No-negative-equity guarantees: Black-Scholes and its discontents

The Prudential Regulation Authority (PRA) has issued a consultation paper CP13-18 on valuation of the no-negative-equity guarantee (NNEG) in equity release mortgages.  I think the use of the Black-Scholes formula in this context is flawed, in ways which are more fundamental than suggested by the PRA’s rather bland observation that “some of the assumptions that allow the mathematical derivation of the formula…are not met.”  The prescribed approach is likely to over-estimate the value of NNEG. 


An equity release mortgage is a product where a home owner typically aged 60+ borrows 25-30% of the valuation of their house from an insurer. A fixed rate of interest is charged to the home owner, but not actually paid. On the home owner’s death or any earlier permanent vacation of the house (e.g. after they move into a care home), the loan plus accumulated interest is repayable from the sale proceeds of the house. The NNEG guarantees that the amount repayable will not exceed the sale proceeds of the house.

Equity release mortgages are typically parcelled up into ‘restructured ERM notes’ on an insurer’s balance sheet. The restructured notes earn a relatively high yield and closely match annuity liabilities, so firms are then allowed the regulatory benefit of the ‘matching adjustment’ – an increase in the discount rate which can be used to discount the liabilities when determining the firm’s regulatory solvency. The quantum of the matching adjustment depends on the spread above risk-free rates earned on the ERM notes. This spread is construed partly as an illiquidity premium, but also partly as a risk premium, including the risk of the NNEG; the latter should not in itself give rise to a matching adjustment benefit. The question then arises: in apportioning the spread between illiquidity and retained risks, how should firms value the NNEG? 

CP 13-16 proposes to amend Supervisory Statement 3/17 to mandate the use of a Black-Scholes type model with prescribed parameter values to value the NNEG. [1]CP13-18 para 2.7 & proposed SS3/17 para 3.20. The previous version of SS3/17 was somewhat less prescriptive. In the proposed amendments, some lip service continues to be given to the possibility of alternative approaches[2]CP13-18 para 2.6., but this is now limited by an implication that any results which differ from the prescribed Black-Scholes model will automatically be regarded as suspect.[3]Eg. proposed SS3/17 para 3.22.

The proposed option valuation formula is the Black (1976) formula for an option on a forward price, slightly restated in CP 13-18 as follows:[4] Proposed SS 3/17 para 3.20



–         N() is the standard Normal cumulative distribution function

–         S is the spot price (current price) of the property

–         T is the term to maturity (the NNEG is evaluated separately for each possible future year of maturity, with deterministic assumptions for mortality, morbidity and voluntary repayments)

–         K is the loan plus rolled-up interest at time T

–         r is the prescribed Solvency II risk-free interest rate for maturity T

–         σ is the volatility of the property price, prescribed as 13%

–         q is the deferment rate, prescribed as 1%.

 Where does Black-Scholes come from?

CP 13-18 includes a brief acknowledgement that “some of the assumptions that allow the mathematical derivation of the formula….are not met”[5]CP13-18 para 2.7., but then effectively mandates that formula anyway.  To assess how important the missing assumptions are in the context of NNEG, we need to step back and recall the fundamental constructs from which Black-Scholes is derived.

Black-Scholes is an intuitively very surprising formula.  A natural first thought is that the value of an option would depend on one’s expectation for the price of the underlying at expiry. But Black-Scholes says that this intuition of expectation-based pricing is wrong: the expected rate of growth of the underlying does not appear in the formula.

This surprising result arises from the constructs of dynamic hedging and arbitrage.  Black-Scholes derives a value for an option by considering a long-short hedge portfolio in the option and the underlying. If this is continuously adjusted as the price of the underlying changes, the value of the portfolio can be kept neutral to rate of growth of the underlying.  Black-Scholes then argues that this risk-free portfolio must earn the risk-free rate. Why? Because profit-seeking arbitrageurs make it so. If it were not so, an arbitrageur could go long the riskless hedge portfolio, short riskless zero coupon bonds (or vice versa) and so earn risk-free profits.  Since risk-free profits in practice seem thin on the ground, we conclude that market prices for options are generally set so as avoid such arbitrages. 

