First, I need to be clear what I mean by “cheap” and “dear”. In this blog, I characterise an option as cheap (dear) if its price today is below (above) its expected value at maturity, discounted at the risk-free rate back to today. The expected value at maturity is calculated using a reasonable assumption for the underlying asset’s risk premium. I am not saying that I wish to buy or sell the options I label cheap or dear (I may wish to with some of them, but the label says nothing definite about that.)
In a recent paper on equity release mortgages, Tony Jeffery and Andrew Smith note that the Black- Scholes formula gives prices for call options which are cheap (and put options dear), in the sense above. They note that this raises a puzzle: why should there be willing buyers for put options at prices which are expected to lose money? Their answer is that put options are a form of insurance, held in conjunction with other assets, which can substantially reduce losses in adverse conditions. So the investor’s reservation price – the highest price he is prepared to pay – is above expected value, at a level which implies a negative expected return on the put option in isolation. This is acceptable to the investor in the context of a reduction in the risk of the portfolio as a whole.
An analogous argument can be made for the buyer of a call option, who is increasing the risk of his portfolio by adding the option, and so requires a positive expected return to justify this increase in risk. This rationalises the buyer having a reservation price somewhat below the expected value of the call option.
The above explanations focus on the demand for options. This blog focuses on the supply of options, and in particular the asymmetric practical impact of margin requirements on sellers of calls and puts.
The supply side: margin requirements
If Black-Scholes prices for puts are typically above expected values, can we make money by selling puts at these prices? There are academic studies which suggest that excess returns are available from doing precisely this: selling out of the money short-term index put options (which are generally priced closed to Black-Scholes). But no, I do not do this. The obstacle is margin requirements, a practical matter which academic studies usually overlook.
Buyers of options pay a premium at outset, but thereafter have no further potential liabilities. Sellers receive a premium, but also have to deposit initial margin with their broker, and then variation margin as the price moves against them. This seems likely to have asymmetric effects on the reservation prices at which sellers are willing to offer calls and puts, as follows:
– The seller of a call suffers margin calls as the index rises. When the index is rising, credit is likely to be easy; the rest of the seller’s portfolio is probably rising and saleable; and his increasing wealth implies declining marginal utility. The margin calls are not a problem.
– The seller of a put suffers margin calls as the index falls. When the index is falling – and especially when it is crashing – credit is likely to be difficult, the rest of the seller’s portfolio may be unsaleable, and declining wealth implies increasing marginal utility. The margin calls may be very difficult.
Margin issues are difficult to quantity. But papers which attempt to do so find that when margin issues are allowed for, the apparent attraction of selling index puts largely disappears.
This asymmetry – margin calls are easier to manage for sellers of calls than for sellers of puts – may help to explain why calls are supplied cheap and puts dear. But as with Jeffrey and Smith’s demand-side argument, this is only directional, not quantitative. It doesn’t show that the discrepancy between Black-Scholes prices and expected values represents fair compensation for margin issues; it only notes that the observed discrepancies – puts priced higher relative to their expected values than calls – are in directions consistent with margin considerations.
If call options are cheap relative to expected values, can we make money by buying call options? If long-dated equity options were offered for sale priced on Black-Scholes, then maybe yes; this looks like it could be a low-cost form of non-recourse leverage, so I might buy some. But as they don’t seem to be offered, I haven’t. (Historically I did occasionally buy investment trust warrants; these were like call options with terms up to a few years, and sometimes very cheap.)
Further observations
A few further observations are in order.
First, to summarise the discussion above: the price at which I am prepared to sell an option depends on the margin requirements. Both expected values and Black-Scholes fail to account for this.
Second, the “calls cheap, puts dear” pattern of Black-Scholes prices is quite general. Section 6 of this paper by David WIlkie (presented at the AFIR Colloquim in 2001) gives more examples, and some algebra which shows that the effect is eliminated only when assuming a negative risk premium. Note that although the paper states at Section 6.9 that the “calls cheap, puts dear” pattern of Black-Scholes prices applies “for an investor with a linear utility function”, it will continue to apply (albeit to a lesser degree) as risk aversion increases, for all but the most risk averse investors.
Third, in the absence of liquid markets in long-dated options, we have little evidence that many investors are actually prepared to pay prices for long-dated put options quite as high as Black-Scholes. Again, the “demand” and “supply” rationales are both directional, not quantitative: they rationalise why buyers and sellers might be prepared to trade at “some deviation” from expected values, but not at the particular “Black-Scholes deviation”.
Fourth, this Eumaeus blog argues that there is no puzzle to explain. It says that that a forward contract can be synthesised by buying a call and selling a put, and observes that the sum of the put and call prices provided by Black-Scholes is equal to forward price discounted at the risk-free rate. This argument shows that Black-Scholes prices are consistent with no-arbitrage between puts, calls and forward prices. But so what? Why should the investor postulated by Jeffrey and Smith be concerned with this? The blog doesn’t seem to engage with the discrepancy between expected value and the prices at which investors may be prepared to trade; it just changes the subject.
Application to NNEG?
How does this discussion of margin issues apply to no negative equity guarantees? The writing of a NNEG doesn’t give rise to any exposure to margin calls. I am therefore willing to write it at a lower price than if margin calls were possible. To the extent the regulator wishes to impose a requirement for “initial margin”, in the form of “required capital” calculated using Black-Scholes, that will increase my price. But this is an artefact of the regulator’s exogenous imposition, not because I place any credence on Black-Scholes. And in the absence of all the apparatus of dynamic hedging – or at the very least, liquid markets in in long-dated puts, calls and forwards – the regulator’s Black-Scholes calculation seems rather arbitrary.
Jeffrey and Smith characterise this type of argument as a “false dilemma”. That is, they note that my observation that Black-Scholes is an arbitrary formula for long-dated unhedgeable options doesn’t show that any particular alternative is correct. But it seems to me that this characterisation can be turned around: in the absence of compelling hedging and no-arbitrage arguments, the reverence currently afforded to Black-Scholes prices amounts to a “false primacy”. Why should Black-Scholes be taken so seriously, and other approaches, such as percentiles from real-world stochastic models, not considered at all?
Conclusion
A “calls trade cheap, puts trade dear” pattern relative to expected values can be explained by demand-side and supply-side arguments as outlined above. But these arguments are directional, not quantitative. They justify “somewhat” cheap and dear, but not the particular amounts “Black-Scholes” cheap and dear.
I agree with Buffett: for long term puts, Black-Scholes prices seem unreasonably far above expected values. And the gap between Black-Scholes prices and reasonable prices is larger for sellers with limited margin requirements, such as NNEG writers – and Buffett.