A Mathematician on Wall Street
Optimal Capital Growth
Versus Black Swan
Insurance Part II:
Extremistan
Edward O. Thorp and Steven Mizusawa
C
onsider an investor who allocates a fixed fraction of his wealth
to a stock index at the start of each period and the remainder to
U.S. Treasury bills. This article was motivated by the question of
whether the investor can be better off buying calls on the index
instead, in order to limit his one-period downside risk (in Nassim
Taleb’s evocative terminology, bad “Black Swans”).
Previously, we assumed that stock index returns follow a stationary
lognormal distribution – the world of Black–Scholes and Long Term Capital
Management. Taleb calls this world where Gaussian statistics prevail
Mediocristan. Here we explore the world of Extremistan by using a distribution with fatter tails than the lognormal.
Assumptions and Formulas
Portfolios are limited to either T-bills plus a stock index or T-bills plus call
options on this stock index. We assume no transactions costs for simplicity. After considering various proposed fat-tailed distributions, we chose to
make up our own, a t-distribution with four degrees of freedom, truncated
at zero (i.e., set equal to zero for negative values of the argument). The arithmetic mean m = 1.12750 and variance σ2 = 0.05188 match those we used for
Mediocristan. The resulting density function was
p(x) = 72.416 (4 + 36.3275(–1.12562 + x)2)–5/2 for x > 0
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26-31_Thorp_Jan_2012.indd 26
(1)
and p(x) = 0 for x < 0
For simplicity, the index pays no dividends. Again, the portfolio is
revised annually. Calls are European, i.e., only exercisable at expiration.
Wilmott magazine
2/24/12 9:41:52 AM
Recall that the Black–Scholes formula for European calls results if we take the
lognormal distribution and replace the expected growth rate by the riskless
rate. If this mean-growth-rate-shifted lognormal distribution is w(x) then
∞
(2)
C = e−rt
(x − K)w(x)dx
Figure 1: Cumulative distribution of maximum drawdowns for indexonly Mediocristan.
x=K
However, because we have only specified a terminal distribution of index
prices, and not one resulting from a process with known transition probabilities, we don’t have a “no arbitrage” model for call option pricing.1 Proceeding
by analogy with equation (2), and choosing T = 1 (year), we priced the options
in a risk-neutral setting using the fat-tailed density function of equation (1).
Let u(x) equal p(x) after shifting its mean so that its expected growth rate
equals the riskless rate, i.e.,
∞
∞
u(x)dx =
u(x)dx = er
−∞
A
Note that when p(x) is mean-shifted to get u(x), the point where u(x)
becomes non-zero is at some number A < 0, whereas for p(x) it was at 0.
Now calculate C from
∞
C = e−r
(x − K)u(x)dx
Figure 2: Cumulative distribution of maximum drawdowns for indexonly Extremistan.
(3)
x=K
Note that K > 0 > A so we don’t ever include the values of x where u(x) = 0.
We obtained the density function u(x) by shifting the mean of p(x) to
m~ = exp (.05) = 1.05127, the one year wealth relative at our assumed riskless
compound growth rate of 5%. The difference between the mean of u and
m~ is 1.12749 – 1.05127 = 0.07622 so we change –1.12562 in (1) to –1.2562–
0.07622 = –1.04940 to get u(x) in equation (4).
u(x) = 72.416(4 + 36.3275(–1.04940 + x)2)–5/2
(4)
for x ≥ –0.07622 and
u(x) = 0 for x < –0.07622.
Alternatively, we could have truncated the risk neutral t-distribution at
zero, for m
~ and σ, finding it by the same process we used to find p(x).
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26-31_Thorp_Jan_2012.indd 27
Geometric Growth, Standard Deviation and Sharpe
Ratio
Proceeding as in Part I of this 2-part article,2 we compute g,s and the Sharpe
ratio from equations (9) – (12), (9a) – (12a), and (12s), replacing the lognormal
^
Table 1: Call Prices Compared for Lognormal and Truncated Student t
K
Lognormal
Student t
.2
.809
.809
.4
.619
.620
.6
.429
.433
.8
.245
.253
1.0
.104
.105
1.2
.0325
.0299
1.4
.00785
.00861
1.6
.00158
.00315
1.8
.000286
.00142
2.0
.0000479
.000748
Table 1 compares the call prices we used for Extremistan with those
for Mediocristan. The differences are small in magnitude across the entire
range.
However, the truncated t distribution leads to much greater maximum
drawdowns than in the lognormal case as Figures 1 and 2 show. Figure 1
shows the simulated cumulative distribution of the maximum drawdown
for the lognormal. The five curves represent fractions of 0.2, 0.4, 0.6, 0.8 and
1.0 in the index. As the fraction increases, the curves of course move progressively to the right. Figure 2 displays the same graphs for the truncated t
distribution.