To an options market-maker, this no-arbitrage argument – avoiding the possibility that arbitrageurs can earn risk-free profits from our prices – is always more compelling than any expectations-based argument.  Even if the market-maker thinks he has some insight into the expected price at expiry, quoting an expectations-based price is perilous, because it creates an exposure to arbitrages against him.  In other words: no-arbitrage prices take precedence over expectations-based prices because for a market-maker, expectations-based prices are too dangerous to quote.

But the Black-Scholes argument just given depends crucially on the idea of dynamic hedging: the existence of liquid markets which give the ability to continuously adjust the hedge portfolio in underlying and the option. It also depends on the existence of arbitrageurs hungry for risk-free profits, and on market makers with limited capital who cannot afford to bleed risk-free profits to those arbitrageurs. All these elements are missing for housing. There are no markets in NNEGs.  There are no markets in the appropriate underlying, that is forwards (or alternatively futures) in housing.  There are no market makers or arbitrageurs.  Dynamic hedging is simply not possible in any shape or form. This is not a failure of ‘some assumptions’ of Black-Scholes; it is a failure of the whole construct of Black Scholes.

Recognising the absence of forward contracts, the PRA prescribed formula recasts the original Black (1976) formula by substituting the forward price discounted at the risk-free rate, F exp{–rt}, with the deferment price, S exp{–qt}, that is a price agreed now and paid now to take possession of the property in future.  The PRA comments that this substitution alleviates the problem of the lack of an observable forward price, because the deferment price can be readily estimated from the spot price (I will have more to say on this below).  But as far as dynamic hedging is concerned, this modification does nothing to help us: there are still no markets in which to construct the dynamic hedge.

It is sometimes argued (eg Derman and Taleb 2005) that although the Black-Scholes formula as formally derived by academics requires dynamic hedging, it can alternatively be justified by the existence of forward contracts, puts and calls, and the constraint of put-call parity (ie the avoidance by market makers of prices which give rise to static arbitrages).  But even this doesn’t help for housing: there are simply no markets in forward contracts, or puts and calls. If there are no puts and calls, it is hard to see how put-call parity could be a constraint.[6]Derman, E. and Taleb, N.N. (2005) ‘The illusion of dynamic replication’, Quantitative Finance, 5(4):323-326.. By way of simple example, consider an underlying which trades at 100, where the … Continue reading

 Geometric Brownian motion is unrealistic for house prices, especially in the lower tail

The PRA argues that although “some of the assumptions” (in my view: foundational assumptions) for Black-Scholes are not met, this doesn’t really matter:

 “The PRA is aware, as noted by respondents to DP 1/16, that some of the assumptions that allow the mathematical derivation of the formula in paragraph 3.20 for option valuation are not met in the residential property market. However, the PRA has not seen evidence that the approach set out in the proposed updated text of SS3/17 would automatically over- or under-estimate the allowance for NNEG, compared with other methods that are consistent with the four principles”.[7]CP13-18 para 2.7.

I think the PRA’s proposed approach will tend to over-estimate the allowance for NNEG (albeit ‘automatically’ is always going to be a stretch, and in that sense the PRA’s wording here may be carefully chosen). 

Black-Scholes assumes when constructing the dynamic hedge that the underlying in which we trade (a forward which doesn’t actually exist for housing!) follows a geometric Brownian motion.  For housing this would have a positive trend above the risk-free interest rate, but the trend then gets eliminated from consideration in option valuation by the construct of the hedge portfolio.  In effect, the value of the option is then evaluated as the expectation under a ‘risk-neutral measure’ where the underlying has an expected return equal to the risk-free rate, but wanders around that as per the Brownian motion, and in particular, can fall arbitrarily close to zero. The NNEG value resides in the possibility of outcomes in the lower tail.

