Download EC372 Economics of Bond and Derivatives Markets Empirical

U NIVERSITY OF E SSEX
D EPARTMENT OF E CONOMICS
EC372 Economics of Bond and Derivatives Markets
Empirical testing of the Black-Scholes Model
The Black-Scholes model should, in principle, be straightforward to apply empirically because –
unlike many models in economics – it makes an unambiguous prediction: the Black-Scholes formula. One of the applications is to test the theory, i.e. to investigate how well empirical evidence
(observed option prices) match up with the model’s predictions (the option price generated by the
Black-Scholes formula).1
Critical evaluation of any model can follow several routes, including:
1. Theoretical – are the assumptions of the theorem plausible (sensible, or reasonable, or justifiable, or acceptable)? In the Black-Scholes model, the assumptions most commonly examined
are: (i) geometric Brownian motion (gBm) of underlying asset prices; and (ii) frictionless markets. Other assumptions limit the scope of the model, in particular (i) European style options;
(ii) no dividend paid during the life of the option; (iii) protection against stock-splits (and other
variations in the definition of the underlying asset).
In some cases, it is possible to obtain empirical evidence directly about the assumptions
but this is not required for testing the Black-Scholes model – just test its predictions.
2. Logical coherence – do the predictions of the model follow from its assumptions? Following
several decades of analysis, this is not an issue in the Black-Scholes model: no one denies that
the formula can indeed be derived from the assumptions (though some textbook expositions
are untrustworthy).
3. Empirical counterparts – how well do the entities of the model (which are necessarily abstract,
purely theoretical) correspond to things that can be observed? For example, what corresponds
the ‘risk-free interest rate’ that appears in the Black-Scholes formula? The identification of
empirical counterparts to theoretical entities is often overlooked (considered to be trivial) but
can be more tricky than it first appears.
4. Empirical – does the evidence (using the empirical counterparts) support the model’s prediction? This is where the methods of statistical inference are needed to evaluate the evidence in
a systmatic way.
Figure 1 (next page) shows a general schema handy for evaluating any scientific theory, including the Black-Scholes model.
In applying the Black-Scholes model, the greatest concerns are with (a) the reasonableness of
the assumptions, and (b) identifying appropriate empirical counterparts.
What role for assumptions?
Given that empirical testing centres on a model’s predictions, why are the assumptions of any
concern? Most importantly the assumptions – if plausible – help to give credibility to the prediction.
But what makes an assumption ‘plausible’? For some it is possible to obtain direct empirical
evidence, e.g. for the gBm assumption in the Black-Scholes model (by studying the pattern of underlying asset returns). Evidence tends not to support the assumption because observed asset return
distributions commonly have ‘thicker tails’ than gBm predicts.
1 There are other applications too, of course, most importantly to calculate the price of an option to be quoted in
negotiations between an option writer and a potential purchaser. Another application is the estimation of implicit volatility,
i.e. to calculate the value of σ such that the Black-Scholes formula would predict the observed option price. It should be
clear that implicit volatility cannot be used to test the theory, because the predicted option price would always be identically
equal to the observed option price. An independent estimate of σ (together with each of the other explanatory variables)
is necessary to test the theory.
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Assumptions:
A1 , A2 , . . . , Ak
- Logical
deduction
Prediction:
P
?
Empirical
counterpart
?
Statistical
inference
Q
Q
Q
Q
QQ
+
s
Reject
Do not reject
hypotheses
hypotheses
Figure 1: Elements in the evaluation of models.
In some cases, an assumption may be considered plausible if it is consistent with other generally
accepted theories or is itself a prediction of another theory. An example could be that of ‘equilibrium’
in the Black-Scholes model, which presumably means the absence of arbitrage opportunities (that
the arbitrage principle is satisfied). Although the markets may be in disequilibrium in the sense that
arbitrage opportunities exist, given the consequences that would follow (unbounded risk-free profits)
it seems plausible to assume that the arbitrage principle is satisfied.
Beware of fallacious arguments, for example that the ‘truth of the assumptions proves that the
correctness of the model’. Certainly the assumptions should logically imply the predictions but the
same predictions could be implied by different sets of assumptions, i.e., different models may imply
the same predictions.
Similarly, it does not logically follow that the empirical support for a prediction implies that the
assumptions are true (i.e., that the prediction implies the assumptions). But it is correct to argue that
if the prediction is rejected empirically, then at least one of the assumptions must be at fault.2
Are the empirical counterparts appropriate?
By far the most problematic assignment is a value for volatility, σ . Estimates of σ almost
invariably rely on past – ‘historical’ – data. But different estimates can be found using different
sample time intervals and different esimation methods. Indeed conflicting evidence obtained by
comparing the predictive accuracy of the Black-Scholes formula for options with different expiry
dates and exercise prices suggests that the assumption of constant volatility (i.e., constant σ ), is
implausible. (The evidence is usually expressed in terms of volatility ‘smiles’ and ‘smirks’.)
Assignment of values for the remaining arguments of the Black-Scholes formula3 – S, X, τ and r
– is regarded as less problematic, although, when markets are in turmoil (with very frequent changes
in securities prices), it can be tricky to ensure that the time at which S is observed is precisely the
same as for the observed option price (as the model requires).
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2 Formally, if assumptions A , A , . . . , A imply prediction P, then not-P implies that at least one of A , A , . . . , A is
1 2
1 2
k
k
false. This is what logicians call a modus tollens argument (or an instance of ‘denying the consequent’). Do not confuse it
with the fallacious claim that if any of the assumptions is false, then their logical implication, P, is also false – logicians
call this the ‘fallacy of denying the antecedent’.
3 The underlying asset price, S, the option’s exercise price, X, its time to maturity, τ, and the risk-free interest rate, r.
Note that Merton showed early on that it is unnecessary to assume that the interest rate is constant – though its values
during the life of the option must be assumed known, i.e. non-stochastic.
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