QuoteTo provide some more clarity about the paper I am attempting to implement (it is not publicly available yet; as far as I can tell, you must either pay to download it from SSRN or wait until its forthcoming journal publication) here is the abstract:Google give relevant link:http://staff.cbs.dk/abtrolle/stochasticVolatility.pdf

Thanks Mars for providing the link to the paper. MJ, I think wires might be getting crossed, which may be due to my not understanding what I'm actually saying. If that's true, then I'd want to know, because that likely means that I'm chasing the wrong rabbit down a hole. So let me try to explain somewhat differently.Let's suppose that I want to build a model for interest rates and I decide to use a HW2. So I go over to Brigo and Mercurio and use their methodology to calibrate a G++ model to swaption volatilities. So now I have a set of calibrated parameters and I know my liability greeks at time zero, so I purchase a set of swaptions to hedge my risk. Now roll forward a week. Swaption prices have changed and the current value of my liabilities and swaption basket have also changed. So now I recalibrate my HW2 to new market data and repeat. I don't think there's any misunderstanding here, but this is the starting point for the original mental exercise that lead to the model I'm trying to build. Now suppose that it's still today and I want to see how well my hedging strategy works ,a priori , by simulating the process above. That is, I have my current liabilities and current assets that make me rho neutral. I want to not only forecast interest rates (hence the stochastic interest rate component), but I want to recalibrate the HW2 model to forecasted swaption prices to mimic what I would actually being doing each day or week or whatever. Based upon my 'new' HW2 parameters I would revalue my swaption portfolio and determine what my new swaption purchases are, etc.What I most definitely do not want to do is keep the same HW2 parameters fixed throughout the simulation and assume that markets are going to remain the same. But I don't see how I can recalibrate a stochastic interest model (within a simulation) without having some kind of forecast for swaption prices. Now, having said that, the above thought experiment strikes me as terribly useless unless I am in possession of either a hugely powerful grid or a series of super computers, which I'm not. This kind of recalibration is too computationally expensive, especially when tied-in to a similar equity/implied vol process. Thus I turn to the paper, that Mars so kindly provided a link to, which I believe satisfactorily addresses both my needs and my limitations.So I guess the answer to the question about whether I need interest rate vol can be concluded by addressing the following: How can I simulate the process of recalibrating an interest rate model to swaptions without some model for interest rate vol? And if the answer to that is "I can't", then my next question is: Does the paper previously mentioned provide a reasonable method for doing that?And if I am still failing to be clear, then I must conclude that I am misunderstanding something fundamental and more study is required before I attempt to post again.In any event, I appreciate the time taken to read my posts and attempt to help me.Best Regards,J

okWell, I have thought about these issues a fair bit and even applied for a research grant recently to a similar analysis (we'll see if i get it...)As I see it you need two models:1. a real-world model to simulate how the market changes from day to day -- this includes the swaps and swaptions. we'll call this model A2. a pricing model . We'll call this model B. Each day B is calibrated to the output of A. Hedges are purchased and funded. We go forward a day. We repeat. We do this until we get to the end. This gives a single path result. We do this many times.Clearly, model A needs a stochastic vol component since the swaption surface moves around. Model B may or may not have such a component. Should we let A=B? If we do then we don't really need to run the test since with perfect knowledge of the parameters and only a small number of state variables we can perfectly hedge, and there will be variance. More realistic is to assume that model A is unknown, and model B is a model that would be used on the trading desk. We then see how it performs for various choices of model A. Note that A should be a real-world model and the objective is really to reproduce statistical properties of rates and surfaces not to price things with it. Whereas B is a risk-neutral model and the objective is to reproduce the prices of traded instruments (outputted by A or the market.)

