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 )−52 for x>0

andp ( 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

∞

( x−K )w ( x )dx

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

1

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=∫A

∞

u ( x )dx=er

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

(3 )C=e−r ∫x=K

∞

( x−K )u ( x )dx

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 ~μ=exp ( .05 )=1.05127 , the one year wealth relative at our assumed riskless compound growth rate of 5%. The difference between u and ~μ 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).

(4 )u (x )=72.416 (4+36.3275 (−1.04940+x )2 )−5

2

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 ~μ 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 .809 .809

.4 .619 .620

.6 .429 .433

.8 .245 .2531.0

.104 .105

1.2

.0325 .0299

1.4

.00785 .00861

1.6

.00158 .00315

1.8

.000286 .00142

2.0

.0000479 .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.

4

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 S h=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.

5

6

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.50000.0400

0.0500

0.0600

0.0700

0.0800

0.0900

0.1000

0.1100

0.1200

0.1300

Figure 3: Geometric growth versus standard deviation for values in the Tables where g>= .05 and v<= .5. Extremistan.

f1f2f3f4f5f6f7f8f9RisklessRateStkIndex

Standard Deviation

Geom

etric

Gro

wth

7

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 0.0518805.7/13/201

1 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 0.09680.1 0.0577 0.0648 0.0713 0.0772 0.0826 0.0873 0.0914 0.0947 0.09710.2 0.0585 0.0663 0.0734 0.0798 0.0854 0.0903 0.0943 0.0974 0.09900.3 0.0596 0.0683 0.0760 0.0829 0.0888 0.0938 0.0976 0.1000 0.10040.4 0.0610 0.0707 0.0793 0.0867 0.0928 0.0977 0.1011 0.1025 0.10090.5 0.0627 0.0738 0.0834 0.0913 0.0976 0.1022 0.1046 0.1043 0.09910.6 0.0652 0.0780 0.0886 0.0971 0.1033 0.1070 0.1076 0.1040 0.09250.7 0.0685 0.0836 0.0954 0.1042 0.1096 0.1113 0.1083 0.0985 0.07560.8 0.0733 0.0910 0.1039 0.1120 0.1150 0.1122 0.1019 0.0802 0.03530.9 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 -0.26231.1 0.0869 0.0950 0.0832 0.0534 0.0043 -0.0685 -0.1751 -0.3391 -0.63711.2 0.0741 0.0565 0.0135 -0.0522 -0.1429 -0.2655 -0.4348 -0.6857 -1.13091.3 0.0493 -0.0003 -0.0763 -0.1768 -0.3052 -0.4707 -0.6920 -1.0126 -1.57231.4 0.0219 -0.0540 -0.1543 -0.2786 -0.4317 -0.6243 -0.8774 -1.2394 -1.86511.5 -0.0008 -0.0940 -0.2089 -0.3467 -0.5134 -0.7204 -0.9901 -1.3732 -2.03201.6 -0.0174 -0.1207 -0.2436 -0.3887 -0.5624 -0.7767 -1.0547 -1.4481 -2.12291.7 -0.0289 -0.1379 -0.2652 -0.4140 -0.5914 -0.8095 -1.0917 -1.4903 -2.17311.8 -0.0366 -0.1489 -0.2786 -0.4296 -0.6089 -0.8290 -1.1134 -1.5148 -2.20181.9 -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

