|
K
|
Number of counties, indexed by
k ∈ {1,…,
K}.
|
|
I
|
Number of players, indexed by
i ∈ {1,…,
I}.
|
|
J
|
Index of the option series,
j ∈ {1, 2, 3}.
|
|
W
i
|
Current wealth of player,
i ∈ {1,…,
I}.
|
|
p
k
|
Conditional probability that a hurricane present in the Atlantic Basin will make landfall in county
k. For
k = 0,
p
0 = 1−∑
k
p
k is the probability that it will not make landfall.
|
|
L
ik
|
Potential (expected) losses of player
i in county
k, where
L
i0 ≡ 0.
|
|
c
|
Nominal value of a HuRLO.
|
|
R
jk
|
Payoff of an option
k in series
j if outcome
k is realized.
|
|
O
jk
|
Decimal odds of outcome
k in series
j.
|
|
q
jk
|
Implied market probabilities for outcome
k in series
j.
|
|
P
jk
|
Probability that option
k of series
j will mature in-the-money.
|
|
n
ijk
|
Number of each option
k of series
j owned by player
i.
|
|
m
jk
|
Number of each option
k of series
j in the market.
|
|
M
j
|
Total mutual reserve of series
j.
|
|
π
jk
|
Market price of option
k of series
j.
|
|
Y
i
|
Expected utility of agent
i.
|
|
γ
|
Risk aversion parameter.
|
|
r
|
Annual discount rate.
|
|
d
|
Time remaining until the end of the hurricane season.
|
|
H
|
Number of series that have already been executed.
|
|
l
|
Indicator of the presence of a hurricane.
|
|
λ
|
Annual expected number of hurricanes.
|
For each of the 11 Atlantic regions of the United States, Table
5.6 presents the probability that a named storm, a hurricane, or an intense hurricane will make landfall in a given year.
Under a Poisson distribution, the probability that at least one storm makes landfall is given by the complement probability that no storm makes landfall: 1 −
P(0) = 1 − exp(−
x/
d), where
x is the number of named storms or hurricanes or intense hurricane to make landfall in that particular region over a span of
d years according to historical records.
Table 5.6
Historical probabilities of hurricane landfall
|
1
|
1880
|
136
|
503
|
72
|
43
|
17
|
43
|
27
|
12
|
0
|
4
|
29
|
|
2
|
1880
|
136
|
257
|
27
|
13
|
4
|
19
|
9
|
3
|
12
|
38
|
75
|
|
3
|
1880
|
136
|
666
|
104
|
51
|
23
|
56
|
31
|
16
|
0
|
2
|
18
|
|
4
|
1880
|
136
|
382
|
44
|
19
|
2
|
29
|
13
|
1
|
3
|
25
|
86
|
|
5
|
1900
|
116
|
373
|
26
|
9
|
5
|
22
|
7
|
4
|
9
|
46
|
65
|
|
6
|
1900
|
116
|
483
|
49
|
35
|
16
|
37
|
26
|
13
|
1
|
5
|
25
|
|
7
|
1880
|
136
|
574
|
25
|
11
|
3
|
18
|
8
|
2
|
14
|
45
|
80
|
|
8
|
1851
|
165
|
673
|
83
|
54
|
15
|
41
|
28
|
9
|
0
|
4
|
40
|
|
9
|
1851
|
165
|
527
|
14
|
5
|
0
|
9
|
3
|
0
|
41
|
74
|
100
|
|
10
|
1851
|
165
|
426
|
26
|
15
|
7
|
15
|
9
|
4
|
19
|
40
|
65
|
|
11
|
1851
|
165
|
447
|
9
|
5
|
0
|
6
|
3
|
0
|
56
|
74
|
100
|
The market is seeded by the purchase of an equal number of each option (say 1) at a price that reflects historical probabilities of the possible outcomes (see Appendix
2). Each time an option of type
b is bought, the market price vector of HuRLOs is updated using an adjustment factor
α that ensures that “pricing probabilities” sum to 1 and that the payoff of the last option bought is close to the time-adjusted nominal value. Denote by
β = exp(
rt) the capitalization factor applied to account for the time elapsed since the initialization of the market, and by
ξ
k the pricing probabilities,
k = 1,…,
K + 1. The updating algorithm is the following:
$$ {\xi}_k\leftarrow \Big\{{\displaystyle \begin{array}{c}{\xi}_b+\alpha {\xi}_b\left(1-{\xi}_b\right)\ \mathrm{if}\ k=b,\\ {}{\xi}_k\left(1-\alpha {\xi}_k\right)\ \mathrm{otherwise},\end{array}} $$
$$ {\pi}_k\leftarrow {\xi}_k\beta c, $$
$$ \alpha \leftarrow \Big\{{\displaystyle \begin{array}{c}0\ \mathrm{if}\ \frac{M}{m_b}>\beta c,\\ {}\frac{1}{m_b+{\pi}_b}\ \mathrm{if}\ \frac{M}{m_b}=\beta c,\\ {}A\ \mathrm{otherwise},\end{array}} $$
where
m
k and
M represent the number of type-
k options and the total mutual reserve immediately before the transaction, and where
A > 1/(
m
b +
π
b). The effect of the smoothing constant
α is to increase the pricing probability of option
b and to reduce the pricing probabilities of the other options. When
M/
m
b >
βc, option
b is overpriced, and no update is made. When
M/
m
b =
βc, the smoothing constant corrects for the dilution effect due to the additional claimer for outcome
b. Otherwise, the option is underpriced, and the smoothing factor should be higher. In our implementation, we used
A = max{(
M −
cβm
b)(1 −
π
b)/
M, 1/(
m
b +
π
b)}, which has the desirable properties of increasing with the imbalance, decreasing with the total mutual reserve, and decreasing with the pricing probability of option
b.