MJ,Yes! That is precisely how I saw the problem. In fact, that is how I've implemented it relative to equities. But the real hurdle I'm facing is that, given that in an ideal world I would would want risk-neutral model B calibrated at each time step to real-world model A, practically I'm not sure about its feasibility. The paper that I've mentioned, however, does something quite similar. It is essentially a 'meta-model' that encompasses both models A and B, but in a manner that seams quite clever, as far as I can tell. The advantage being that I can model the real world scenarios while simultaneously modeling the risk-neutral implied volatility. In the long-run it may not as good as a fully partitioned model, but I think that the balance of the equities may fall in its favor (I have a tendency to describe the tension between mathematical beauty and corporate necessity in legal terms; I suppose it operates as a nod to my law degree, which I don't seem to use nearly as much as I would like!).I would like to express my thanks to you, Mark, for not only providing me feedback, but forcing me to thoroughly think through what I wanted to do so as to express it more clearly. I am very wet behind the ears in the quant game, but I am still well aware of your name and greatly appreciate the interaction.And I wish you the best of luck in your pursuit of the grant!Best,J

I am curious what interest rate model you use for pricing liability.

I'm late to this conversation but I'm happy to see VA guarantees on this website at along last!I've been hedging these for a bunch of different companies for the last several years and have a few thoughts to throw in. First, proper valuation requires some surface model for the equities. I prefer time-dependent Heston calibrated to a 10 year dealer vol surface. Stochastic rates are necessary too since the products are so long term. If its a GMAB or GMDB, a HullWhite 1F is enough. These two put-like guarantee types are only interest rate sensitive through the discounting mechanism. Their payoffs occur at one fixed point in time. If its a GMIB, where the payoff at maturity is sensitive to the whole curve, you need a 2factor model to get twists. If its a GMWB, I would prefer a 2factor model but a one factor model might be sufficient. WBs are much more interest rate sensitive because their cash flows are stretched out over a spectrum of time (eg 7% for life). Such cash flow structures have more rate convexity and require realistic rate dynamics. It's also important to not that the guarantees are on a basket of equity, bond and money markets. The risk-neutral equity fund returns are simulated by some combination of S&P, Russell, Nasdaq, EAFE indices (which are each calibrated to vol surfs). The risk-neutral money market return is linked to the short rate in the rate model (which is calibratred to swaptions). Risk neutral bond fund returns are linked to the yield curve movements (make some assupmtion about the bond fund's key rate duration profile) AND credit spreads (whose dynamics can be calibratred to CDX swaps and swaptions). So, the put-like guarantee has EXPLICIT rate (and credit) optionality through the fund returns, as well as having rate exposure through the discounting process (and curve shape in the case of GMIBs. Some new products dynamically shift the bond/equity fund allocations. For example, Pru's Highest Daily WB for life product does this. Last fall, they underweighted equities and overweighted bonds. Because of this kind of over allocation to bonds, the rate exposure can be a LOT more delicate. So some insurers are moving to LMM to hedge the rate exposure. In any event, you need a hybrid model that consistently calibrates to equity options and rate and credit curves and swaptions simultaneously. In terms of "real world" scenarios for the outter loop in the stocahstic on stochastic projection: LeRoux presents a great method for simulation equity vols. There has been some work on swaption surface simulation as well. There are lots of principal components in the swaption surface, so research has focused on structural models, regime-switching models, to explain the dynamics.

I'm late to this conversation but I'm happy to see VA guarantees on this website at along last!I've been hedging these for a bunch of different companies for the last several years and have a few thoughts to throw in. First, proper valuation requires some surface model for the equities. I prefer time-dependent Heston calibrated to a 10 year dealer vol surface. Stochastic rates are necessary too since the products are so long term. If its a GMAB or GMDB, a HullWhite 1F is enough. These two put-like guarantee types are only interest rate sensitive through the discounting mechanism. Their payoffs occur at one fixed point in time. If its a GMIB, where the payoff at maturity is sensitive to the whole curve, you need a 2factor model to get twists. If its a GMWB, I would prefer a 2factor model but a one factor model might be sufficient. WBs are much more interest rate sensitive because their cash flows are stretched out over a spectrum of time (eg 7% for life). Such cash flow structures have more rate convexity and require realistic rate dynamics. It's also important to not that the guarantees are on a basket of equity, bond and money markets. The risk-neutral equity fund returns are simulated by some combination of S&P, Russell, Nasdaq, EAFE indices (which are each calibrated to vol surfs). The risk-neutral money market return is linked to the short rate in the rate model (which is calibratred to swaptions). Risk neutral bond fund returns are linked to the yield curve movements (make some assupmtion about the bond fund's key rate duration profile) AND credit spreads (whose dynamics can be calibratred to CDX swaps and swaptions). So, the put-like guarantee has EXPLICIT rate (and credit) optionality through the fund returns, as well as having rate exposure through the discounting process (and curve shape in the case of GMIBs. Some new products dynamically shift the bond/equity fund allocations. For example, Pru's Highest Daily WB for life product does this. Last fall, they underweighted equities and overweighted bonds. Because of this kind of over allocation to bonds, the rate exposure can be a LOT more delicate. So some insurers are moving to LMM to hedge the rate exposure. In any event, you need a hybrid model that consistently calibrates to equity options and rate and credit curves and swaptions simultaneously. In terms of "real world" scenarios for the outter loop in the stocahstic on stochastic projection: LeRoux presents a great method for simulation equity vols. There has been some work on swaption surface simulation as well. There are lots of principal components in the swaption surface, so research has focused on structural models, regime-switching models, to explain the dynamics.

