Optimal Capital Growth Versus Black Swan Insurance. - Part 2:
Extremistan
by Edward O. Thorp*† and Steven Mizusawa*
1. Introduction
Consider 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.
2. 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 𝜇 = 1.12750 and variance 𝜎 2 = 0.05188
match those we used for Mediocristan. The resulting density function was
(1)
𝑝(𝑥) = 72.416(4 + 36.3275(−1.12562 + 𝑥)2 )−
5⁄
2
for 𝑥 > 0
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. 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
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 fattailed 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.
1
∞
∞
∫ 𝑢(𝑥)𝑑𝑥 = ∫ 𝑢(𝑥)𝑑𝑥 = 𝑒 𝑟
−∞
𝐴
Note that when 𝑝(𝑥) is mean-shifted to get 𝑢(𝑥), the point where 𝑢(𝑥) becomes nonzero 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
(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
𝜇̃ = exp(. 05) = 1.05127, the one year wealth relative at our assumed riskless
compound growth rate of 5%. The difference between 𝑢 and 𝜇̃ is 1.127491.05127=0.07622 so we change -1.12562 in (1) to -1.2562-0.07622=-1.04940 to get
𝑢(𝑥) in equation (4).
(4)
𝑢(𝑥) = 72.416(4 + 36.3275(−1.04940 + x)2 )−
for
𝑥 ≥ −0.07622
𝑢(𝑥) = 0 for
5⁄
2
and
𝑥 < −0.07622.
Alternatively, we could have truncated the risk neutral t-distribution at zero, for 𝜇̃
and 𝜎, finding it by the same process we used to find 𝑝(𝑥).
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.
2
Table 1. Call Prices Compared for Lognormal and Truncated Student t.
K
Lognormal
Student t
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
.809
.619
.429
.245
.104
.0325
.00785
.00158
.000286
.0000479
.809
.620
.433
.253
.105
.0299
.00861
.00315
.00142
.000748
Figure 1.
3
Figure 2.
3. Geometric Growth, Standard Deviation and Sharpe Ratio
Proceeding as in Part 1 we compute 𝑔, 𝜎 and the Sharpe ratio from equations (9) –
(12), (9a) – (12a), and (12s), replacing the lognormal 𝑔(𝑥) by the truncated Student t
𝑢(𝑥) throughout. The results, in Figure 3 and Tables 2-4, correspond to those for
Mediocristan in Part 1, Figure 1 and Tables 1-3.
Figure 3 shows that for 𝜎 ≤ 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 < 𝜎 < 0.24, the option portfolios have a greater 𝑔 for a given 𝜎, or
alternatively, less 𝜎 for a given 𝑔, with the exception of 𝑓 = 0.1. Above 𝜎 = 0.24, the
option portfolios have an efficient frontier peak at 𝑓 = 0.4, 𝐾 = 0.9, with 𝑔 = 0.1168,
𝜎 = 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, 𝜎 = 0.1926, and 𝑆ℎ = 0.195, versus the index portfolio 𝑓 =
1.0, which has 𝑔 = 0.0968, 𝜎 = 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.
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.
4
Figure 3: Geometric growth versus standard deviation for values in the Tables where
g>= .05 and v<= .5. Extremistan.
0.1300
0.1200
0.1100
f1
Geometric Growth
0.1000
f2
f3
f4
0.0900
f5
f6
0.0800
f7
f8
0.0700
f9
RisklessRate
0.0600
StkIndex
0.0500
0.0400
0.0000
0.0500
0.1000
0.1500
0.2000
0.2500
0.3000
Standard Deviation
5
0.3500
0.4000
0.4500
0.5000
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.
7/13/2011
f-->
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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
0.0570
0.0577
0.0585
0.0596
0.0610
0.0627
0.0652
0.0685
0.0733
0.0796
0.0861
0.0869
0.0741
0.0493
0.0219
-0.0008
-0.0174
-0.0289
-0.0366
-0.0418
-0.0454
0.0635
0.0648
0.0663
0.0683
0.0707
0.0738
0.0780
0.0836
0.0910
0.0999
0.1055
0.0950
0.0565
-0.0003
-0.0540
-0.0940
-0.1207
-0.1379
-0.1489
-0.1561
-0.1608
0.0695
0.0713
0.0734
0.0760
0.0793
0.0834
0.0886
0.0954
0.1039
0.1122
0.1118
0.0832
0.0135
-0.0763
-0.1543
-0.2089
-0.2436
-0.2652
-0.2786
-0.2872
-0.2928
0.0750
0.0772
0.0798
0.0829
0.0867
0.0913
0.0971
0.1042
0.1120
0.1168
0.1059
0.0534
-0.0522
-0.1768
-0.2786
-0.3467
-0.3887
-0.4140
-0.4296
-0.4393
-0.4456
0.0801
0.0826
0.0854
0.0888
0.0928
0.0976
0.1033
0.1096
0.1150
0.1135
0.0871
0.0043
-0.1429
-0.3052
-0.4317
-0.5134
-0.5624
-0.5914
-0.6089
-0.6197
-0.6267
6
0.0847
0.0873
0.0903
0.0938
0.0977
0.1022
0.1070
0.1113
0.1122
0.1007
0.0529
-0.0685
-0.2655
-0.4707
-0.6243
-0.7204
-0.7767
-0.8095
-0.8290
-0.8410
-0.8486
0.0887
0.0914
0.0943
0.0976
0.1011
0.1046
0.1076
0.1083
0.1019
0.0753
-0.0026
-0.1751
-0.4348
-0.6920
-0.8774
-0.9901
-1.0547
-1.0917
-1.1134
-1.1266
-1.1349
0.0922
0.0947
0.0974
0.1000
0.1025
0.1043
0.1040
0.0985
0.0802
0.0301
-0.0930
-0.3391
-0.6857
-1.0126
-1.2394
-1.3732
-1.4481
-1.4903
-1.5148
-1.5295
-1.5387
0.0951
0.0971
0.0990
0.1004
0.1009
0.0991
0.0925
0.0756
0.0353
-0.0580
-0.2623
-0.6371
-1.1309
-1.5723
-1.8651
-2.0320
-2.1229
-2.1731
-2.2018
-2.2188
-2.2293
0.0968
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.
