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Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management
Vincenzo Bochicchio, Niklaus Bühlmann, Stephane Junod and Hans-Fredo List Swiss Reinsurance Company
Mythenquai 50/60, CH-8022 Zurich Telephone: +41 1 285 2207 Facsimile: +41 1 285 4179
Mark H.A. Davis Tokyo-Mitsubishi International plc 6 Broadgate, London EC2M 2AA
Telephone: +44 171 577 2714 Facsimile: +44 171 577 2888
Abstract. Asset/Liability management, optimal fund design and optimal portfolio selection have been key issues of interest to the (re)insurance and investment banking communities, respectively, for some years - especially in the design of advanced risk- transfer solutions for clients in the Fortune 500 group of companies. Recently, the securitization of (re)insurance claims portfolios has also attracted considerable attention among (re)insurance companies and their clients. It turns out that the new concept of limited risk arbitrage (LRA) investment management in a diffusion type liabilities, securities and derivatives market introduced in our papers Baseline for Exchange Rate – Risks of an International Reinsurer, AFIR 1996, Vol. I, p. 395, and Risk/Arbitrage Strategies: A New Concept for Asset/Liability Management, Optimal Fund Design and Optimal Portfolio Selection in a Dynamic, Continuous-Time Framework Part I: Securities Markets and Part II: Securities and Derivatives Markets, AFIR 1997, Vol. II, p. 543, is immediately applicable to all of the above mentioned practical problems. The competitive advantage of applying LRA strategies in the design of advanced risk transfer solutions for Fortune 500 clients lies in the fact that these techniques achieve an efficient allocation of risk in an overall portfolio context rather than eliminating (at a high price) derivatives risk exposure on a single- instrument basis by replication (hedging) with underlying securities. The main quantities of practical interest (i.e., the optimal LRA asset allocation, etc.) can be derived by solving a (quasi-) linear partial differential equation of the second order (e.g., by using a finite difference approximation with locally uniform convergence properties, see Part III: A Risk/Arbitrage Pricing Theory) or (in our more sophisticated impluse control approach, see Part IV: An Impulse Control Approach to Limited Risk Arbitrage) by using an efficient Markov chain approximation scheme [i.e., essentially the same (formal) finite difference techniques (with weak convergence properties)]. However, in many practical applications there are much simpler numerical solution techniques, see Part V: A Guide to Efficient Numerical Implementations. We present here such an alternative lattice-bared options portfolio
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management methodology which allows the determination of the main LRA quantities by simply solving a linear program at each lattice node.
Key Words and Phrases. Risk/Arbitrage platform, dynamic programming procedure, contingent claim price/sensitivity forecast, LRA optimization program, state dependent linear optimization.
Contents.
1. Introduction - Swiss Re Registered Share Model - Risk-free Interest Rate - Volatility - Dividend Yield - European Call Options - European Put Options - One Period LRA Strategies - Base Value Scenario - Minimum Premium Scenario - Maximum Premium Scenario
2. LRA Option Strategies 3. “Mitarbeiter-Option” Trading
Base Value Scenario Minimum Premium Scenario Maximum Premium Scenario
Appendix: References
3 3 3 3 3 5 8 11 11 21 23 25 36 37 43 45 47
1. Introduction
In order to gain a first insight into how limited risk arbitrage (LRA) trading and portfolio management strategies work in practice and can be successfully used in modern (re)insurance and corporate and investment banking applications, we consider long-dated European call and put options on the Swiss Re registered share (RUKN) with a current market value of CHF 1218.00 (as of 18 October 1995, see also Bühlmann, Davis and List [1], [2] and [3]).
Swiss Re Registered Share Model. In a first approximation, the Swiss Re registered share can be assumed to follow an Ito process
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----------
---
risk -averse evolution
risk-neutral evolution
standard Brownian motions
with constant expected rate of return µ = 17% and constant volatility σ (geometric Brownian motion). The risk-free rate of interest r and the dividend yield y can also be assumed to be constant over the 5 year option maturity horizon.
Risk-free Interest Rate. The risk-free rate of interest applicable during the 5 year option maturity period is estimated to be 4% p.a. (continuously compounded). The sensitivity of the European call and put option characteristics with respect to changes in the risk- free interest rate is however examined for a rate variation range from 3% to 5% (p.a.).
Volatility. The volatility of the Swiss Re registered share applicable during the 5 year option maturity period is estimated to be 22.5% p.a.; the sensitivity of the European call and put option characteristics with respect to changes in RUKN price volatility is however examined for a volatility variation range from 20% to 25% (pa.).
Dividend Yield. The dividend yield of the Swiss Re registered share applicable during the 5 year option maturity period is calculated as follows:
Current Dividend Value (18 October 1995):
CHF 15.00
Current Share Value (18 October 1995):
CHF 1218.00
Recent Dividend Growth Rate Estimates (18 October 1995):
UBS 13.3% pa. SBC Warburg 26.0% p.a James Capel 21.6% pa
Average 20.3% p.a
Dividend Yield:
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dividend in year i
futures price of registered share in year i increase in share position in year i
(reinvestment of dividends)
equation for dividend yield y
Note that this equation is simplified for ease of calculation. A more accurate (and more complicated) expression would be
The effects of this simplification are compensated for in the choice of the dividend yield variation range outlined below.