Hi Mark, Can you recomend any articles on "a hybrid model that consistently calibrates to equity options and rate and credit curves and swaptions simultaneously"?Thanks,YQuoteOriginally posted by: markhadleyI'm late to this conversation but I'm happy to see VA guarantees on this website at along last!I've been hedging these for a bunch of different companies for the last several years and have a few thoughts to throw in. First, proper valuation requires some surface model for the equities. I prefer time-dependent Heston calibrated to a 10 year dealer vol surface. Stochastic rates are necessary too since the products are so long term. If its a GMAB or GMDB, a HullWhite 1F is enough. These two put-like guarantee types are only interest rate sensitive through the discounting mechanism. Their payoffs occur at one fixed point in time. If its a GMIB, where the payoff at maturity is sensitive to the whole curve, you need a 2factor model to get twists. If its a GMWB, I would prefer a 2factor model but a one factor model might be sufficient. WBs are much more interest rate sensitive because their cash flows are stretched out over a spectrum of time (eg 7% for life). Such cash flow structures have more rate convexity and require realistic rate dynamics. It's also important to not that the guarantees are on a basket of equity, bond and money markets. The risk-neutral equity fund returns are simulated by some combination of S&P, Russell, Nasdaq, EAFE indices (which are each calibrated to vol surfs). The risk-neutral money market return is linked to the short rate in the rate model (which is calibratred to swaptions). Risk neutral bond fund returns are linked to the yield curve movements (make some assupmtion about the bond fund's key rate duration profile) AND credit spreads (whose dynamics can be calibratred to CDX swaps and swaptions). So, the put-like guarantee has EXPLICIT rate (and credit) optionality through the fund returns, as well as having rate exposure through the discounting process (and curve shape in the case of GMIBs. Some new products dynamically shift the bond/equity fund allocations. For example, Pru's Highest Daily WB for life product does this. Last fall, they underweighted equities and overweighted bonds. Because of this kind of over allocation to bonds, the rate exposure can be a LOT more delicate. So some insurers are moving to LMM to hedge the rate exposure. In any event, you need a hybrid model that consistently calibrates to equity options and rate and credit curves and swaptions simultaneously. In terms of "real world" scenarios for the outter loop in the stocahstic on stochastic projection: LeRoux presents a great method for simulation equity vols. There has been some work on swaption surface simulation as well. There are lots of principal components in the swaption surface, so research has focused on structural models, regime-switching models, to explain the dynamics.