This model of the underlying seems reasonable for an option on a hedgeable single stock. The expected return on the stock above the risk-free rate (i.e, the equity risk premium) is eliminated for option valuation purposes by the dynamic hedging argument; and the price of a single stock can indeed fall to zero, because companies can indeed go bust.  But there are two reasons why a model in which price can fall arbitrarily close to zero is not a reasonable model for forwards on house prices.

First, the long-term experience in the UK has been that house prices have tended to increase ahead of risk-free interest rates (and also ahead of inflation and even earnings). We do not know to what extent this will continue, but it seems unreasonable – again, in the absence of the hedging argument – to give it no weight at all.[8]One might wonder about a possible inconsistency of housing increasing ahead of earnings indefinitely.  Do housing costs eventually absorb 100% of earnings?  David Miles has thought … Continue reading

Second, a deep and prolonged fall in house prices, with the attendant collapse in mortgage lending, widespread repossessions and distress in the electorate, seems overwhelmingly likely to induce a policymaker response. This is illustrated by the policymaker response to the modest fall in house prices following the 2008 financial crisis: policies such as purchasing first gilts and then corporate bonds (and even equities in Japan); the term funding scheme to revive mortgage lending; and policies such as help-to-buy and associated schemes providing blatantly direct support for house prices. And all this was in response to a quite modest and short fall in prices!  In a deeper or more prolonged slump, there are many more steps which policymakers can (and I believe will) take.  In a country with its own currency, the government can (and I believe will) ultimately print money and buy houses. The activities and statements of central bankers worldwide in relation to asset purchases in recent years provide further general support for this notion of a policy response to deep and prolonged falls in asset (and particularly house) prices.

This is not the same as saying that I think house prices will always go up, or that housing is a better investment than shares, or that you can never lose by buying a house.  I do not believe any of that. I just believe that overwhelmingly likely policymaker responses now provide a reflecting barrier under house prices, which makes geometric Brownian motion an unreasonable assumption for house prices in the lower tail. The level and firmness of that barrier is matter for reasonable debate, but it seems unrealistic to pretend – as the prescribed NNEG formula does – that it does not exist at all.

Update: (29 Jan 2020) These ideas are developed in more detail in my paper “Valuation of no negative equity guarantees with a lower reflecting barrier”, available on my main page.

A couple of decades ago, I did not have this belief in a ‘policy put’.  My views have slowly changed, based on observation of public policy over the past 25 years, and especially the policy response to the relatively minor decline in house prices after 2008. I now recognise that for better or worse, I live in a country where most MP’s own more than one property, former prime ministers buy whole apartment blocks to let,[9] ‘Tony and Cherie Blair’s property portfolio worth estimated £27m’ The Guardian, 14 March 2016. senior Bank of England policymakers assert in unguarded moments that ownership of property is a far superior form of personal investment to pensions[10]‘Property is better bet than a pension says Bank of England economist’ The Guardian, 28 August 2016., etc etc. Policymaker ideologies and preferences can of course slowly change, for example as new generations of MPs are elected; if and when they do, I might slowly change my mind about the reflecting barrier.  But any change is likely to be very gradual, because the ideology amongst policymakers which substantiates the reflecting barrier runs much deeper than the political hue or personalities of any particular government.

One response to my beliefs about a ‘policy put’ on houses prices might be to say that whilst everyone is entitled to their own opinions, different people will have different opinions, and none of these should enter into consideration in option valuation. We should instead stick to the risk-free rate as a ‘neutral’ view. But in the absence of hedging, this purported neutrality is an illusion: assuming (in effect) growth at the risk-free rate is just another belief, and I see no reason to give it primacy.  In the absence of hedging, one has to take a view on the trend in prices; it cannot be conjured away as in the standard Black-Scholes. 

One notable investor has made statements suggesting a belief that analogous arguments hold good for long-term equity indices as well. In his 2008 annual report, Warren Buffett discussed long-term put options written by Berkshire Hathaway on various stock indices.  He gave an example of a 100-year put option on the Dow Jones index, and suggested that the Black-Scholes formula (with typical assumptions at the time of writing) very substantially over-valued this option. Cornell (2009) interprets Buffett’s commentary as reflecting  “ the belief that future nominal stock prices are not well approximated by a lognormal distribution, because inflationary policies of governments and central banks will limit future declines in nominal stock prices compared with those predicted by an historically estimated lognormal distribution”  (I agree with this interpretation.)[11]Cornell, B. (2009) ‘Warren Buffett, Black-Scholes, and the valuation of long-dated options.’ Journal of Portfolio Management, Summer 2010: 107-11.

