The posterior density

6 Bayesian analysis of the Black–Scholes option price

6.2 Derivation of the prior and posterior densities

6.2.3 The posterior density

�

126

0

Forecasting Expected Returns in the Financial Markets

Integrating out P_t gives us the prior density of the option price:

pdf�c� = pdf�c� P_t��P_t

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).

127 Bayesian analysis of the Black–Scholes option price

Proposition 3: The unconditional density of the statistic s is given by 2 v

v ₂

s^v−1 pdf�s�= �^�

B ^v� � 2 _{2 2} ^v _+�

2 �+ ^vs₂

Proof: From Assumptions 4 and 3, we know pdf��²�and pdf�s²\�²�. Then pdf�s²� �²�= pdf��²�pdf�s²\�²�. We can now integrate out �² to obtain the marginal probability density of s², i.e. pdf�s²�=

0

� pdf�s²� �²���². Finally, pdf�s�= 2spdf�s²�. The complete proof and all the relevant calculations are presented in Section A.2 in the Appendix.

Now take pdf�P_t� �� s� and consider the transformation:

c = C���≡ C�P_t� �� ⇔ � =��c�≡ ��c� P_t� P_t = P_t

s =s

where � is the inverse of the option price with respect to �. The Jacobian of the trans­

formation is given by:

�c/�� =vega �c/�P_t =��d₁� �c/�s =0

1 = �P_t/�� =0 �P_t/�P_t =1 �P_t/�s = 0 = vega (6.10)

J

�s/�� =0 �s/�P_t = 0 �s/�s =1 Then

pdf�P_t� c� s� =pdf�P_t� ��c� P_t�� s��J� (6.11) We can now obtain an expression for pdf�P_t� c\s�:

pdf�P_t� c� s�

pdf�P_t� c\s�= (6.12)

pdf�s�

vs^{2 v} ₂ +�

�J^∗� �+

= √ 2

�tP_t� v

2 +�

��c� P_t�^2�+v+2

×exp

− 2�+vs²

2��c� P_t�² − 1 4��c� P_t�²t

ln

P_t P₀

−

m− 1

2 ��c� P_t�²

t 2

128 Forecasting Expected Returns in the Financial Markets

⎛ ⎛ P_t ��c� P_t�²� ⎞ ⎞

!⎜ ⎜⎜ ⎜ln Ke^−r� + 2 ⎟⎟ √ ⎟⎟ where J^∗=1 ⎝ ⎝� ��c� P_t� � √ ⎠P_t � ⎠.

Having obtained pdf�P_t� c\s�; the posterior density of the option price is given by:

pdf�P_t� c\s�

pdf�c\Pt� s� = (6.13)

pdf�P_t\s�

We derived pdf�P_t� c\s� in equation (6.12), but we need an expression for pdf�P_t\s�.

Proposition 4: The conditional (on the sample and prior information) density for the asset price is given by

1"

K_�+2 t

4 2�2�+vs²�t+�ln�P_t/P₀�−mt�²

pdf�P_t\s�= √ v

�tP_t� �+ 2 �+ ²

v ^{2 v} t _�+^t

2�2�+vs²�t+�ln�P_t/P₀�−mt�^{2 2�4 +t}

× �+ s ²

2 16 64

ln�P_t/P₀�−mt

×exp − 4 1 "

where K_�+t

2 4 2�2�t+vs²�t+�ln�P_t/P₀�−mt�² is the modified Bessel function¹⁴ of the second kind of order �+�t/2�.

14Modified Bessel functions are solutions to the differential equation x²y^��+xy^�−�x² +a²�y =0.