markhadley:Your general advice is reflected in several US consulting firms' annuity hedging platforms: multi-factor Hull-White, emphasis on fund basis mapping, Heston equity volatility, emerging interest in credit spread basis, etc.This generic platform misses the heart of insurance-linked risk, however:1. GMxB liabilities have far more rate optionality than simple discount exposure. Some, like GMIB, even have hard rate optionality. THIS MEANS THAT INSURANCE LIABILITES ARE SIGNIFICANLTY MORE SENSITIVE TO RATE VOLATILITY THAN YOU HAVE PRESUMED. This was the heart of haojiew's initial posting in this thread.2. Heston can't begin to capture the long-dated skews you see in practice when you go to hedge these risks. Furthermore, your dealers aren't using Heston and you will be blown out of the water when you see actual hedge costs.3. Most consulting firms superficially model credit spreads rather than the underlying default risk. This is a useful approach for modeling asset portfolios but a rather dangerous and inadequate approach for modeling derivative portfolios. The distribution matters.4. Unlike financial derivatives where underlying, exercise, and maturity are known with certainty, the structure of insurance-linked GMxB guarantees is random and market linked. These dynamics create additional market exposures that are invisible to the market models you have proposed. This may be why some insurance companies are reporting poor hedging results.5. The whole concept of real-world stochastic projections is overblown. Because it is difficult to do, it is assumed to be really valuable, when in fact you are simply computing statistics about a model that has little to do with reality in the first place. Carefully constructed sensitivity tests and single-scenario analysis are far easier to calculate, far more easily understood, far more accurate/useful, and far less expensive to produce.I recommend that yqaz and jdobiac talk to their dealers and see how they value/manage hybrid risk. They will tell you a lot, although you have to get them past the sales pitch and on to the quant-talk.To jdobiac, I would recommend that you reach out to one of the primary GMWB dealers and see if you can arrange a risk partnership. This is a complex risk and you don't want to cut your hedging teeth on that kind of product. Far better to learn side-by-side with good partner.

MarkHadley,Thanks for your input. It actually sounds a lot like advice I've gotten from some Milliman cosultants which, based upon your location, wouldn't surprise me if you were with that particular organization. In terms of your comments regarding the "outer loop", I agree that LeRoux does a good job, but that comparatively there are a lot of principal components in the swaption surface, which was part of my hesitation of essentially replicating the equity portion of the model on the interest rate side by using HW2 and some vol surface forecast. Hence the paper that I was trying to implement. Also I think our product is particularly sensitive to some of the criticism that ClosetChartist has mentioned. It just seems that a HW2 factor will not adequately model the underlying risk exposures. ClosetChartist,Your criticism is well taken and already a big discussion piece both within my group and with the VA group here. We are one of the largest VA sellers in the US, so that side has had a lot of experience with their hedging program and we've had the advantage of learning from their mistakes. Our product is slightly different, howver, because while it has a GMWB, it is not a VA per se. In many ways our product is more complex and so we've had to try to break new ground in how to accurately model it. What I've been working on is but one part of a complete revision of our modeling methodology, but we are working hard at trying to capture the hidden optionalities of which you speak. And we have had many discussions with other parties in terms of risk sharing, but mostly on the equity side so that we can directly hedge a large portion of that particular risk. Dealing with the hybrid risk movements is something we're trying to quantify and we have been working with others on how best to capture that.

