My paper Long-term option pricing with a lower reflecting barrier has just been published by Annals of Actuarial Science. This post gives background context, and touches on some points not covered in the paper.
The intended application is pricing of options over “actuarial” timescales – at least one decade, and typically two or three. The most obvious example is no-negative-equity guarantees (NNEG) in equity release mortgages.
In an earlier blog, I explained the rationale for assuming a lower reflecting barrier (say 50% under today’s house prices) as follows:
“[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….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.”
Similar arguments about the prospect of intervention after a large fall in prices might also apply for equity indices. But I am a bit less enthusiastic about this. For housing, there are two types of argument to substantiate the notion of a reflecting barrier: the prospect of electorally-motivated intervention as detailed in the quote above; and the economic nature of the asset, which has some differences compared to equities. To be specific:
- freehold land is an absolute claim (cf. equities, a residual claim);
- the supply of new land is less elastic than the supply of new equity; and
- houses in good neighbourhoods are positional goods.
These economic arguments do not apply for equities, and hence the barrier notion may be somewhat less compelling. (The new paper gives no detail on these arguments, but for a fuller discussion see Thomas 2021, sections 2 and 3.)
Although the asset model is non-standard, the paper otherwise follows a typical paradigm for option pricing: an option price is assumed to be the lowest initial cost from which the payoff can reliably be replicated. Two interlocking issues not considered in this typical paradigm, and hence not considered in the paper, are credit risk and margin requirements. These issues may be particularly important for longer terms.
Perhaps the best-known writer of long-term puts is Berkshire Hathaway. From the descriptions in Berkshire’s annual reports, Warren Buffett’s views on the long-term equity index puts Berkshire wrote between 2002 and 2007 appeared to encompass the following:
- Credit costs and credit risk were more salient constructs than replication or volatility.[i]
- It was an essential feature that negligible collateral (margin) payments were required.[ii] After the rules on this changed, Berkshire wrote no more puts.[iii]
- Black-Scholes “produces wildly inappropriate values when applied to long-dated options”.[iv]
For the Buffett puts, there was credit risk for the buyer, but no margin payments required from the writer.
For NNEG, there is no credit risk for the buyer, because the option payoff is netted off a loan redemption the borrower is otherwise obliged to pay. But there is credit risk for third parties: annuitants and other customers of the insurer. Hence the regulator imposes “margin requirements” on the NNEG writer, by requiring Solvency II capital to be appraised by reference to a specific option pricing model (i.e. the Black-Scholes model specified in the PRA’s Effective Value Test).
The contrast between the Buffett puts and NNEGs highlights a potentially reflexive or circular aspect of option pricing, which is not captured in standard models (including the barrier model):
- Buffett assigned a low value to his puts partly because there were no margin requirements.
- Insurers charge high prices for NNEG (equivalently, high interest rates on lifetime mortgages) partly because high “margin requirements” are imposed by the regulator.
In both cases, the prices at which options are supplied depends on the contractual or regulatory model used to determine the margin requirements; and so the price will tend to validate the model. That is, if the margin requirements are based on a model which places a higher value on the option, the prices at which sellers are actually prepared to supply options will be higher, and vice versa. (I touched on this idea in a previous blog, see here.)
The standard escape from this circularity is to appeal to replication arguments: an option price is the lowest initial cost for which we could reliably replicate the payoff. But this is compelling only for an option writer who either intends to do the replication, or is vulnerable to arbitrage by others who can do it. For options on house prices, nobody expects to do the replication, and indeed nobody can. So the replication arguments are less compelling.
How then should we view a replication price when replication isn’t expected to be done, and indeed can’t be done? One way is to think of it is as a Schelling point: a weakly justified consensus which tends to prevail in the absence of anything more compelling.
For long-term puts (at least one decade, typically two or three), there currently doesn’t seem to be a successful Schelling point:
- Some people advocate Black-Scholes, but it is not obvious that they have thought about it much (try searching phrases like “long term option pricing”).
- Buffett says that this is “wildly inappropriate”, but hasn’t suggested anything in its place.
- Insurers writing NNEGs tend to use the “real-world” method (answers up to an order of magnitude smaller than Black-Scholes).
- Others complain, with some justification, that this doesn’t seem to have any specific rationale.
The barrier model can be thought of as the “halfway house” sought in this ARC research project some years ago. It keeps the replication principle of Black-Scholes, but with a different model for the asset price. With my preferred assumption of a barrier 50% below the spot price, it typically gives values for long-dated puts somewhere in between “real-world” and Black-Scholes.
Perhaps there is some better Schelling point, predicated on a different principle that nobody has thought of yet. In the meantime, relative to real-world and Black-Scholes, the barrier model does at least typically split the difference!
Here’s a link to full paper (if you prefer a pdf version, the button is just below the author’s name)
[i] Berkshire Hathaway Chairman’s Letter, 2008, p20:
“But if we had received our theoretical premium of $2.5 million up front, we would have only had to invest it at 0.7% compounded annually to cover this loss expectancy. Everything earned above that would have been profit. Would you like to borrow money for 100 years at a 0.7% rate?…
…Remember that 99% of the time we would pay nothing if my assumptions are correct. But even in the worst case among the remaining 1% of possibilities – that is, one assuming a total loss of $1 billion – our borrowing cost would come to only 6.2%.
…The ridiculous premium that Black-Scholes dictates in my extreme example is caused by the inclusion of volatility in the formula and by the fact that volatility is determined by how much stocks have moved around in some past period of days, months or years. This metric is simply irrelevant in estimating the probability-weighted range of values of American business 100 years from now.”
[ii] Berkshire Hathaway Chairman’s Letter, 2008:
“Our derivative dealings require our counterparties to make payments to us when contracts are initiated. Berkshire therefore always holds the money, which leaves us assuming no meaningful counterparty risk.” (p18).
[iii] Berkshire Hathaway Chairman’s Letter, 2011:
“One important industry change, however, must be noted: Though our existing contracts have very minor collateral requirements, the rules have changed for new positions. Consequently, we will not be initiating any major derivatives positions. We shun contracts of any type that could require the instant posting of collateral.” (p17).
[iv] Berkshire Hathaway Chairman’s Letter, 2010:
“Both Charlie and I believe that Black-Scholes produces wildly inappropriate values when applied to long-dated options…We believe the true liability of our contracts to be far lower than that calculated by Black-Scholes, but we can’t come up with an exact figure…”