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A MATHEMATICIAN ON WALL STREET
Table 2: Geometric Growth for Extremistan, r = .05, T = 1 year, using truncated Student T distribution with 4 degrees of freedom, Mean of 1.12749, Variance
of 0.0518805
f-->
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
K
0.0
0.0570
0.0635
0.0695
0.0750
0.0801
0.0847
0.0887
0.0922
0.0951
0.0968
0.1
0.0577
0.0648
0.0713
0.0772
0.0826
0.0873
0.0914
0.0947
0.0971
0.2
0.0585
0.0663
0.0734
0.0798
0.0854
0.0903
0.0943
0.0974
0.0990
0.3
0.0596
0.0683
0.0760
0.0829
0.0888
0.0938
0.0976
0.1000
0.1004
0.4
0.0610
0.0707
0.0793
0.0867
0.0928
0.0977
0.1011
0.1025
0.1009
0.5
0.0627
0.0738
0.0834
0.0913
0.0976
0.1022
0.1046
0.1043
0.0991
0.6
0.0652
0.0780
0.0886
0.0971
0.1033
0.1070
0.1076
0.1040
0.0925
0.7
0.0685
0.0836
0.0954
0.1042
0.1096
0.1113
0.1083
0.0985
0.0756
0.8
0.0733
0.0910
0.1039
0.1120
0.1150
0.1122
0.1019
0.0802
0.0353
0.9
0.0796
0.0999
0.1122
0.1168
0.1135
0.1007
0.0753
0.0301
–0.0580
1
0.0861
0.1055
0.1118
0.1059
0.0871
0.0529
–0.0026
–0.0930
–0.2623
1.1
0.0869
0.0950
0.0832
0.0534
0.0043
–0.0685
–0.1751
–0.3391
–0.6371
1.2
0.0741
0.0565
0.0135
–0.0522
–0.1429
–0.2655
–0.4348
–0.6857
–1.1309
1.3
0.0493
–0.0003
–0.0763
–0.1768
–0.3052
–0.4707
–0.6920
–1.0126
–1.5723
1.4
0.0219
–0.0540
–0.1543
–0.2786
–0.4317
–0.6243
–0.8774
–1.2394
–1.8651
1.5
–0.0008
–0.0940
–0.2089
–0.3467
–0.5134
–0.7204
–0.9901
–1.3732
–2.0320
1.6
–0.0174
–0.1207
–0.2436
–0.3887
–0.5624
–0.7767
–1.0547
–1.4481
–2.1229
1.7
–0.0289
–0.1379
–0.2652
–0.4140
–0.5914
–0.8095
–1.0917
–1.4903
–2.1731
1.8
–0.0366
–0.1489
–0.2786
–0.4296
–0.6089
–0.8290
–1.1134
–1.5148
–2.2018
1.9
–0.0418
–0.1561
–0.2872
–0.4393
–0.6197
–0.8410
–1.1266
–1.5295
–2.2188
2
–0.0454
–0.1608
–0.2928
–0.4456
–0.6267
–0.8486
–1.1349
–1.5387
–2.2293
g(x) by the truncated Student t u(x) throughout. The results, shown in
Figure 3 and Tables 2 through 4, correspond to those for Mediocristan in
Part I, Figure 1 and Tables 1–3.
Figure 3 shows that for s ≤ 0.10, g ≤ 0.08, the option and index portfolios
lie close to a common curve. To prefer one over the other requires a different metric. In the range 0.10 < s < 0.24, the option portfolios have a greater
g for a given s, or alternatively, less s for a given g, with the exception of
f = 0.1. Above s = 0.24, the option portfolios have an efficient frontier peak at
f = 0.4, K = 0.9, with g = 0.1168, s = 0.3593.
We illustrate some possible tradeoffs. It is helpful to use Figure 3 to compare the points for the pairs of portfolios in our examples.
Figure 4 shows the MDD distribution for the option portfolio K = 1.1,
f = 0.1, T = 32 years, where g = 0.0869, s = 0.1926, and Sh = 0.1915, versus the
index portfolio f = 1.0, which has g = 0.0968, s = 0.2333, and Sh = 0.2005.
Although the option portfolio has an annualized growth rate that is 1% less,
and is well inside the efficient frontier, the reduction in MDD is enormous
– a great comfort to the portfolio manager who wants to retain his clients
and his job. The index portfolio has a 50% MDD with probability about
40% while the option portfolio rarely has an MDD this large. The investor
Michael Korns has followed a similar strategy for more than a decade.