Hi ClosetChartist,I thought you had some very interesting comments, and I was hoping to discuss these points a little further... QuoteOriginally posted by: ClosetChartistmarkhadley:Your general advice is reflected in several US consulting firms' annuity hedging platforms: multi-factor Hull-White, emphasis on fund basis mapping, Heston equity volatility, emerging interest in credit spread basis, etc.This generic platform misses the heart of insurance-linked risk, however:1. GMxB liabilities have far more rate optionality than simple discount exposure. Some, like GMIB, even have hard rate optionality. THIS MEANS THAT INSURANCE LIABILITES ARE SIGNIFICANLTY MORE SENSITIVE TO RATE VOLATILITY THAN YOU HAVE PRESUMED. This was the heart of haojiew's initial posting in this thread.Could you elaborate? It's well-known that a GMIB has hard rate optionality. Are you saying that the generic platforms from the consultants ignore the life annuity rate guarantees embedded in GMIB options? That would be surprising.Also, how does your comment apply to products that do NOT have hard interest rate optionality (such as GMDB and GMAB)? In every model, the risk-free rate scenarios are used as the drift in risk-neutral underlying dynamics and the same scenarios are used for discounting cash-flows. What else is there? It would seem to me that this is fully capturing the sensitivity of liabilities to movements in interest rates. Now the the chosen stochastic interest rate model might be flawed but that is a separate issue from the nature of option value's exposure to interest rates.QuoteOriginally posted by: ClosetChartistmarkhadley:2. Heston can't begin to capture the long-dated skews you see in practice when you go to hedge these risks. Furthermore, your dealers aren't using Heston and you will be blown out of the water when you see actual hedge costs.From what I understand, you are saying that the historical implied volatility surface for equities has statistical behavior in the long-end that cannot be properly captured by the Heston model. Could you explain exactly which empirical feature of this "long-dated skew" makes the Heston model inappropriate? Is there something special about the long end of the vol surface that makes the Heston model inappropriate, or are you saying that the Heston model just doesn't work at any maturity?Also, what model ARE the dealers using? Why does it even matter what model they are using? Is the dealer's model the "one true model" since it is the the dealer who sets bid/ask prices? What if different dealers are using different models? Finally, you say that the dealer quotes for long-dated volatility are very expensive, which means that hedging the implied vol exposure is very costly. That may be true nowadays, but why would high cost of hedging by itself be an indicator of a correct or incorrect model? If the models and dealer quotes indicate that the cost of hedging the risks are very high then doesn't simply indicate that the direct writers need to raise GMxB insurance premiums? QuoteOriginally posted by: ClosetChartistmarkhadley:3. Most consulting firms superficially model credit spreads rather than the underlying default risk. This is a useful approach for modeling asset portfolios but a rather dangerous and inadequate approach for modeling derivative portfolios. The distribution matters.Where do these credit spreads enter the model? A portion of the underlying basket in a GMxB option is often a bond fund (mutual fund). Are you are talking about using credit spreads to model the bond fund NAV stochastic process? And are you proposing that the bond fund NAV dynamics should be based on a model of corporate bond defaults? Roughly, what would such a default risk model look like? What is wrong with just assuming that the bond fund value is determined by the risk-free term structure of interest rates plus a credit spread? I would wager that the most common approach in the insurance industry is to just pretend that the bond funds are equity funds (i.e. they pretend that bond fund value behaves just like a stock price). I think that simply moving away from pretending that the bond fund as a portfolio of stocks would be one giant leap for the industry. Of course another solution is to never sell options on managed bond funds in the first place. QuoteOriginally posted by: ClosetChartistmarkhadley:4. Unlike financial derivatives where underlying, exercise, and maturity are known with certainty, the structure of insurance-linked GMxB guarantees is random and market linked. These dynamics create additional market exposures that are invisible to the market models you have proposed. This may be why some insurance companies are reporting poor hedging results.Not knowing the underlying with certainty is an insane problem that the industry has created for itself. The solution is very simple: know the underlying with certainty (but good luck convincing the actuaries who dream up these monstrosities). QuoteOriginally posted by: ClosetChartistmarkhadley:5. The whole concept of real-world stochastic projections is overblown. Because it is difficult to do, it is assumed to be really valuable, when in fact you are simply computing statistics about a model that has little to do with reality in the first place. Carefully constructed sensitivity tests and single-scenario analysis are far easier to calculate, far more easily understood, far more accurate/useful, and far less expensive to produce.If real-world stochastic projections provide statistics about a model that has little to do with reality then wouldn't sensitivity tests and single-scenario real-world projections also just be statistics about a model that has little to do with reality? Either the model is useful and then we can produce useful statistics about that model, or the model is useless and has nothing to do with reality. If the model is useless then there is no point in computing the parameter sensitivities of the model since that information would be useless by definition. The fact of the matter is that real-world stochastic projections (i.e. stochastic-on-stochastic) are required by the insurance regulators and credit rating agencies, even though it can leads to an astronomically huge simulation that typically takes several weeks to complete. So in a purely practical sense these projections are very necessary and useful, if only due to regulations. From a theoretical standpoint, if a single scenario can give you some kind of useful information, then how could multiple scenarios possibly degrade the accuracy or usefulness of that information? QuoteOriginally posted by: ClosetChartistmarkhadley:I recommend that yqaz and jdobiac talk to their dealers and see how they value/manage hybrid risk. They will tell you a lot, although you have to get them past the sales pitch and on to the quant-talk.I'm curious as to what is meant by "hybrid risk?" From the context, it sounds like it just refers to recognizing that an option position is a bet on interest rates, implied vols, etc, and is not simply just a bet on the underlying. QuoteOriginally posted by: ClosetChartistmarkhadley:To jdobiac, I would recommend that you reach out to one of the primary GMWB dealers and see if you can arrange a risk partnership. This is a complex risk and you don't want to cut your hedging teeth on that kind of product. Far better to learn side-by-side with good partner.By "primary GMWB dealers" do you mean the direct writer insurance companies? And by "risk partnership" do you mean one insurance company sharing its trading desk, option models, hedging strategies, software, etc, with another insurance company? Why would any insurance company want share this kind of core intellectual property? The only such partnership that is currently possible in practice, as far as I know, is to enter into a deal with one of the insurance brokers / consultants who offer VA hedging services / platforms to direct writers.

