Users/rallyschwachofer/Desktop/vshedging/hedgepnl.png[/img]

I derived a more accurate formula for the hedge ratio, and I will update the arXiv paper accordingly. See the formula below and also a histogram of the hedge p/l in volatility points. I.e. if final value of volatility swap is 20% and the hedge is 20.2%, then p/l is 0.2%.

I ran 500 simulations of the Heston model of daily hedging of a 1 year volatility swap until maturity with variance swap using my formula below for the hedge ratio.

Formula:

[$]

d \mathcal{V}(t) = \frac{1}{ \mathcal{V}(t) + V^2(t) / \mathcal{V}(t)} d V^2(t)

[$]

where [$] \mathcal{V}(t) [$] is the (seasoned) volswap price at time [$] t [$] and [$] V^2(t) [$] is the (seasoned) varswap price at time [$] t [$]. Implication of my paper is that this is robust for practically any SV model.

I am quite happy with this actually. Approximations can be elegant too

The histogram of the PNL:

[img]file:///Users/rallyschwachofer/Desktop/vshedging/hedgepnl.png[/img]

Congratulations.

How sensitive is the p/l distribution width to the vol-of-vol? For example, how does it do when the Heston V-process can reflect off the origin?

The mean-reverting SABR case would still be a good test, too.

How sensitive is the p/l distribution width to the vol-of-vol? For example, how does it do when the Heston V-process can reflect off the origin?

The mean-reverting SABR case would still be a good test, too.

Thanks Alan.

I actually want to try the 3/2 model with mean reversion - as in your book volume I.

At first I thought that I would need to recalculate the whole smile at each time step (which is a Monte Carlo in a Monte Carlo), find the new (adjusted) zero vanna IV and varstrike and then update the hedge ratio for the next hedge interval.

But the I realised that my hedge formula does not require that: I just need the initial zero vanna to start with, and then at each time step update the replicating portfolio value using the hedge formula. So that considerable reduces the amount of computation.

However, I still need to calculate K_var at each time step. I suppose there is no closed form formula for Kvar in the 3/2 model right?

I actually want to try the 3/2 model with mean reversion - as in your book volume I.

At first I thought that I would need to recalculate the whole smile at each time step (which is a Monte Carlo in a Monte Carlo), find the new (adjusted) zero vanna IV and varstrike and then update the hedge ratio for the next hedge interval.

But the I realised that my hedge formula does not require that: I just need the initial zero vanna to start with, and then at each time step update the replicating portfolio value using the hedge formula. So that considerable reduces the amount of computation.

However, I still need to calculate K_var at each time step. I suppose there is no closed form formula for Kvar in the 3/2 model right?

Last edited by frolloos on January 20th, 2020, 6:04 am, edited 1 time in total.

So, you want [$]f(T,V_0) \equiv E[\int_0^T V(t) \, dt][$] for [$]dV = (\omega V - \theta V^2) \, dt + \xi V^{3/2} dW[$], right? There is a closed-form for the mgf:

[$]H(T,V_0;c) \equiv E[\exp\{-c \int_0^T V(t) \, dt\}][$],

developed in Chapt 11 of the volume I book you mention. You can take [$]\rho = 0[$] in (2.2) there. With that, just differentiate [$]H[$] w.r.t. c, multiply by -1, and set c=0.

Likely will require derivatives of [$]M(a,b,x)[$] with respect to a or b. But Mathematica, for example, will treat those as closed-forms until the appropriate point where you need to get a number.

[$]H(T,V_0;c) \equiv E[\exp\{-c \int_0^T V(t) \, dt\}][$],

developed in Chapt 11 of the volume I book you mention. You can take [$]\rho = 0[$] in (2.2) there. With that, just differentiate [$]H[$] w.r.t. c, multiply by -1, and set c=0.

Likely will require derivatives of [$]M(a,b,x)[$] with respect to a or b. But Mathematica, for example, will treat those as closed-forms until the appropriate point where you need to get a number.

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