7/14/2011
f-->
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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
0.0214
0.0236
0.0263
0.0297
0.0339
0.0395
0.0470
0.0577
0.0737
0.0988
0.1379
0.1926
0.2511
0.2924
0.3071
0.3004
0.2815
0.2579
0.2337
0.2108
0.1901
0.0425
0.0469
0.0522
0.0588
0.0670
0.0778
0.0924
0.1128
0.1426
0.1878
0.2547
0.3401
0.4178
0.4559
0.4505
0.4185
0.3765
0.3339
0.2949
0.2607
0.2312
0.0635
0.0700
0.0779
0.0877
0.1000
0.1160
0.1374
0.1671
0.2100
0.2733
0.3638
0.4723
0.5595
0.5874
0.5604
0.5057
0.4447
0.3874
0.3374
0.2950
0.2593
0.0845
0.0932
0.1038
0.1169
0.1333
0.1546
0.1830
0.2221
0.2781
0.3593
0.4719
0.6003
0.6926
0.7071
0.6576
0.5812
0.5028
0.4326
0.3731
0.3236
0.2827
7
0.1056
0.1168
0.1302
0.1467
0.1676
0.1945
0.2303
0.2796
0.3494
0.4494
0.5845
0.7318
0.8267
0.8249
0.7515
0.6531
0.5576
0.4748
0.4063
0.3502
0.3043
0.1272
0.1409
0.1575
0.1779
0.2036
0.2368
0.2810
0.3417
0.4271
0.5481
0.7081
0.8752
0.9707
0.9492
0.8491
0.7270
0.6134
0.5177
0.4398
0.3769
0.3261
0.1496
0.1662
0.1864
0.2113
0.2427
0.2834
0.3377
0.4121
0.5166
0.6631
0.8529
1.0427
1.1371
1.0909
0.9589
0.8093
0.6753
0.5650
0.4767
0.4063
0.3500
0.1731
0.1935
0.2182
0.2488
0.2877
0.3382
0.4058
0.4987
0.6291
0.8103
1.0405
1.2599
1.3510
1.2708
1.0967
0.9118
0.7519
0.6233
0.5221
0.4424
0.3792
0.1989
0.2249
0.2567
0.2962
0.3467
0.4129
0.5022
0.6258
0.7996
1.0400
1.3384
1.6063
1.6897
1.5521
1.3098
1.0689
0.8687
0.7119
0.5908
0.4970
0.4235
1
0.2333
Table 4: Sharpe 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.
7/14/2011
f-->
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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
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
8
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
0.9
1
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 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).
When 𝐾 = 0.9, 𝑓 = 0.2, 𝑇 = 32 the option portfolio has 𝑔 = 0.0999, 𝜎 = 0.1878,
𝑆ℎ = 0.2656 whereas the index portfolio with 𝑓 = 1.0 is inside the efficient frontier at
𝑔 = 0.0968, 𝜎 = 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 peaks of
the efficient frontier, it yields 𝑔 = 0.1168, 𝜎 = 0.3593, and 𝑆ℎ = 0.2274. 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.
4. 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 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 𝜇, 𝜎, 𝑎𝑛𝑑 𝑟, etc.
Figure 5.
Comparison of the cumulative distribution of Maximum Drawdowns
9
For Options with K = .9, f=.2, T = 32 years ( in Blue) and Stock Index (fraction = 1.0 in
Red).
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. ¼ per quarter, to smooth out costs,
payoffs, and the need to replace – if American options are used – options which are
exercised early.
10
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).
See Ekstrom, et al, Quantitative Finance Vol. II, No. 8, August 2011, page 1125, for a
discussion of when we have no-arbitrage pricing or risk-neutral pricing models. Also
see J. Huston McCulloch, “The Risk-Neutral Measure and Option Pricing Under LogStable Uncertainty,” http://www.econ.ohio-state.edu/jhm/papers/rmn.pdf.
1
11