# PLOTTING THE GRIM REAPER’S PATH

Elie Ayache

Mortality tables have produced the concept of actuarial rate of return. It connects the actuarial value of the death insurance contract with its future contingent payments. The term has caught on in fixed-income analysis.

In actuarial terms, the fair value of a bond is the result of multiplying the probability that the bond would experience premature death at a certain “age” prior to its expiration by the interest that would have accrued from coupon payments until that age increased by the fraction of principal that would be recovered at that point and of summing these probability-weighted returns over the different ages of possible death. The actuarial rationale is that anyone having initially invested that sum would break even on average, given the various probabilistic scenarios. The problem is that, while the insurance company does indeed break even on average because the statistics predicted by the mortality table are eventually effectuated, the bond investor will almost certainly never break even because the issuing entity is one of a kind and its collapse is a single event.1

Now consider the opposite end of the spectrum. Consider the valuation of a financial contingent claim in a register where statistics abound. Equity options are underlain by the equity price and the event triggering their contingent payment is the underlying equity moving above (or below) a certain strike at a certain maturity. Contrary to the default event which happens only once in a lifetime, such equity-option triggering events happen countless times every day.

Naturally, the idea emerged that the population from which to sample the next tiny price movement is the past population of such tiny movements (in basically the same way that the probability of individual death was deduced from the statistical tables of already dead people). Future volatility was believed to be observable from the historical series of underlying prices (as historical volatility) and the temptation was indeed great to start off the option valuation problem as an actuarial problem.2

When it was later observed that equity options (or generally derivatives) were different from insurance claims and bonds precisely in that the underlying triggering their contingent payoff was traded in a market, the idea naturally imposed itself that they should be valued in the same market.

The problem, however, is that those underlying prices are not themselves the result of actuarial valuation. The present market price of the equity is not manufactured in such a way as to make its buyer break even on average under the probability distribution implied in the statistical series (or real probability). On the contrary, he would make money on average because of the risk-premium attaching to risky assets. It is in a changed probability measure that he would break even or, equivalently, that the expected value of his investment would be the same whether he elected to hold the equity or to invest it in the riskless bond. This symbolic probability is eponymously called risk-neutral probability. It is equivalent to the real probability measure in the sense that the price it sets on probabilistically impossible events is zero.

Option pricing and hedging

Enforcing non arbitrage amongst the option, its underlying equity and the discount bond has the consequence that the option price would also be equal to its expected value under the same risk-neutral probability, discounted by the riskless interest rate.

When the real probability distribution of underlying equity returns is Gaussian, the risk-neutral probability is uniquely determined. Indeed Brownian volatility is preserved by the change of measure (Girsanov’s theorem) and the drift is by definition changed into the riskless interest rate. When jumps or stochastic volatility are involved, however, the pricing system is no longer unique. There will be as many risk-neutral probability measures as there are ways of transforming the frequency of the jumps or the drift of the volatility process though the change of measure.

The Black-Scholes-Merton argument gave the exact pricing of options under Brownian motion a more operational turn. It no longer was by virtue of remote impossible events that the option price would be uniquely determined, but more immediately by virtue of dynamic replication of the option payoff and the law of one price.3

More importantly, a whole chain of arguments that still rested on an unwarranted metaphysical foundation – the assumed existence of an underlying random generator – was replaced by the local and mundane preoccupation of hedging the derivative against the next movement of the underlying price. This paved the way for the materiality of trading finally taking over the question of probability. What makes the market after all is not probability, real or unreal, but the surface observation that derivative and underlying trade alongside each other and move together.