The probability distribution of the number of new hurricanes that will form until the end of the season is denoted by
ϕ
d, where
d is the remaining time in the season. Assuming that hurricane formation is a Poisson process of intensity
λ, the probability that
y hurricanes will form until the end of the season is then given by:
$$ {\phi}_d(y)=\frac{{\left(\lambda d\right)}^y\exp \left(-\lambda d\right)}{y!}. $$
The probability
P
jk (
d,
h,
l) that option
k in series
j will be the winning option depends on the remaining time in the season, on the number of series that have already been settled, and on the presence or absence of a hurricane. For
j = 1 and
h = 0, it is given by:
$$ {P}_{1k}\left(d,0,0\right)=\sum \limits_{y=1}^{\infty }{\phi}_d(y){p}_k\sum \limits_{s=0}^{y-1}{p}_0^s,\mathrm{for}k=1,\dots, K, $$
$$ {P}_{1k}\left(d,0,1\right)={p}_k+{p}_0\mathrm{E}\left[{P}_{1k}\left(d+D,0,0\right)\right],\mathrm{for}k=1,\dots, K, $$
where
D is a random variable representing the presence time of a hurricane. Here, an option
k in the first series will be executed if one hurricane or more is formed, and the first one to land does so in county
k. Clearly,
P
1k (
d,
h,
l) = 0 for
h ≥ 1. Similarly, for
j = 2 and
h = 0 we get:
$$ {P}_{2k}\left(d,0,0\right)=\sum \limits_{y=2}^{\infty }{\phi}_d(y){p}_k\sum \limits_{s=0}^{y-2}{C}_{s+1}^1{p}_0^s\left(1-{p}_0\right),\mathrm{for}\ k=1,\dots, K, $$
$$ {P}_{2k}\left(d,0,1\right)={p}_0\mathrm{E}\left[{P}_{2k}\left(d+D,0,0\right)\right]+\left(1-{p}_0\right)\mathrm{E}\left[{P}_{2k}\left(d+D,1,0\right)\right],\mathrm{for}\ k=1,\dots, K, $$
$$ {P}_{3k}\left(d,h,l\right)={P}_{2k}\left(d,h-1,l\right),\mathrm{for}\ h\ge 1. $$
Notice that there is exactly one winning option in each series, so that
$$ {P}_{j0}\left(d,h,l\right)=1-\sum \limits_{k=1}^K{P}_{jk}\left(d,h,l\right). $$
The computation of the marginal impact of buying an option is presented for the case of a single series (
J = 1). When more than one series are offered, the computation is done in a similar way, accounting for the fact that an option
k ≠ 0 in series
j can only be a winning option if the null option is not the winning option in series
j − 1.
If Player
i does not buy option
b, expected utility is given by:
$$ A=\sum \limits_{k=0}^K{P}_k\left(d,h,l\right){U}_i\left({W}_i-{L}_{ik}+\frac{n_{ik}}{m_k}M\right), $$
where
L
i0 = 0.
On the other hand, if Player
i buys option
b, expected utility is given by:
$$ {\displaystyle \begin{array}{l}B=\sum \limits_{k=0,k\ne b}^K{P}_k\left(d,h,l\right){U}_i\left({W}_i-{\pi}_b-{L}_{ik}+\frac{n_{ik}}{m_k}M\right)\\ {}\kern1.5em +{P}_b\left(d,h,l\right){U}_i\left({W}_i-{\pi}_b-{L}_{ib}+\left({n}_{ib}+1\right)\frac{M+{\pi}_b}{m_b+1}\right).\end{array}} $$
The marginal value of investing in option
b is therefore:
$$ {\mu}_{ib}\left(d,h,l,M,\pi, m,{n}_i,{W}_i\right)=B-A. $$
1.
Read parameters and initialize the market using
p, yielding
M
0 and
m
0. Initialize the options held by all players to
n
ik
0 = 0 for
I = 1,…,
I and
k = 0,…,
K. Set
d = 1,
h = 0 and
l = 0.
2.
Using the hurricane model, generate hurricane dates, durations, and outcomes during the hurricane season.
3.
Compute probabilities
P
jk at (
d,
h,
l) and select randomly an ordering
O of the players.
4.
For
I = 1,…,
I
a.
Determine the purchase order for Player
O(
i) according to her strategy,
b.
For each transaction by player
O(
i), update market variables,
c.
When purchase order of Player
O(
i) is completed, set
i =
i + 1.
5.
If
d = 0, stop. Otherwise, update
h and
l according to the hurricane scenario realization. Set
d =
d − 1/365 and go to 3].
Read parameters and initialize the market using
p, yielding
M
0 and
m
0. Initialize the options held by all players to
n
ik
0 = 0 for
I = 1,…,
I and
k = 0,…,
K. Set
d = 1,
h = 0 and
l = 0.
Using the hurricane model, generate hurricane dates, durations, and outcomes during the hurricane season.
Compute probabilities
P
jk at (
d,
h,
l) and select randomly an ordering
O of the players.
For
I = 1,…,
I
a.
Determine the purchase order for Player
O(
i) according to her strategy,
b.
For each transaction by player
O(
i), update market variables,
c.
When purchase order of Player
O(
i) is completed, set
i =
i + 1.
Determine the purchase order for Player
O(
i) according to her strategy,
For each transaction by player
O(
i), update market variables,
When purchase order of Player
O(
i) is completed, set
i =
i + 1.
If
d = 0, stop. Otherwise, update
h and
l according to the hurricane scenario realization. Set
d =
d − 1/365 and go to 3].