I also note that the Bank of England ‘stress tests’ for banks involve a scenario where house prices fall one-third in 3 years, but then resume their trend rate of growth. This completely rules out the deep and prolonged falls in house prices for which insurers are being asked to reserve on NNEGs.  It’s not obvious why the reserving requirements for insurers writing NNEGs should be much more stringent than those for banks underwriting ordinary mortgages.

 Deferment price less than spot price? Not necessarily.

In the prescribed NNEG formula quoted above, the PRA calls the quantity S exp{-qt}  the ‘deferment price’ of the property, that is the price payable now to take possession on a future date. There is no meaningful market in deferment prices over the periods of 20-40 years most relevant to NNEGs.[12]One might possibly refer to freehold reversions on short leaseholds, but this seems a feeble argument, because any market is realistically negligible. The PRA nevertheless asserts that the deferment price must always be lower than the spot price of the property, on the following rationale:

“This statement is equivalent to the assertion that the deferment rate for a property is positive. The rationale can be seen by comparing the value of two contracts, one giving immediate possession of the property, the other giving possession (‘deferred possession’) whenever the exit occurs. The only difference between these contracts is the value of foregone rights (eg to income or use of the property) during the deferment period. This value should be positive for the residential properties used as collateral for ERMs.”[13]SS 3/17 proposed para 3.16.

In isolation, this appears a reasonable argument. But there are also reasonable counter-arguments.

Housing today is owned mainly by owner-occupiers.  They have a preference for a current interest to a deferred interest, because they need a roof over their heads, they like long-term security of occupation, they like being able to make their own choices on extensions and repairs, etc. In other words, they like the practical and sentimental benefits of home ownership.  A minority of owners are buy-to-let landlords: they like understandable form of the investment, the unusual ability to finance it largely with borrowed money, and perhaps the disengagement it facilitates from the distrusted pensions and savings industry.

For an insurer, on the other hand, these practical and sentimental benefits of a current interest in a house have no relevance.  The main potential benefit of a current (as opposed to deferred) interest is the potential income from letting. But a current interest also has several disbenefits: tenants need to be managed, houses need to be maintained, from time to time there are costs (Including possibly PR costs) of evicting tenants in arrears, and there is a possibility (through existing or new legislation) that tenants might acquire new rights. If on the other hand houses are kept vacant, this gives another set of problems: council tax, security and maintenance costs, and possibly very considerable PR costs of owning substantial amounts of empty housing.  These disbenefits are not fanciful; their materiality can be inferred from the observable fact that despite the excellent long-term performance of housing as an investment, neither insurers nor any other financial institutions have shown any enthusiasm over the past several decades for housing as an asset class.

So current interests in houses are evidently not attractive to insurers and other institutional investors. Deferred interest might well be more attractive, particularly if in the form of cash-settled financial contracts, so that all the problems of current interests are permanently avoided. Even if a deferred interest is not strictly preferred, the relative valuation of a deferred interest compared to a current interest seems very likely to be much higher for an insurer than a typical individual owner. 

Now if there were a substantial market for deferred interests, the money weight of individuals’ preference for current interests versus insurers’ preference for deferred interests would determine the relative market prices for the two types of interest (i.e. what the PRA calls the ‘deferment rate’). But we have the same problem as with the hedging arguments: the market for deferred interests does not exist on any meaningful scale.  And this is not mere happenstance or oversight; to create such a market would require the development of legal and governance frameworks covering maintenance, insurance, the rights of occupiers during and on maturity of deferred interests, etc.  In the absence of such a framework, the relative values of current interests and deferred interests remain a matter of conjecture. The PRA’s argument is a reasonable one, but not the only reasonable one, and therefore not as conclusive as CP 13-18 asserts.[14]Eg SS3/17 proposed para 3.16.

