6 Bayesian analysis of the Black–Scholes option price
6.2 Derivation of the prior and posterior densities
6.2.2 The prior density
122 Forecasting Expected Returns in the Financial Markets
distribution of the option price, not just as a transformation of the posterior den
sity of the variance but by incorporating randomness in the underlying asset price as well.
Remark 1:
• Our Bayesian framework has been established on the basis of geometric Brownian motion for the stock price process. It is just as easy to establish a Bayesian framework for an Ornstein–Uhlenbeck (O–U) price process: �dPt = −��Pt −��dt+�dWt�, since it is also a normal Markovian process with stationary transition probabilities.
• It can be shown (see, for example, Nelson, 1990) that typical discrete-time models of heteroscedasticity, including certain ARCH and EGARCH models, converge in a natural way as the time intervals shrink to the continuous-time stochastic volatility model in which the log of volatility is well defined and satisfies the Ornstein–Uhlenbeck differential equation. Thus an interesting extension of this chapter could be a Bayesian analysis of a stochastic volatility model.
123 Bayesian analysis of the Black–Scholes option price
not large and they are more concerned with the dispersion induced in the option price.
Interestingly, they suggest that a Bayesian approach may be usefully employed to improve on the precision of option price estimates.
Butler and Schachter (1986), on the other hand, are concerned with the variance- induced option price bias, and investigate potential remedial measures. In fact they con
struct a uniformly minimum-variance unbiased estimator for the BS option price. The estimator is derived by taking a Taylor series expansion of the pricing formula and the moments of the estimated variance.
In a discussion of the Butler and Schachter paper, Knight and Satchell (1997) re-examine the question of statistical bias in the BS option price. They show that the only unbiased estimated option is an at-the-money option. However, they argue that the importance of bias in option pricing seems minor compared with other obvious sources of mispricing.
Noh et al. (1994) assess the performance of ARCH models for pricing options. Rather than comparing implied volatilities with GARCH volatilities, their study compares pre
dictions of options prices from GARCH with predictions of the same options prices from forecasting implied volatility. The results indicate that both methods can effectively fore
cast prices well enough to profit by trading if transaction costs are not too high. The GARCH models are considerably more effective.
Remark 3: Unconditionally, the option price depends both on the volatility � and the stock price process Pt = P0 exp���−�1/2��2�t+��Wt −W0��. We write c = C�Pt� �� to denote that fact. Bayesian theory mandates that this should be taken into account when deriving the posterior density. In other words we should treat prices and volatility as unknown random variables and identify a prior density for them.
The building block for the derivation of the prior density of the BS option price11 is therefore the joint density function of Pt and �; i.e. pdf�Pt� �\P0� �� �� t�. Then, by transforming pdf�Pt� �\P0� �� �� t� to pdf�c� �\P0� �� �� t� and integrating out �, we obtain pdf�c\P0� �� �� t�. Note that when the distributions are conditional on any prior parameters (i.e. �� � and m), and/or on P0, and/or on t (it is not unreasonable to assume that the sample size is known before the sample is drawn), we will refer to these distributions as prior or unconditional. In what follows, for ease of notation, we shall ignore the dependence on P0� �� � and t.
Proposition 1: The joint unconditional density of Pt and � is given by
1 �� 1
� pdf�Pt� �� = √
�tPt ���� �2��+1� exp −
�2
×exp − 1 ln
Pt
−
m− 1
�2
t 2
4�2t P0 2
11When we refer to the probability density of the BS option price, we mean the probability density of a European call option.
�
� = �
124 Forecasting Expected Returns in the Financial Markets
Proof: From Assumptions 4 and 2 we have pdf ��� and pdf��\�� respec
tively. pdf��� �� = pdf���pdf��\��. Similarly, pdf�Pt� �� �� = pdf��� ��pdf�Pt\�� ��
with pdf�Pt\�� �� given in Assumption 1. Finally, pdf�Pt� �� = −� pdf�Pt� �� ����.
Section A.1 in the Appendix contains the analytic proof and all the relevant calculations.
We are now in a position to derive the unconditional density function of the option price. Let us first obtain pdf�c� ��: Take pdf�Pt� �� and consider the transformation
c = C�Pt� ≡C�Pt� �� ⇔Pt =��c� ≡ ��c� ��
where � is the inverse of the option price with respect to Pt.12 The Jacobian J of the transformation is given by:
�c �c = ��d1� = vega
1 �Pt ��
= =��d1� (6.6)
J �� = 0 �� = 1
�Pt ��
where d1 = ln KeP−tr� + �22 �
� √
� and vega=�
ln�Ke P−
� tr�
√�
� +�2
2� Pt √
�, and ��� � ��=���� � ��
denotes the standard normal probability density function. Then
pdf�c� �� =pdf���c� ��� ���J� (6.7)
1 �� 1
= √
��d1∗� �t��c� �� ���� �2��+1�
� 1 ��c� �� 1 2
×exp − − ln − m− �2 t
�2 4�2t P0 2
��c� �� �2� √ where d1 ∗= ln + /� �.
Ke−r� 2
12We invert the option pricing formula in terms of Pt, hence obtaining Pt as a function of c and �. It should, however, be noted that there is no analytic expression (with the exception of an at-the-money option) for Pt = ��c� �� and a Newton–Raphson numerical approximation is required.
�
!
125 Bayesian analysis of the Black–Scholes option price
Integrating out � will give us the prior density of the option price:
pdf�c� = pdf�c� ���� (6.8)
0
Note, however, that there is no closed form solution for this integral and it will have to be evaluated numerically (more of that in Section 6.2).
Remark 4: We can utilize another procedure to obtain the marginal density of the option price pdf�c�. We start again from pdf�Pt� �� but this time we consider the transformation
c =C���≡C�Pt� �� ⇔� =��c�≡��c� Pt� Pt =Pt
where � is the inverse of the option price with respect to �.13 This is commonly referred to as the implied volatility of the option price. The Jacobian J of the transformation is given by
�c �c =vega =��d1�
1 �� �Pt
= =vega
J �P�� t = 0 �P�Pt t =1
Thus we obtain:
pdf�c� Pt�=pdf���c� Pt�� Pt��J��
�J�� �� 1
= √ �tPt ���� ��c� Pt�2��+1�
×exp − �
− 1 ln
Pt
−
m− 1
��c� Pt�2
t 2
��c� Pt�2 4�2t P0 2
⎛ ��c� Pt�2� ⎞
⎜⎜
where J�= 1 � ⎝ ��c� Pt� � ⎠
13We invert the option pricing formula in terms of �, hence obtaining � as a function of c and Pt. Note again that there is no analytic expression (with the exception of an at-the-money option) for � = ��Pt� c� and a Newton–Raphson numerical approximation is required.
ln Pt Ke−r� +
√ 2 ⎟⎟Pt √
�.
�
126
0
Forecasting Expected Returns in the Financial Markets
Integrating out Pt gives us the prior density of the option price:
pdf�c� = pdf�c� Pt��Pt
Needless to say, either procedure gives the same (numerical) values for pdf�c�. Unless otherwise stated, in our later calculations we shall use the first procedure, given by equation (6.8).