Category: derivatives

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Arbitrage Problems with Reflected Geometric Brownian Motion

Good news!

 

Our article, “Arbitrage Problems with Reflected Geometric Brownian Motion” by Dean Buckner, Kevin Dowd and Hardy Hulley, has just been accepted for publication by the prestigious journal Finance and Stochastics.

The abstract of the article states:

 

Contrary to the claims made by several authors, a financial market model in which the price of a risky security follows a reflected geometric Brownian motion is not arbitrage-free. In fact, such models violate even the weakest no-arbitrage condition considered in the literature. Consequently, they do not admit numeraire portfolios or equivalent risk-neutral probability measures, which makes them totally unsuitable for contingent claim valuation. Unsurprisingly, the published option pricing formulae for such models violate classical no-arbitrage bounds.

 

What this means in plain English is that our friend Guy Thomas’s article “Valuation of no-negative-equity guarantees with a lower reflecting barrier” in the Annals of Actuarial Science in 2020 is wrong, dead wrong.

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Actuarial fallacy again

Actuary predicting the future

The fallacy is very clearly articulated here.

The author correctly states that “the historical evidence can soundly reject the hypothesis that the expected rate of house price inflation is equal to the risk free rate,” then incorrectly states that “the Black Scholes priced put option gives you that expected value if and only if the expected value of house price inflation equals the risk free rate”.

Kevin and I address the fallacy in a forthcoming paper, but I will briefly discuss it here.

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The Discounted Projection Approach

The standard approach used by UK ERM actuaries to value NNEGs and ERMs is the Discounted Projection (DP) approach, which its adherents like to call the ‘real world’ approach.

This approach is based on the use of a projection of future house price growth to value the NNEG. In particular, it replaces the forward house price as the underlying in the Market Consistent (i.e., correct) approach with some expected future house price ‘forecast’ that a cynic (not us!) might say was indistinguishable from a convenient guess.

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Just Group interim

The Just Group 2019 interim statement is out today. It has already received coverage in mainstream financial media, and we don’t normally repeat what is already said.

What caught our eye, however, was the mention on p.50 of the put option on property index, the NNEG hedge we discussed earlier. We speculated whether the firm has put the hedge in place, or whether they are still waiting to establish ‘appropriate regulatory treatment’ with PRA. Turns out (and we should have spotted this from the 2018 financial statement – 2018 p.72, section 25 ) that it was already in place by 2018, so it is merely the regulatory treatment they are waiting for .

But here’s the interesting thing.

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Quids in

Kevin writes here

It also gets interesting if the firms use different valuation approaches from each other. In that case it would be theoretically possible for both parties to post a profit on the transaction or for both parties to post a loss on it.

He is referring to the approaches used to value the embedded put option in the ERM, and the actual put option used to hedge the ERM.

There is a troubling reminder here of what happened to AIG Financial Products in the run-up to the last financial crisis.

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Just a NNEG Securitisation

In a recent (July 30) posting, Dean wrote that

Since Just Group’s trading update last week, there has been much speculation about the ‘pioneering no negative equity guarantee (NNEG) hedging transaction’ announced by the firm. It is not clear whether the firm has put the hedge in place, or whether they are still waiting to establish ‘appropriate regulatory treatment’ with PRA. The current thinking is that the hedge will be transacted through a major reinsurer, and that it will be a purchase of some form of long dated put option on the housing index.

A related possibility is that the hedge, if there is one, is some kind of NNEG swap, which then raises the question: what is Just’s NNEG valuation?

Short answer: we don’t know and it would be naughty of us to speculate. We don’t have enough information about the firm’s ERM portfolio or enough information about the firm’s valuation approach.

But this got us thinking …

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Just hedging their bet

Since Just Group’s trading update last week, there has been much speculation about the ‘pioneering no negative equity guarantee (NNEG) hedging transaction’ announced by the firm. It is not clear whether the firm has put the hedge in place, or whether they are still waiting to establish ‘appropriate regulatory treatment’ with PRA. The current thinking is that the hedge will be transacted through a major reinsurer, and that it will be a purchase of some form of long dated put option on the housing index.

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The forward paradox

Jeffery and Smith (Equity Release Mortgages: Irish & UK Experience, p.30) discuss the apparent paradox that when we use a ‘real world’ model to project a forward price, then calculate the expected value of put and call options at different strikes, the internal rate of return of those options is considerably different from that obtained using the Black formula. See their table which I have copied below. Put options even have negative discount rates.

Taking the case of the put options, how can we rationalise these negative discount rates? Why would an investor even consider an asset that is expected to lose money, let alone one as risky as a put option which has a chance of expiring worthless, losing everything?

They continue.

The answer is that few rational investors hold a portfolio 100% in a put option. Rather, a put option is a form of insurance held in connection with other assets. An investor in shares can, sometimes with a modest outlay, acquire a put option that substantially mitigates losses in a market crash. The willingness to accept a negative expected return on the put option reflects the reduction of risk to the portfolio as a whole. This is the same reason that buyers of household or motor insurance would not expect (or hope) to make a profit on that insurance.

Are they right? Does the ‘willingness to accept a negative expected return’ really reflect the need to reduce the risk?

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Brownian dilapidation

Source: Aviva

(Geeks only). Thanks to T Pocock for pointing out this amazing page of data on Aviva ERM securitisations, some of them dating back to 2001. Lots of stuff to dig out or back out, including data on NNEG claims. The chart above shows cumulative NNEG claims on ERF4, which was set up in 2004.

No surprise at first. Most ERMs start with a loan to value of lower than 50%, and property prices have gone up in most areas of the UK since 2004, so it takes a few years for the compound loan amount to reach current property value and reach NNEG territory.

But there is a real surprise in store.

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