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26-31_Thorp_Jan_2012.indd 28
Just as in the Mediocristan examples, the MDD distribution at shorter
times typically favors an index portfolio for small MDD and a “comparable”
option portfolio for large MDD, with the option MDD becoming more dominant as T increases.
When K = 0.9, f = 0.2, T = 32 the option portfolio has g = 0.0999,
s = 0.1878, Sh = 0.2656 whereas the index portfolio with f = 1.0 is inside the
efficient frontier at g = 0.0968, s = 0.2333 and Sh = 0.2005. The dramatic
reduction in “tail risk” is illustrated in Figure 5.
The portfolio with the highest growth is K = 0.9, f = 0.4. Sitting at
the peak of the efficient frontier, it yields g = 0.1168, s = 0.3593, and
Sh = 0.1860. Although g and Sh are much better than the index, the MDD
graphs in Figure 6 show a “ride” so wild few investors are likely to stay
with it.
Caveats and Conclusions
As there is no generally accepted fat-tailed distribution for describing
equity index returns, we have made an arbitrary illustrative choice. Our
truncated t-distribution is not inconsistent with the observed annual
extreme returns over the last 86 calendar years. Some will argue that, even
Wilmott magazine
2/24/12 9:42:16 AM
A MATHEMATICIAN ON WALL STREET
Table 3: Std. Deviation for Extremistan, r = .05, T = 1 year, using truncated Student T distribution with 4 degrees of freedom, Mean of 1.12749, Variance of
0.0518805
f-->
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
K
0.0
0.0214
0.0425
0.0635
0.0845
0.1056
0.1272
0.1496
0.1731
0.1989
0.2333
0.1
0.0236
0.0469
0.0700
0.0932
0.1168
0.1409
0.1662
0.1935
0.2249
0.2
0.0263
0.0522
0.0779
0.1038
0.1302
0.1575
0.1864
0.2182
0.2567
0.3
0.0297
0.0588
0.0877
0.1169
0.1467
0.1779
0.2113
0.2488
0.2962
0.4
0.0339
0.0670
0.1000
0.1333
0.1676
0.2036
0.2427
0.2877
0.3467
0.5
0.0395
0.0778
0.1160
0.1546
0.1945
0.2368
0.2834
0.3382
0.4129
0.6
0.0470
0.0924
0.1374
0.1830
0.2303
0.2810
0.3377
0.4058
0.5022
0.7
0.0577
0.1128
0.1671
0.2221
0.2796
0.3417
0.4121
0.4987
0.6258
0.8
0.0737
0.1426
0.2100
0.2781
0.3494
0.4271
0.5166
0.6291
0.7996
0.9
0.0988
0.1878
0.2733
0.3593
0.4494
0.5481
0.6631
0.8103
1.0400
1
0.1379
0.2547
0.3638
0.4719
0.5845
0.7081
0.8529
1.0405
1.3384
1.1
0.1926
0.3401
0.4723
0.6003
0.7318
0.8752
1.0427
1.2599
1.6063
1.2
0.2511
0.4178
0.5595
0.6926
0.8267
0.9707
1.1371
1.3510
1.6897
1.3
0.2924
0.4559
0.5874
0.7071
0.8249
0.9492
1.0909
1.2708
1.5521
1.4
0.3071
0.4505
0.5604
0.6576
0.7515
0.8491
0.9589
1.0967
1.3098
1.5
0.3004
0.4185
0.5057
0.5812
0.6531
0.7270
0.8093
0.9118
1.0689
1.6
0.2815
0.3765
0.4447
0.5028
0.5576
0.6134
0.6753
0.7519
0.8687
1.7
0.2579
0.3339
0.3874
0.4326
0.4748
0.5177
0.5650
0.6233
0.7119
1.8
0.2337
0.2949
0.3374
0.3731
0.4063
0.4398
0.4767
0.5221
0.5908
1.9
0.2108
0.2607
0.2950
0.3236
0.3502
0.3769
0.4063
0.4424
0.4970
2
0.1901
0.2312
0.2593
0.2827
0.3043
0.3261
0.3500
0.3792
0.4235
Figure 3: Geometric growth versus standard deviation for values in the Tables
where g ê .05 and v Ä .5. Extremistan.
Figure 4: Comparison of the cumulative distribution of maximum drawdowns
for options with K = 1.1, f = .1, T = 32 years (in blue), and stock index (fraction
= 1.0 in red).
