6 Bayesian analysis of the Black–Scholes option price
6.5 Concluding remarks and issues for further research
140 Forecasting Expected Returns in the Financial Markets
Table 6.5 Summary statistics of distributions exhibited in Figure 6.6 Quantiles
0.025 0.975 Mean SD Skewness Excess kurtosis pdf�c\Pt� s�
pdf�c\Pt�
1�7 1�1
8�9 10�1
4�27 4�22
1�73 2�30
1�15 1�45
2�14 3�24
Note that pdf�c\Pt� s� and pdf�c\Pt� are defined within the support of the distribution.
For example, for the case K =2025 the density has the support given by the no-arbitrage bounds of the option price: 187�066 < c = C�Pt� �� < 2206. For the cases K = 2225 and K =2425, the support is 0 < c = C�Pt� �� < 2206.
Finally, and perhaps most importantly, note that the posterior distributions for all three cases generally exhibit excess kurtosis and are positively skewed when compared with the normal distribution. In particular for the at-the-money and near-the-money cases the distributions are close to normal, however as we move progressively out- or in- the money the posterior distribution of the option price exhibits an increasingly thinner left tail than the normal distribution (see Tables 6.3–6.5).
141 Bayesian analysis of the Black–Scholes option price
This will have important implications for forecasting. Although this chapter is not about forecasting, we see this analysis as a necessary prelude to establishing a Bayesian theory of option price forecasting. Existing theories (such as those of Karolyi, 1993;
Noh et al., 1994; Hwang and Satchell, 1998; etc.) use only the implied volatility (or other measures of volatility (e.g. GARCH)) to forecast option prices while keeping the price of the underlying fixed. Our theory will allow us to consider forecasting when both prices and volatility can vary, as they do in many practical applications. For a forecasting application/extension of the underlying theory developed in this chapter, we refer the reader to Darsinos and Satchell (2001a). There we utilize the Bayesian approach to combine implied and historical volatility information and forecast the prices of FTSE 100 Index European options.
Furthermore, our results have potential uses in risk management as we can report VaR (Value at Risk) and other distributional measures. Having derived in analytic form the ‘true’ distribution of the Black–Scholes option price, we have provided a viable and potentially superior alternative to the approaches that use linear (delta normal) or quadratic (delta-gamma) approximations for the calculation of VaR. We illustrated that the true distribution of a long call option tends to have an increasingly thinner left tail than the normal distribution (similarly, the distribution of a short call will tend to have an increasingly fatter tail). The VaR for an asset or portfolio of assets is criti
cally dependent on the left tail of their distributions. If, for example, one assumes that the distribution of a long call option is normal, then the tendency will be to calcu
late a VaR that is higher than the true VaR. Similarly, for a short call the calculated VaR will be too low. Although we do not consider portfolio problems, it is possible to carry out such extensions. For an application/extension of our theory in value-at-risk (VaR) calculations and a comparison with existing VaR approaches, see Darsinos and Satchell (2001b).
Likewise, one could use our methodology in option pricing models other than the Black–Scholes. Thus, at least in principle, we could incorporate randomness due to inter
est rates (Merton, 1973) or specific models of volatility (such as Duan, 1995; Bauwens and Lubrano, 2000). Finally, there is a class of financial problems involving the valua
tion of warrants and corporate bonds. These problems require the determination of the distribution of the return of an asset, which is the combination of a value process and an option on that value process. Our analysis will allow us to address such questions, and we refer the reader to Darsinos and Satchell (2001c), where we derive the distribution of the stock price for a firm that issues warrants and/or executive stock options and thus provide an alternative approach for warrant valuation.
Acknowledgements
Theofanis Darsinos gratefully acknowledges financial support from the A.G. Leventis Foundation, the Wrenbury Scholarship Fund at the University of Cambridge, and the Economic and Social Research Council (ESRC).
)
�
, -
0 * +
142
A.1
Forecasting Expected Returns in the Financial Markets
Appendix
Proposition 1: The joint unconditional density of the price Pt and volatility � is given by:
pdf�P � ��= √ 1 �� 1 exp
− �
exp − 1 ln
Pt
−
m− 1
�2
t 2
t �tPt���� �2��+1� �2 4�2t P0 2
Proof of Proposition:
From Assumptions 4 and 2 we have expressions for pdf��� and pdf��\�� respectively.
Then pdf��� �� is just pdf���pdf��\��:
2t �� 1 t��−m�2 �
pdf��� ��=
� ���� �2��+1� exp −
2�2 exp −
�2 Similarly pdf�Pt� �� �� is given by pdf��� ��pdf�Pt\�� ��.
We have just obtained pdf��� ��, and in Assumption 1 we state pdf�Pt\�� ��. Then 1 �� 1
pdf�Pt� �� �� =
�Pt ���� �2�+3
* 2 +
×exp −
�
�
2 exp − 2�
1
2t ln P P
0 t + �
2
2t
−�t +t2��−m�2
To get the proposed result we therefore need to integrate out �:
pdf�Pt� �� = pdf�Pt� �� ����
−�
=K � exp ,
− 1
�z−�t�2 +t2��−m�2-
−� 2�2t ��
where we have set K = 1 �� 1 exp
− �
and z= ln Pt
+ 1
�2t.