8

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 1

K 0.0 0.0214 0.0425 0.0635 0.0845 0.1056 0.1272 0.1496 0.1731 0.1989 0.23330.1 0.0236 0.0469 0.0700 0.0932 0.1168 0.1409 0.1662 0.1935 0.22490.2 0.0263 0.0522 0.0779 0.1038 0.1302 0.1575 0.1864 0.2182 0.25670.3 0.0297 0.0588 0.0877 0.1169 0.1467 0.1779 0.2113 0.2488 0.29620.4 0.0339 0.0670 0.1000 0.1333 0.1676 0.2036 0.2427 0.2877 0.34670.5 0.0395 0.0778 0.1160 0.1546 0.1945 0.2368 0.2834 0.3382 0.41290.6 0.0470 0.0924 0.1374 0.1830 0.2303 0.2810 0.3377 0.4058 0.50220.7 0.0577 0.1128 0.1671 0.2221 0.2796 0.3417 0.4121 0.4987 0.62580.8 0.0737 0.1426 0.2100 0.2781 0.3494 0.4271 0.5166 0.6291 0.79960.9 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 1.33841.1 0.1926 0.3401 0.4723 0.6003 0.7318 0.8752 1.0427 1.2599 1.60631.2 0.2511 0.4178 0.5595 0.6926 0.8267 0.9707 1.1371 1.3510 1.68971.3 0.2924 0.4559 0.5874 0.7071 0.8249 0.9492 1.0909 1.2708 1.55211.4 0.3071 0.4505 0.5604 0.6576 0.7515 0.8491 0.9589 1.0967 1.30981.5 0.3004 0.4185 0.5057 0.5812 0.6531 0.7270 0.8093 0.9118 1.06891.6 0.2815 0.3765 0.4447 0.5028 0.5576 0.6134 0.6753 0.7519 0.86871.7 0.2579 0.3339 0.3874 0.4326 0.4748 0.5177 0.5650 0.6233 0.71191.8 0.2337 0.2949 0.3374 0.3731 0.4063 0.4398 0.4767 0.5221 0.59081.9 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

9

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 0.9 1

K 0.0 0.3265 0.3173 0.3073 0.2965 0.2850 0.2726 0.2591 0.2441 0.2266 0.20050.1 0.3255 0.3152 0.3040 0.2919 0.2788 0.2646 0.2489 0.2312 0.20930.2 0.3247 0.3132 0.3006 0.2869 0.2721 0.2559 0.2378 0.2170 0.19080.3 0.3240 0.3111 0.2969 0.2814 0.2645 0.2459 0.2252 0.2011 0.17020.4 0.3235 0.3089 0.2927 0.2750 0.2556 0.2344 0.2105 0.1826 0.14670.5 0.3231 0.3063 0.2877 0.2673 0.2450 0.2204 0.1928 0.1605 0.11880.6 0.3225 0.3030 0.2812 0.2575 0.2315 0.2028 0.1706 0.1330 0.08470.7 0.3208 0.2977 0.2720 0.2439 0.2132 0.1794 0.1415 0.0973 0.04090.8 0.3154 0.2877 0.2567 0.2229 0.1861 0.1457 0.1005 0.0481 -0.01840.9 0.3000 0.2656 0.2274 0.1860 0.1413 0.0925 0.0382 -0.0246 -0.1039

1 0.2622 0.2180 0.1698 0.1184 0.0635 0.0041 -0.0617 -0.1375 -0.23331.1 0.1915 0.1324 0.0703 0.0056 -0.0624 -0.1354 -0.2159 -0.3089 -0.42771.2 0.0961 0.0156 -0.0653 -0.1476 -0.2334 -0.3250 -0.4264 -0.5446 -0.69891.3 -0.0025 -0.1103 -0.2151 -0.3207 -0.4306 -0.5485 -0.6802 -0.8362 -1.04531.4 -0.0914 -0.2308 -0.3645 -0.4996 -0.6410 -0.7941 -0.9671 -1.1757 -1.46221.5 -0.1693 -0.3439 -0.5119 -0.6825 -0.8627 -1.0598 -1.2852 -1.5609 -1.94771.6 -0.2396 -0.4533 -0.6603 -0.8724 -1.0983 -1.3477 -1.6358 -1.9925 -2.50151.7 -0.3058 -0.5626 -0.8135 -1.0727 -1.3507 -1.6602 -2.0206 -2.4713 -3.12291.8 -0.3706 -0.6744 -0.9738 -1.2854 -1.6218 -1.9986 -2.4405 -2.9973 -3.81111.9 -0.4356 -0.7905 -1.1430 -1.5120 -1.9126 -2.3637 -2.8956 -3.5702 -4.5647

2 -0.5021 -0.9118 -1.3217 -1.7530 -2.2234 -2.7554 -3.3857 -4.1893 -5.3825

10

11

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 ,∧r, etc.

Figure 5.

Comparison of the cumulative distribution of Maximum Drawdowns

12

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.

13

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).

14

1 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 Log-Stable Uncertainty,” http://www.econ.ohio-state.edu/jhm/papers/rmn.pdf.