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Valuation of Spread and Basket Options

2. The Model and Johnson Distribution Family

This section first presents the model setup, and then introduces the Johnson (1949)
distribution family and shows how it can be used to approximate the unknown distribution

of a basket/spread of underlying assets.

2.1 The Basket/Spread of Underlying Assets
Assume that trading takes place continuously over a time interval �0, �, 0 < < . The uncertainty is
Assume that trading takes place continuously over a time interval �0, �, 0 < < . The uncertainty is
],
Assume that trading takes place continuously over a time interval [
.
� � ��, � �, where the filtration is generated by the correlated
� � ��, � �, where the filtration is generated by the correlated
described by a filtered probability space � , , , �ℱ �
The uncertainty is described by a filtered probability space
described by a filtered probability space � , , , �ℱ � ,
standard Brownian motions denoted by ( , , , , , and their instantaneous correlations between
standard Brownian motions denoted by ( , , , , , and their instantaneous correlations
where the filtration is generated by the correlated standard Brownian motions denoted between
�
�
( and ( , , are denoted by . The measure represents the risk-neutral probability
, and their instantaneous correlations between
and
by , are denoted by . The measure represents the risk-neutral probability
�
�
( and ( , �,�
�,�
�
�
measure. , are denoted by . The measure Q represents the risk-neutral
measure.
probability measure.
Consider N underlying assets whose dynamics under the risk-neutral probability
Consider underlying assets whose dynamics under the risk-neutral probability measure are assumed to
Consider underlying assets whose dynamics under the risk-neutral probability measure are assumed to
measure Q are assumed to be the following stochastic differential equations:
be the following stochastic differential equations:
be the following stochastic differential equations:
�� � (� ( , , , , , (1)
(1)
�� � (�
� � (� ( , , , , ,
�
�
�
� � (� � � �
where and represent the drift and diffusion terms, respectively. Their prices at time conditional on time-
5
5
where μ and σ represent the drift and diffusion terms, respectively. Their prices at time T
5
where and represent the drift and diffusion terms, respectively. Their prices at time conditional on time-
i
i
�
�
�
�
0 information can be derived by using the It � Lemma as follows:
conditional on time-0 information can be derived by using the Itȏ Lemma as follows:
0 information can be derived by using the It � Lemma as follows:
( (0 exp �� − � ( �. (2)
�
�
�
�
�
�
�
�
( (0 exp �� − � � ( �. (2)
�
�
�
�
�
�
�
�
�
Therefore, within the model setting, the time- price of the underlying asset follows a lognormal distribution.
Therefore, within the model setting, the time- price of the underlying asset follows a lognormal distribution.
Therefore, within the model setting, the time-T price of the underlying asset follows a
The model, presented by equation (1), can be straightforwardly generalized by including other risk factors,
The model, presented by equation (1), can be straightforwardly generalized by including other risk factors,
lognormal distribution.
such as stochastic interest rates (e.g., Kijima and Muromachi, 2001; Bernard, Le Courtois, and Quittard-Pinon,
such as stochastic interest rates (e.g., Kijima and Muromachi, 2001; Bernard, Le Courtois, and Quittard-Pinon,
The model, presented by equation (1), can be straightforwardly generalized by
2008; Wu and Chen, 2007a, 2007b), and price jumps (e.g., Merton, 1976; Metwally and Atiya, 2002; Flamouris
including other risk factors, such as stochastic interest rates (e.g., Kijima and Muromachi,
2008; Wu and Chen, 2007a, 2007b), and price jumps (e.g., Merton, 1976; Metwally and Atiya, 2002; Flamouris
and Giamouridis, 2007; Ross and Ghamami, 2010). Within the framework of these two extended models, the
2001; Bernard, Le Courtois, and Quittard-Pinon, 2008; Wu and Chen, 2007a, 2007b), and
and Giamouridis, 2007; Ross and Ghamami, 2010). Within the framework of these two extended models, the
time- price of the underlying asset remains a lognormal distribution, and thus, their pricing methods for
time- price of the underlying asset remains a lognormal distribution, and thus, their pricing methods for
basket/spread options are similar to those derived within the model setting given in equation (1). Our purpose is
basket/spread options are similar to those derived within the model setting given in equation (1). Our purpose is
to examine the performance of the Johnson (1949) distribution family, which is used to approximate the
The model can be easily applied to a variety of underlying assets by adjusting the setting of the drift
5
to examine the performance of the Johnson (1949) distribution family, which is used to approximate the
terms. For example, if S stands for the foreign exchange rate, then μ = r d ─ r f , where r d and r f represent
distribution of a basket/spread of underlying assets (or simply lognormal variates). Hence, to focus on the
distribution of a basket/spread of underlying assets (or simply lognormal variates). Hence, to focus
the domestic and foreign risk-free interest rates, respectively. If S denotes an equity index, then μ = r d on the
purpose of this study, we confine our model setting of the underlying assets to a geometric Brownian motion
─ q, where q represents the dividend yield rate. If S represents a forward or futures price, then μ =0,
purpose of this study, we confine our model setting of the underlying assets to a geometric Brownian motion
presented by equation (1). which is the same with the model setting specified in Borovkova et al. (2007). In addition, σ i can be
presented by equation (1).
replaced by a time-varing process.

2.2. The Basket/Spread of Underlying Assets 6
2.2. The Basket/Spread of Underlying Assets
Since both a basket or spread of underlying assets can be expressed in the same form, we integrate them
Since both a basket or spread of underlying assets can be expressed in the same form, we integrate them
hereafter as a generalized basket ( ) defined as follows:
hereafter as a generalized basket ( ) defined as follows:
( ∑� ( , �0, �, (3)
�
(3)
( ∑
���
� �
��� ( , �0, �,
� �

5 The model can be easily applied to a variety of underlying assets by adjusting the setting of the drift terms. For example, if stands
5 The model can be easily applied to a variety of underlying assets by adjusting the setting of the drift terms. For example, if stands
for the foreign exchange rate, then � − � , where � and � represent the domestic and foreign risk-free interest rates, respectively.
for the foreign exchange rate, then � − � , where � and � represent the domestic and foreign risk-free interest rates, respectively.
If denotes an equity index, then � − , where represents the dividend yield rate. If represents a forward or futures price,
If denotes an equity index, then � − , where represents the dividend yield rate. If represents a forward or futures price,
then 0, which is the same with the model setting specified in Borovkova et al. (2007). In addition, � can be replaced by a time-
then 0, which is the same with the model setting specified in Borovkova et al. (2007). In addition, � can be replaced by a time-
varing process.
varing process.
4
4