�Pt ���� �2�+3 �2 P0 2 Let us now evaluate the integral:
� �z−�t�2 +t2��−m�2
exp − ��
−� 2�2t
= � exp
− 1 .
�z2 +t2 m 2�+�2�2t2�−2�t�z+mt� /
−� 2�2t ��
= � exp − 1
z 2 +t2 m 2 +2t2 �2 −2�
z+mt +
z+mt 2
−� 2�2t 2t 2t
1
)
)
143
A.2
Bayesian analysis of the Black–Scholes option price
2
−2t2 z+mt 2t ��
� 1 1 2 z+mt 2
= exp − �z−tm�2 +2t �− ��
−� 2�2t 2 2t
= exp
− 1
�z−tm�2 �
exp − t
�− z+mt 2
4�2t −� �2 2t ��
= exp
− 1
�z−tm�2 √
2�
√ �
4�2t 2t
⎡ ⎤
1 � 1
z+mt 2
×⎣√ exp − �− �� ⎦ 2� √� −� 2��2/2t� 2t
2t
= exp − 1
�z−tm�2 �
4�2t t �
Therefore:
pdf�Pt� �� =Kexp
− 1
�z−tm�2 �
4�2t t �
Substituting in the values for K and z, we get the proposed result for pdf�Pt� ��.
Proposition 3: The unconditional density of the statistic s is given by:
2 v sv−1
pdf�s� = B v
2 � � �� 2 v 2
�+ vs 2
v +�
2 2
Proof of Proposition:
We start by obtaining pdf�s2� �2� = pdf��2�pdf�s2\�2�, where pdf��2� is given in Assumption 4 and pdf�s2\�2� in Assumption 3. Then:
�� vv 2 �s2�2 v −1
1 2 v +�+1
2�+vs2
pdf�s2� �2�= v exp −
���� 2 v 2 ��2 � �2 2�2 Now let
�� v2 v�s2�2 v −1 v 2�+vs2
K = � m= +�+1� a = �
���� 22 v��2 v� 2 2
�+
144
0
Forecasting Expected Returns in the Financial Markets
Then
� � 1 m a
pdf�s2�= pdf�s2� �2���2 =K exp − ��2
0 0 �2 �2
We need to evaluate:
�
1
a
m
exp − ��2
�2 �2
Let �12 =x ⇒ −��12 �2��2 =�x. Then
� 1 m a 0 �
exp − ��2 = − xm−2 exp�−ax��x = xm−2 exp�−ax��x�
0 �2 �2 � 0
Now
� 1 �
m−2 �
xm−2 exp�−ax��x = − xm−2 exp�−ax� − xm−3 exp�−ax��x =
0 a 0 −a 0
note that limx→��xke−x� =0
= m−2 �
xm−3 exp�−ax��x a 0
m−2 ,
1 m−3 �
m−3 � m−4 -
= − x exp�−ax� − x exp�−ax��x
a a 0 −a 0
�m−2��m−3� � m−4 �m−2�! �
= a2 0
x exp�−ax��x =� � � � � � �� =
am−2 0 exp�−ax��x
�m−2�! � �m−2�!
= −
am−1 �exp�−ax��0 = am−1
v +�−1 !
= 22�+vs2 v2 +�
2
v
⇒pdf�s2� =K� 22 +� 2 v +� vs
2
Substituting for K we get:
�� v2 v�s2�v 2 −1 � v+� pdf�s2�=
���� 22 v� v
2 �+
2 vs
2 2 2 v +�
1 vv �s2�2 v −1
= v �� 2
�+vs2 v 2 +��18 B 2� � 2
2
18This implies that vs22 is unconditionally distributed inverted-beta fi�2 �vs22 \2
v� �� ��≡B�v 2 1
����� �vs2
2 2
�2 v v 2
− + 1
� .
��+vs2 �
145
A.3
Bayesian analysis of the Black–Scholes option price
Having obtained pdf�s2�, it is straightforward to get pdf�s�:
v v 2
2 sv−1
pdf�s� = 2spdf�s2�= B
2
v� � �� 2
�+ vs2 2 v +�
2
Proposition 4: The conditional (on the sample and prior information) density of the underlying asset price is given by:
1 "
t 4 t
K�+
2 2�2�+vs2�t+�ln�P /P0�−mt�2
pdf�Pt\s� = √ v
�tPt� �+ 2
�+v �+ t 2�+t
v 2 t 2 2�2�+vs2�t+�ln�Pt/P0�−mt�2 4
× �+ s 2
2 16 64
ln�Pt/P0�−mt
×exp − 4
where K � � n is the modified Bessel function of the second kind of order n.
Proof of Proposition:
Let us first illustrate how to derive the marginal density of the asset price (i.e. pdf�Pt�).
