Corrections 27/08/07 ( C.O.Ewald ) :
1)pg 2, ln 13 : concept of ( like “theory of…” )
2)pg. 2, ln 28 : added a reference to Bakshi et al. Can’t edit bib file with my texnic center, please add reference ( indicated by 1st referee )
3)pg. 2, ln 15 : suggestion of 2nd referee, I moved the definition of the cost process from section 2 to section 1
4)pg 3, ln. 1 : changed the header of section 2
5)pg 3, ln. 19 : introduced probability space and measure P
6)pg 3, ln. 42 : changed “will seeks to reduce” to “and will seek to reduce”
7)pg 4, ln. 40 : added “but does not include jumps”
8)pg 4, ln. 49 : changed “stock” to “asset” according to referee’s suggestion
9)pg 4, ln. 54 : changed “formally similarity” to “formally similar”
10)pg 5, table 1 : replaced abbreviations and added Hagan. Reference needs to be added.
11)Pg 6, ln 30 : I removed the min’s at BM’s. The dynamic displayed here is under the original measure.
12)Pg 7, ln57 : as suggested by referee, I added a comment on FX options
13)Pg 7, ln 32 : changedH(S(t)) to H(S(T))
14)Pg 7, ln. 40: added S(t)^\gamma in the denominator of the drift
15)Pg 8, ln 35 : added a reference to Bakshi et al.
16)We were inconsitent with the use of time as a subindex or argument ( as function of time ), I tried to bring everything in line with Rolf’s notaion ( such as V(t), dW(t) etc ), please check if there are still occurneces of t as a subindex.
Reply to the 2nd referee’s specific comments :
Page 2, line 15: The ‘cost process’ is not well-known, so it should be defined properly here r replaced by a simpler concept.
Done, see point 3.) above
Page 2, 2nd paragraph: The authors admit that the result on the local risk-minimizing hedge ratio is not original, but offer an alternative derivation ‘in a rigorous and novel fashion’. k, this may be interesting but you need to explain, briefly in the introduction and fully fterwards, why your derivation is really novel and why I should be interested in this new ay of deriving a result that is already known. For instance, is it more general than the existing ones? Could I apply your derivation to models in which hedge ratios have not been derived yet?
I tried to do this by the various new comments I made. We already agreed that we can not claim real novelty, El Karoui’s approach would also apply tp path dependent options. I am working with Zhaojun on applications within the framework of Asian options. There is a bit of theory about hedging of Barrier options. But in my opinion to much work and different in spirit to our existing paper.
Page 2, line 27: Instead of writing that Frey ‘derives the result in a different way’, you hould write precisely what the difference is. You have to convince the reader that your pproach is really novel.
Novel in a way, that no one has applied El Karoui’s result to find locally risk minimizing hedges in stochastic volatility models.
Page 2, paragraph 3: What did Ahn & Wilmott (2003) do? From this paragraph, I conclude hat they (a) provided evidence that risk-minimizing hedge ratios are not well-known in the practical/applied literature or (b) mistakenly ignored the volatility correction. Again, please be precise.
Rolf, you have added this reference. Could you make a comment. I don’t know Ahn’s paper.
Page 2, line 39: Here the authors make clear that the numerical exercise only applies to theHeston model. But this comes too late, since the abstract was more general than that.
Rolf is working on this.
Page 2, 4th paragraph: Very interesting paragraph. It provides a good motivation to read the paper, but I was frustrated when I realised that the results on model risk are not proved for he general class of SV models.
In a rigorous way? That could mean parametrizing the functions g and f and taking derivatives in the way of generalized Greeks and studying the effects on the hedge. Again that would be an entire new paper.
Page 2, last paragraph: Please be precise. What do you mean by a ‘general class of stochastic volatility model’? You should state your assumptions, for instance: Are themodels price- or time-homogeneous? How many Brownian motions are allowed? Arejumps allowed? How do I know if some model I have in mind is included in this class?
I clarified this point. There is a nice paper on locally risk minimizing hedging in jump models ( BS based ) by Trautman… I’ll see him next week.
Section 2: Here the authors present a review of Fölmer and Schweizer (1990) and Schweizer (1991), but I doubt that this section is really necessary. My impression is that it just complicates things. The section is too formal, not original and hard to follow. In other words, it is waste of time and space. Instead, it would be enough to define the cost process and the notion of a locally risk-minimizing hedge ratio in a simpler way, perhaps verbally,
AND
Section 3: The results from El Karoui et al. (1997) are relevant to this section but they have not been properly introduced. In fact, rather than having a whole section on Schweizer’s results (as in section 2), it would be more interesting to see a good review of El Karoui et al. (1997).
Okay, I removed the technical part and instead included a bit more of El Karoui’s approach in section 2.
Page 3, line 50: The probability P is not defined.
See point 5.) above
Page 4, line 49: your results should hold when S is a traded asset, not only for a stock.
Yes, point 8.) above.
Page 5, table 1: I feel uncomfortable with the use of the terminology local variance (or local volatility) to denote the stochastic instantaneous variance of the price process. This is non-standard terminology and may be mistakenly interpreted as the same as the local variance derived via Dupire’s forward equation.
Rolf? For me local volatility is fine. Do you know a good reference?
Page 5, table 1: It is odd to write SABR (2002). The correct reference would be Hagan et al. (2002). Besides, why use the abbreviations fct. and tech.?
See point 10.) above.
Page 5, line 45: When completing the market, it would make more sense to introduce a volatility-sensitive asset to the market, instead of another ‘stock’. This is because the vega of the new asset must be non-zero.
To think of the market completion as pricing a particular volatility derivative is generally a good idea. I introduced such an asset. It is not a simple asset and there are not many choices. The vega of the asset and the derivative are clearly non zero, but the vega of the original asset is also non zero. I don’t know quite what to make of this remark.
Page 6, line 10: the variable r has already been used in the end of section 2.
The first r (actually r^\tau) has been removed.
Page 6, line 16: the variable sigma_t^1 is not defined. Besides, the derivations here are far from obvious. Please add proof to appendix or explain exactly what you are doing.
The variable is now being defined in section 2. The derivation is simple linear algebra, but I guess we can’t tell the referee. I suggest we write that with the added material in section 2 these derivations should now be more accessible.
Page 6, line 30: I would expect the drift of S to change after the change of probability.
Right, there was a mistake in pg 6, ln 30. The dynamics there must hold under the original measure, no W^min’s, in Proposition 1 (now 2) the dynamic is right.
Page 7, proposition 1: Equation 3 is very interesting and indeed very general. But you need to state clearly if this is an original result, explain why this is different from the hedge ratio derived by other authors, and make clear whether El Karoui et al. have also derived a similar result.
I wrote something which may sound interesting.
Page 7, lines 32-44: I am confused here. You define the minimal martingale measure as the same as the risk-neutral measure since you use the money market account to discount the expected payoff and use r for the drift of S. I think this is different from the definition I have seen in Schweizer’s work, in which both the stock price and the option price become local martingales. Please explain why you have used a different definition for the minimalmartingale measure or if I am making a mistake here.
Schweizer assumes r=0, after discounting they become (local) martingales, see also El Karoui et al. and others who have worked with positive interest rates.
Page 7, last paragraph: The correlation coefficient is typically negative for equities, but maybe not for FX or commodities.
See point 12 above.
Page 8, last paragraphs: Interesting discussion on the shortcomings of the simpler way of deriving equation 3.
No comment necessary