Definition 2: Bessel Functions

The differential equation x²y^�� +xy^�+�x² −a²�y = 0 is known as the Bessel equation where a is a non­

negative constant. Some of its solutions are known as Bessel functions. The function J_a defined by J_a�x� = � �−1�ⁿ x 2n+a

n=0 n!�n+a�! 2 for x > 0 and a a non-negative integer is called the Bessel function of the first kind of order a. The function K_a defined for x > 0 by

1 x −a a−1 �a−n−1�! x 2n 1 x a � hn +hn+a x 2n

a a

K �x� =J �x�ln x− − �−1�ⁿ

2 2 ⁿ⁼⁰ n! 2 2 2 ⁿ⁼⁰ n!�n+a�! 2

is called the Bessel function of the second kind of order a. The general solution of the Bessel equation in this case for x > 0 isy = c₁J_a�x�+c₂K_a�x�. For an exposition of Bessel functions and their relevance in diffusion theory, see, for example, Feller (1971). For a detailed discussion, see Watson (1944).

129 Bayesian analysis of the Black–Scholes option price

Proof: Let us first illustrate how to obtain the marginal density of the asset price – i.e. pdf�P_t�. Then it is straightforward to also obtain pdf�P_t\s� using a similar procedure. In Proposition 1 we have obtained pdf�P_t� ��, which we can straightfor­

wardly transform to pdf�P � �²�. Then the result follows from the fact that pdf�P � =

_� ^{t t}

pdf�P_t� �²���² can be written as A ^{� c�} �_�¹2 �^�+1 exp�−_�^c 2 �exp�−p�²���² where A

0 0 ����

is a constant. It is easy now to observe that the integral is the Laplace transform of an inverted-gamma function �L�f_i���: L�f_i�� = F�p� =

0

� f_i���²�e^−p�2 d�². For the complete proof, see Section A.3 in the Appendix.

We have now completed the derivation of the posterior density of the BS option price.

Let us present here the full expression:

2�+t

2�2�+vs²�t+�ln�Pt/P₀�−mt�² 4

�J^∗�

pdf�c\P_t� s� = 64 _�+ ^t ₂

1 " t K_�+ t 2�2�+vs²�t+�ln�P_t/P₀�−mt�² ��c� P_t�^2�+v+2

2 4 16

2�+vs² ln�P_t/P₀�−mt

×exp −

2��c� P_t�² + 4

×exp

− 1 4��c� P_t�²t

ln�P_t/P₀��−

m− 1

2 ��c� P_t�²

t 2

(6.14)

where J^∗ =1

!⎛

⎜⎜

⎝�

⎛

⎜⎜

⎝ ln

P_t Ke^−r�

+ ��c� P_t�²� 2

��c� P_t� √

�

⎞

⎟⎟

⎠P_t √

�

⎞

⎟⎟

⎠.

Corollary 1: If the option is at-the-money, i.e. P_t =Kexp�−r��, then certain simplifications √ occur: ��c� P_t�=^√2_��⁻¹�₂¹ �_P^c

t +1�� and J^∗=1 ����⁻¹�¹₂ �_P^c

t +1���P_t ��.

Proof: When the option is at-the-money, the _√BS formula (given _√ in equation (6.5)) _√ sim­

plifies to c = C�P_√t = Kexp�−r��� �� = P_t���^� ₂ ^��−��−^� ₂ ^��� = P_t�2��^� ₂ ^��−1�. This then implies that: ��^� ₂ ^��= ₂¹ �_P^c

t +1�⇒� =��c� P_t�=^√2_��⁻¹�₂¹ �_P^c

t +1�� where �⁻¹�� � �� denotes the inverse cumulative normal distribution function.

Also for P_t = Kexp�−r�� we have J^∗= 1

� ^��c�P₂ ^{t� √ �} P_t √

�

. Substituting in, the analytic expression for ��c� P_t� we get the proposed result for J^∗ .

Remark 5: The at-the-money case is best interpreted as a stochastic exercise price where K =P_t exp�r��.

130 Forecasting Expected Returns in the Financial Markets

Remark 6: It is interesting to observe that the posterior density of the option price does depend on the expected rate of return � through the hyperparameter m (m represents our prior beliefs about �). The true unknown � has been integrated out. The existence of m in the formula is due to randomness in prices prior to sampling.