Then, following the same procedure it is straightforward to obtain the conditional density of the asset price (i.e. pdf�Pt\s�), we have that pdf�Pt�=
0
� pdf�Pt� �2���2 where
2 2
1 �� 1 �+3
� 1 Pt 1
pdf�Pt� �2�= 2 √
�tPt ���� �2 exp −
�2 exp −
4�2t ln
P0 − m− 2 �2 t (this follows straightforwardly from Proposition 1)
1 ��
Pt
Now let A= √ and z =ln −mt. Then 2 �tPt ���� P0
��+2
1 �+1 2
pdf�Pt� �2�=A 1
exp − �
exp − 1
z− 1
�2t
�2 �2 4�2t 2
��+2
1 �+1 2
1 � z z �2t
=A exp − exp − − −
�2 �2 4�2t 4 16
�
2 34 5
�
"
�
146 Forecasting Expected Returns in the Financial Markets
z
1 ��+12 �+1
�+z2/4t
t �2
=Aexp − exp − �2 exp −
4 �2 16
z2 t
Also let c = �+ and p = . Then
4t 16
pdf�Pt�= pdf�Pt� �2���2
0
�+1 �+ 1
+1
z � �+ 1 � c 2 1 2 c
=Aexp −
4 �+12 2
0 � �+ 1 �2 exp −
�2 exp�−p�2���2
c 2
Inverted−Gamma−1 function
Observe now that the required integral is the Laplace transform of an inverted-gamma function:
L fi� = F �p� = fi���2�e−p�2 d�2
0
Omitting some tedious algebra to calculate the Laplace transform, we arrive at the result:
4
pdf�Pt�=Aexp − z 2
2p�+1/2�cp�−2�+1 K�+1 �2 "
4 cp�
Substituting in the values for A, z, p and c we get the marginal density of the asset price:
1 "
K�+1
2 4 4�t+�ln�Pt/P0�−mt�2
��
t �+1
4�t+�ln�Pt/P0�−mt�2 2�+1
2 4
pdf�Pt�= √
�tPt���� 16 64
ln�Pt/P0�−mt
×exp − 4
where K�+1 �41 4�t+�ln�Pt/P0�−mt�2� is the modified Bessel function of the second
2
kind of order �+ 21.
Now to derive pdf�Pt\s� we need to follow a similar procedure to the one outlined above. This time we calculate
pdf�Pt\s� = pdf�Pt� �2\s���2
0
� = �
147
A.4
Bayesian analysis of the Black–Scholes option price
pdf�Pt��2�s�
Note that pdf�Pt� �2\s� = pdf�s� , where the numerator can be straightforwardly obtained from Proposition 2 and the denominator is given in Proposition 3. Again omit
ting the algebra, the result is:
1 "
K�+t
2 4 2�2�+vs2�t+�ln�Pt/P0�−mt�2
pdf�Pt\s� = √ v
�tPt� �+ 2
�+v 2 �+ t 2 2�4+t v t 2�2�+vs2�t+�ln�Pt/P0�−mt�2
× �+ s 2
2 16 64
ln�Pt/P0�−mt
×exp − 4
1. We want to derive the density of the option price conditional on the sample estimate of volatility: i.e. pdf�c\s�.
Consider first pdf�Pt� �\s�= pdf�Pt� �� s�
pdf�s� .
pdf�Pt� �� s� is given in Proposition 2 and pdf(s) in Proposition 3. Hence 1 �+ vs2 2 v +�
2�+vs2
pdf�Pt� �\s�=√ 2 exp −
�tPt� v 2 +� �2�+v+2 2�2
1 P 1 2
×exp − ln t − m− �2 t
4�2t P0 2
Using now the same transformation as we did for the prior density:
c =C�Pt� �� ⇒��c�= ��c� �� = Pt
we get
2 v
1
�+ vs 2 +�
2�+vs2
pdf�c� �\s� = √ 2 exp −
��d1∗� �t��c� ��� v 2 +� �2�+v+2 2�2
×exp − 1 ln
��c� ��
−
m− 1
�2
t 2
4�2t P0 2
� �
148 Forecasting Expected Returns in the Financial Markets
��c� �� �2�
ln +
where d1 ∗= Ke−r�
√ 2
Integrating out � numerically will give us pdf�c\s�.
2. We also want to derive the density of the option price conditional on the price: i.e.
pdf�c\Pt�.
Consider first pdf��\Pt�= pdf�Pt� ��
pdf�Pt� .
pdf�Pt� �� is given in Proposition 1 and pdf�Pt� in Section A.3 above. Applying the transformation:
c =C��� ⇔ � = ��c�
we get:
pdf�c\Pt� = 6 vega
2
K�+1
2 �+
2 1 �
4
1 4�t+ ln P P
0
t −mt
exp − �
− 1 ln
Pt
−
m− 1
��c�2
t 2
+
ln Pt
−mt !
��c�2 4��c�2t P0 2 P0 4
× ⎡ 2 ⎤−2�4 +1
�+12 ⎢4�t+ ln Pt
−mt ⎥
t ⎢ P0 ⎥
��c�2��+1� 16 ⎢⎣ 64 ⎥⎦
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