Hello Pannini.Thank you for the careful reading of my earlier comments. Here are some partial replies...1. The interest rate models in commercial insurance-hedging software do not allow very rich rate volatility structure (i.e. vol as a function of calendar time and maturity). GMxB are quite sensitive to this structure but most people do not realize this because they are using overly tame interest rate models. A more reasonable interest rate model will generate much larger option values and spur much larger hedge budgets.2. The Heston model implied vol surface goes flat after about 7 years or so and you cannot "fix" this problem without making short dated skews obscenely large. The model simply does not reflect real quotes. This is a problem for product pricing because you cannot model the costs you will actually incur during hedging.3. Implied vol doesn't affect the payout on a GMxB one iota. Whether 5yr imp vol was 25% or 25,000% doesn't affect the policyholder payout one dime. So why hedge implied vol? Because it affects quarterly reporting. But this reporting hedge is in-addition to the economic hedge and has an incremental cost associated with it. But unless your valuation model includes stochastic implied vol, you have assumed static implied vol and your valuation includes no cost provision implied vol hedging.For example, the Black-Scholes model says that ONLY equities and bank accounts move over time and I can replicate any put payout by trading equity and a bank account. I CAN hedge implied vol vega, but the models says that vega is constant and that vega hedging is extra and unnecessary. The cost of the extra-model hedge is greater than most people appreciate.4. Modeling bonds and risk-free plus a spread is free money. I get extra income from bonds with no additional risk. This throws a serious wrench in the whole risk-neutral pricing framework. In some cases, the impact can be quite severe.5. If you are going to be wrong be wrong in a simple, transparent way. Scenario analysis isn't inherently more accurate than real-world simulation analysis, but it is much easier to understand and respect the inherent limitations. The last thing you want to do is make a decision off some nuance of your real-world scenario generator that just happened to be leveraged by your derivative portfolio.6. Hybrid risk means instruments that combine multiple sources of optionality, such as rates and equities combined in a GMIB.7. A partnership can work well when one company has the expertise but lacks the ability to originate the business. If I can hedge what you sell but can't originate the business myself (due to market penetration, system constraints, capital constraints, etc.) then I might be interested in collaborating with you. Just because I hedge for you doesn't mean I give my secrets away.

The structures of gmxb are becoming more and more complex over the past 10 years. So, I totally agree that one needs rich interest rate volatility structure and better hybrid market models for hedging the optionality of gmxb products. However, my question is whether the hedging performance can be improved by adopting super market models. I think another source of the uncertainty of the gmxb produces comes from the uncertainty of policyholder behavior. eg. when a gmib owner will utilize his gmib option? Variable annuities are insurance products that are quite different from financial derivatives. A policyholder's decision is not only based on market condition. Instead, sales agents can make policyholders make sub-optimal decision. Currently different policyholder behavior models can generate quite different gmxb option values. The difference can be as huge as 100 times. Due to the long waiting period for gmxb, the insurance companies has no creditable history on policyholder behavior to calibrate their models. How can this problem be factored into a hedging program? How can the hedging effectiveness be evaluated when actuarial models are changing?

Rich models at least ensure that you are charging for the risk you take on. The hedging decision is complex. For example, do you hedge rate vol or not? You need to consider what product features are driving your rate vol exposure (does it come from the benefit structure, from policyholder behavior, etc.) and whether you are confident enough in your modeling of this exposure to put real hedge dollars on the table.A reasonable approach to modeling and hedging variable annuities is to "Do no harm." My model may be overly simplistic and off by 100x but so long as I can consistently deliver hedge results that reliably offset a fraction of the benefit I am ahead of the game. I don't try to hedge to the Truth, because I don't know what the Truth is. I only try to be consistently helpful.As policyholder experience comes in, maybe I change my hedge, maybe I don't.