The October 1987 crash

As a matter of fact, the single event that would shake the belief in the reality of probability was soon to hit the market. It was the October 1987 crash. Surely, nobody questions the existence of historical price series; but how does that warrant the existence of a random generator? It is well known that the passage from a frequency-based interpretation of probability à la von Mises to a single-case, or propensity-based, interpretation à la Popper is philosophically very hazardous.4

For instance, is there a “generator” of the default event underlying corporate debt? Can the probability of default be given an objective meaning? Structural models of the firm (KMV, Creditgrades) have suggested that default was objectively lying out there in space, under the form of a default barrier. The “generator” of default was thus tied back to the stochastic process ruling the value of the firm, or to a random generator of the kind we are currently questioning. Evidence of substantial spreads for very short-term credit default swaps (CDS), however, imposed the necessity of hitting the default barrier through a jump, thus referring default back to a single-case event.

The 1987 crash severed any remaining link between option prices and historical volatility. It taught the market-makers that even volatility should be implied from forward-looking option prices and that there was ultimately no difference between an equity option and a credit default swap. Both relate to a single-case event because every event is singular in a forward-looking perspective.

Implied volatility introduced option volatility skew, or evidence that the market was discounting downward jumps of non negligible frequencies and magnitudes often comparable to the 1987 crash, even in equity indices. As there was no evidence of such massive jumps in the contemporaneous statistics, this meant that historical analysis was bankrupt for all pricing purposes. The so-called “real probability” underlying option pricing theory was finally exposed as the residual legacy of actuarial science.

Regime-switching

If, by credit risk, we understand not only corporate default but collapse in general, or the capital and terminal event, then my claim is that credit risk has intertwined with equity options ever since the 1987 October crash. All metaphysics aside, we find there are only two pressing realities in the market: business as usual and the fear of sudden death. Probability as frequency and statistics has bequeathed on us the minute routine of option hedging, whereas probability as eventful and single-case has taught us to read off from the market the major signals of failure of the routine. As a result, the market manifests itself simultaneously as a local place and an apocalyptic outpost.

The option volatility skew is the hybrid progeny of the two intertwining realities. It registers both the wish to retain a dynamic model in order to compute a hedge and the imperative of inverting that model against the market prices of options. If we emblematise market routine with σ, or Brownian volatility, and the disrupting event with λ, or the hazard rate (also known as instantaneous probability of collapse), then we may say that (σ, λ), or jump-diffusion, is the paradigmatic shape of today’s derivatives market. Anything that happens in the market happens in between that shape.

In the simplest case, two traded instruments are sufficient to infer (σ, λ). These can be an equity option and a CDS, or two equity options, with a deep-out-of-the-money put typically acting as the credit instrument. The default event is so extreme however, that we should expect gradations. Intermediate events interpose themselves and the typical transition is (σ, λ) →(σ’, λ’). It can be described as a regime shift whereby volatility and default risk are liable to change dramatically.

Regime shifts are jumps in their own right. They are also characterised by a frequency and typically induce a jump in the underlying. The dichotomy between life and death, or between routine and the event, thus nests a series of rescaling events that take centre stage and differentiate (like an organism) the morphology of the market.

The right credit-equity model subsequently unfolds as a Markov chain (σ, λ) →(σ’, λ’) →(σ’’, λ’’)... whose multiple parameters can be inferred from joint calibration to the market prices of the full vanilla option surface, the full CDS term-structure (when it exists), and generally derivatives bearing either on volatility (variance swaps, variance options, etc.) or jumps (barrier options, gap options, etc.).5

Elie Ayache is CEO of ITO 33 and author of The Blank Swan: The End of Probability. The contents of this article are those of the author and do not represent the views of his employer

1 This is what led insurance companies such as AIG to stray into the credit market.

2 Emmanuel Derman and Nassim Nicholas Taleb, The illusions of dynamic replication, Quantitative Finance, 5(4), August 2005, 323—326.

3 Fisher Black and Myron Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81(3), May–June 1973, 637–654.

4 Donald Gillies, Philosophical Theories of Probability (London and New York: Routledge, 2000).

5 Elie Ayache, The Blank Swan: The End of Probability (London: Wiley, 2010).