Corollary 2: In relation to the above remark, market completeness implies that we can also derive the risk-neutral posterior density of the option price. Under the risk-neutral measure the variance of the price process remains the same but the drift changes and is equal to the risk-free rate r. Under the assumption of a constant risk-free rate this effectively implies that in Assumption 1 we substitute r = m and also Assumption 2 is no longer required since r is constant and not a random variable. Following the same procedure as above (without Assumption 2 this time), we can therefore show that the risk-neutral posterior density is given by:

2�+t

�2�+vs²�t+�ln�P_t/P₀�−rt�² 4

�J^∗�

pdf^RN�c\P_t� s� = 16 _�+ ^t

1 " t ² K_�+ t �2�+vs²�t+�ln�P_t/P₀�−rt�² ��c� P_t�^2�+v+2

2 2 8

2�+vs² ln�P_t/P₀�−rt

×exp − +

2��c� P_t�² 2

1

1 ²

×exp − ln�P_t/P₀�− r− ��c� P_t�² t 2��c� P_t�²t 2

Note, however, that this is as far as we choose to go with the risk-neutral measure. In the context of the problem we examine, it does not make sense to derive a risk-neutral predictive density of the unconditional price process. In other words the predictive density is not invariant in m. The same applies, of course, to the prior density as well.

Having derived the prior and posterior densities of the BS option price (i.e. pdf�c� and pdf�c\Pt� s��, it is interesting, for comparative purposes in particular, to derive expressions for pdf�c\s� and pdf�c\Pt�. This way we can illustrate how the dispersion of the density of the option price changes as we condition on more information: from the prior pdf�c�, to conditioning only on the sample estimate of volatility pdf�c\s�, to conditioning on the price pdf�c\P_t�, to the posterior density pdf�c\P_t� s�. We refer the reader to Section A.4 in the Appendix for the derivation of pdf�c\s� and pdf�c\P_t�.

It should be stressed that randomness in prices and volatility has been assumed through­

out our analysis. We write c = C�P_t� �� to denote that fact. For fixed prices but random volatility, we would write c=C���. Note for example that pdf�c=C�P_t� ��\s� and pdf�c= C���\s� represent two very different densities with dramatically different shapes.¹⁵

15The former represents the density of the option price conditional on the sample estimate of volatility but with prices unknown, while the latter represents the density of the option price conditional on the sample estimate of volatility but with prices known and fixed. Thus the dispersion of the former distribution is expected to be much larger than the latter.

131 Bayesian analysis of the Black–Scholes option price

Remark 7: So far it has been assumed that c = C�P_t� ��. Karolyi (1993) assumes that prices are non-random, i.e. c =C���, and suggests that the posterior density of the option price can be derived as a non-linear transformation of the posterior density of volatility. Let us briefly illustrate how pdf�c =C���\s�can be obtained. The posterior density of volatility is given by

^� pdf��� s�

pdf��\s� = �pdf���L��\s�� pdf��� s��� =

0 pdf�s�

4�^� v v/2 s^v−1 2�+vs² where pdf��� s� =

������v/2� 2 �^2�+v+1 exp − 2�² 2 v v₂ s^v−1

pdf�s� = �^� �

B ^v� � 2

vs² ₂ ^v ₊�

2 �+ ₂

Then

2 2 v +�

2 �+ ^vs₂

2�+vs²

pdf��\s�= _v exp − �

� ₂ +� �^2�+v+1 2�²

This is the posterior density of �. Using the transformation c = C���, i.e. inverting the Black–Scholes formula in terms of � =C⁻¹�c� =��c�, we obtain pdf�c =C���\s�:

₂ v ₂ +� ₂

vs 2�+vs

2 �+ exp −

2 2���c��²

pdf�c =C���\s� = ⎛ P_t ���c��²� ⎞ ln +

v ⎜⎜ Ke^−r� 2 ⎟⎟ √

� 2 +� ���c��^2�+v+1� ⎝ ��c� � √ ⎠P_t �

Again, for the at-the-money case, the simplifications outlined in Corollary 1 apply.