Negative deferment rates might offset omission of the reflecting barrier

The PRA argument that the deferment price should always be less than the corresponding spot price is sometimes characterised as a ‘positive deferment rate’ (i.e. the rate ‘q’ in the deferment price = S exp{–qt) is positive). The PRA says that some insurers may be using a deferment rate that is ‘too low’.  Separately, I also noted above that the assumption of geometric Brownian motion seems unrealistic for long-term house prices in the lower tail, and that it would be more realistic to have a reflecting barrier under prices to represent the likely policy response to a deep and prolonged fall in house prices.  Either the ‘error’ of omitting this reflecting barrier, or the ‘error’ of using a deferment rate that is ‘too low’, will act to reduce the NNEG valuation.  So a low (or even negative) deferment rate combined with omission of the reflecting barrier might arrive at approximately the right answer, albeit arguably for the wrong reasons.


(1)           In the context of NNEGs, the complete inapplicability of dynamic hedging (or even put-call parity) makes the prescribed Black-Scholes formula somewhat arbitrary.  At the very least, it seems unjustified to label this ‘correct’, and all other approaches ‘incorrect’.

(2)           A deep and prolonged fall in house prices is almost certain to lead to a policymaker response.  This means that that the geometric Brownian motion assumed in Black-Scholes is too heavy in the lower tail. Since most of the NNEG value arises from this tail, the prescribed approach seems likely to over-value the NNEG. There should be some implicit or explicit allowance in NNEG valuation for this policymaker response.    

(3)           Reserving requirements for insurers underwriting NNEGs should not be more stringent than those for banks underwriting ordinary mortgages.

(4)           The PRA’s argument that the (hypothetical, unobserved) deferment price should always be less than the spot price (specifically: a minimum ‘deferment rate’ of 1%pa over the term of the NNEG) is not as obvious as CP-13-18 suggests.  There are good counter-arguments, which may justify a lower (perhaps even negative) deferment rate.

Update ( May 2020): My paper Valuation of no-negative-equity guarantees with a lower reflecting barrier deals with point (2).


1 CP13-18 para 2.7 & proposed SS3/17 para 3.20.
2 CP13-18 para 2.6.
3 Eg. proposed SS3/17 para 3.22.
4  Proposed SS 3/17 para 3.20
5, 7 CP13-18 para 2.7.
6 Derman, E. and Taleb, N.N. (2005) ‘The illusion of dynamic replication’, Quantitative Finance, 5(4):323-326.. By way of simple example, consider an underlying which trades at 100, where the call option with a 105 strike trades at 8 and the put at 3 (i.e. a time value of 3 for each). Now suppose the underlying starts trending upwards.  Intuitively, we night guess that the call is now worth say 10 and the put is worth 2.  But given puts, calls and a forward  contract, this would create an arbitrage. Put-call parity in effect requires that the put and call are both independent of the trend in the underlying.
8 One might wonder about a possible inconsistency of housing increasing ahead of earnings indefinitely.  Do housing costs eventually absorb 100% of earnings?  David Miles has thought about this hbbv6 watch. He suggests there are no compelling economic reasons why houses shouldn’t eventually become assets like jets: utilised by many people, but owned by only a very few. See Miles, D. and Sefton, J. (2018)  ‘Houses across time and across place’.
9  ‘Tony and Cherie Blair’s property portfolio worth estimated £27m’ The Guardian, 14 March 2016.
10 ‘Property is better bet than a pension says Bank of England economist’ The Guardian, 28 August 2016.
11 Cornell, B. (2009) ‘Warren Buffett, Black-Scholes, and the valuation of long-dated options.’ Journal of Portfolio Management, Summer 2010: 107-11.
12 One might possibly refer to freehold reversions on short leaseholds, but this seems a feeble argument, because any market is realistically negligible.
13 SS 3/17 proposed para 3.16.
14 Eg SS3/17 proposed para 3.16.

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