^
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A MATHEMATICIAN ON WALL STREET
Table 4: Sharpe
0.0518805
f−>
K
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
for Extremistan, r = .05, T = 1 year, using truncated Student T distribution with 4 degrees of freedom, Mean of 1.12749, Variance of
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3265
0.3255
0.3247
0.3240
0.3235
0.3231
0.3225
0.3208
0.3154
0.3000
0.2622
0.1915
0.0961
-0.0025
-0.0914
-0.1693
-0.2396
-0.3058
-0.3706
-0.4356
-0.5021
0.3173
0.3152
0.3132
0.3111
0.3089
0.3063
0.3030
0.2977
0.2877
0.2656
0.2180
0.1324
0.0156
-0.1103
-0.2308
-0.3439
-0.4533
-0.5626
-0.6744
-0.7905
-0.9118
0.3073
0.3040
0.3006
0.2969
0.2927
0.2877
0.2812
0.2720
0.2567
0.2274
0.1698
0.0703
-0.0653
-0.2151
-0.3645
-0.5119
-0.6603
-0.8135
-0.9738
-1.1430
-1.3217
0.2965
0.2919
0.2869
0.2814
0.2750
0.2673
0.2575
0.2439
0.2229
0.1860
0.1184
0.0056
-0.1476
-0.3207
-0.4996
-0.6825
-0.8724
-1.0727
-1.2854
-1.5120
-1.7530
0.2850
0.2788
0.2721
0.2645
0.2556
0.2450
0.2315
0.2132
0.1861
0.1413
0.0635
-0.0624
-0.2334
-0.4306
-0.6410
-0.8627
-1.0983
-1.3507
-1.6218
-1.9126
-2.2234
0.2726
0.2646
0.2559
0.2459
0.2344
0.2204
0.2028
0.1794
0.1457
0.0925
0.0041
-0.1354
-0.3250
-0.5485
-0.7941
-1.0598
-1.3477
-1.6602
-1.9986
-2.3637
-2.7554
0.2591
0.2489
0.2378
0.2252
0.2105
0.1928
0.1706
0.1415
0.1005
0.0382
-0.0617
-0.2159
-0.4264
-0.6802
-0.9671
-1.2852
-1.6358
-2.0206
-2.4405
-2.8956
-3.3857
0.2441
0.2312
0.2170
0.2011
0.1826
0.1605
0.1330
0.0973
0.0481
-0.0246
-0.1375
-0.3089
-0.5446
-0.8362
-1.1757
-1.5609
-1.9925
-2.4713
-2.9973
-3.5702
-4.1893
0.2266
0.2093
0.1908
0.1702
0.1467
0.1188
0.0847
0.0409
-0.0184
-0.1039
-0.2333
-0.4277
-0.6989
-1.0453
-1.4622
-1.9477
-2.5015
-3.1229
-3.8111
-4.5647
-5.3825
0.2005
Figure 5: Comparison of the cumulative distribution of maximum drawdowns
for options with K = .9, f = .2, T = 32 years (in blue) and stock index (fraction
= 1.0 in red).
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Figure 6: Comparison of the cumulative distribution of maximum drawdowns
for options with K = .9, f = .4, T = 32 years (in blue) and stock index (fraction
= 1.0 in red).
Wilmott magazine
2/24/12 9:42:53 AM
A MATHEMATICIAN ON WALL STREET
so, it is much too tame. We have also made the simplifying assumptions
that options are only exercisable at expiration and that they can be priced
by using the expected terminal value of the payoff, discounted at the riskless rate. Anyone who wanted to apply the methods given here would need
to make their own set of choices for return distribution, option pricing
model, type of option, portfolio revision period, parameters like m, s, and
r, etc.
Nonetheless, we would expect certain qualitative features of our results
to persist more generally, such as: (1) Options portfolios can attain regions
of the geometric mean-variance efficient frontier beyond the reach of the
index portfolios. (2) If we regard one maximum drawdown distribution as
better than another if its right tail dominates (i.e., less chance of extreme
MDDs), then some options portfolios are both on the efficient frontier and
have better MDDs than the index portfolio which is the closest in geometric
mean, standard deviation, and Sharpe ratio.
In any practical application we need to include transactions costs and
the impact of taxes, either or both of which could offset the perceived advantages. One also might want to stagger option expiration dates, e.g., 1/4 per
quarter, to smooth out costs, payoffs, and the need to replace – if American
options are used – options which are exercised early.
Endnotes
1. See Ekstrom, E., Lötstedt, P., Von Sydow, L., and J. Tysk. 2011. Numerical option pricing in
the presence of bubbles. Quantitative Finance 11(8), p. 1125, for a discussion of when we have
no-arbitrage pricing or risk-neutral pricing models. See also J. Huston McCulloch. 2003. The
risk-neutral measure and option pricing under log-stable uncertainty. Available at www.econ.
ohio-state.edu/jhm/papers/rnm.pdf.
2. E. O. Thorp and S. Mizusawa. 2011. Optimal capital growth versus black swan insurance Part I:
Mediocristan. Wilmott magazine, Sept.
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