Numerical Evaluation (Mathematics):
With the initial values D0 = 15.00 and S0 = 1218.00 Mathematica calculates the
implied dividend yields for the above interest rate and dividend growth rate scenarios as follows:
g= 13% p.a r = 3% p.a 4% p.a 5% p.a
y = 1.66% p.a 1.60% p.a 1.55% p.a
g = 20% p.a r= 3% p.a 4% p.a 5% p.a y = 2.02% p.a 1.96% p.a 1.89% p.a
g = 26% p.a r = 3% p.a 4% p.a 5% p.a y = 2.39% p.a 2.31% p.a 2.23% p.a
The dividend yield of the Swiss Re registered share applicable during the 5 year option maturity period is therefore taken to be 2% p.a. (continuously compounded). The sensitivity of the European call and put option characteristics with respect to changes in dividend yield is however examined for a yield variation range from 1.6% to 2.4% (p.a.).
European Call Options. With the above (Black & Scholes) stock price model futures prices of the Swiss Re registered share (RUKN) and the values of corresponding European call options can be analytically calculated as follows:
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Futures:
T futures maturity
European Call:
T option maturity X option strike
N[ ] standard normal distribution
Specifically, we consider the call options (strike schedule)
European Call Option Option Price Exercise Price Base Value Minimum Premium Maximum Premium
1218.0000 265.0653 204.7840 326.4088 1268.0000 246.1182 186.7104 307.1242 1318.0000 228.4387 170.1043 288.9330 1368.0000 211.9629 154.8749 271.7880 1418.0000 196.6260 140.9314 255.6410 1468.0000 182.3631 128.1841 240.4437 1518.0000 169.1102 116.5454 226.1481 1568.0000 156.8048 105.9308 212.7071 1618.0000 145.3866 96.2598 200.0744 1668.0000 134.7972 87.4559 188.2053 1718.0000 124.9809 79.4474 177.0566
Base Value Minimum Premium: Maximum Premium: volatility = 22.5% volatility = 20% volatility = 25% dividend yield = 2% dividend yield = 2.4% dividend yield = 1.6% interest rate = 4% interest rate = 3% interest rate = 5%
or in graphical form
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The associated option risk parameters are
30
(note the relatively modest first and second order risk exposure with respect to changes in the value of the underlying Swiss Re registered share, RUKN)
31
as a function of the exercise or strike price. While the call option’s first and second order risk exposure with respect to changes in the value of the underlying Swiss Re registered share (delta and gamma) is relatively modest, its volatility (vega), interest rate (rho rate) and dividend yield (rho yield) risk exposures are quite significant.
European Put Options. The price of a European put option is similarly
Like above, we consider the put options (strike schedule)
European Put Option Option Price Exercise Price Base Value Maximum Premium Minimum Premium
1218.0000 1268.0000 1318.0000 1368.0000 1418.0000 1468.0000 1518.0000 1568.0000 1618.0000 1668.0000 1718.0000
160.0896 182.0700 205.3181 229.7699 255.3606 282.0253 309.6999 338.3221 367.8314 398.1696 429.2809
172.8269 197.7817 224.2039 252.0029 281.0876 311.3687 342.7583 375.1721 408.5293 442.7538 477.7736
150.4712 170.1159 190.8541 212.6385 235.4209 259.1529 283.7867 309.2750 335.5717 362.6320 390.4127
Base Value: Maximum Premium: Minimum Premium: volatility = volatility = 20% volatility = 25%
dividend yield = 2% dividend yield = 2.4% dividend yield = 1.6%
interest rate = 4% interest rate = 3% interest rate = 5%
or in graphical form
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22.5%
(note that of course now the maximum and the minimum premium scenarios have the opposite meanings). The associated option risk parameters are
33
(note again the relatively modest first and second order risk exposure with respect to changes in the value of the underlying Swiss Re registered share, RUKN)
34
as a function of the exercise or strike price. Again, the option’s volatility, interest rate and dividend yield risk exposures are quite significant.
One Period LRA Strategies. As a next step, therefore, we ask ourselves whether by “wisely” (i.e., in a limited risk arbitrage sense) choosing among RUKN and all the above European options, significantly better investment opportunities could be created.
Specifically, we use the linear program
in our analysis (see also Part I: Securities Markets, Part II : Securities and Derivatives Markets and Part V: A Guide to Efficient Numerical Implementations). Furthermore, we consider LRA strategies under the above three scenarios, where we define the maximum premium and the minimum premium scenarios as in the call options case.
Base Value Scenario. In the base value scenario, the portfolio components’ parameters are
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Instrument Parameters
Term Period 18, 10, 95 Price Delta Gamma Theta Vega Rho Rate Rho Yield ?? ?? ?? ??
Instrument RUKN 1218.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -100.0000 100.0000
2 C1218 265.0653 0.6097 0.0005 -24.2209 888.6284 2390.0997 -3716.8785 -100.0000 100.0000
3 C1268 246.1182 0.5831 0.0005 -24.9993 918.2728 2323.3277 -3555.2671 -100.0000 100.0000
4 C1318 228.4387 0.5569 0.0006 -25.6010 942.0132 2251.8725 -3395.3176 -100.0000 100.0000
5 C1368 211.9629 0.5311 0.0006 -26.0369 060.0800 2176.8897 -3237.8656 -100.0000 100.0000
6 C1418 196.6260 0.5058 0.0006 -26.3194 972.7741 2099.4068 -3083.6143 -100.0000 100.0000
7 C1468 182.3631 0.4811 0.0006 -26.4611 980.4463 2020.3278 -2933.1427 -100.0000 100.0000
8 C1518 169.1102 0.4571 0.0006 -26.4751 983.4800 1940.4392 -2786.9169 -100.0000 100.0000
9 C1568 156.8048 0.4339 0.0006 -26.3745 982.2762 1860.4182 -2645.3014 -100.0000 100.0000
10 C1618 145.3866 0.4115 0.0006 -28.1717 977.2426 1780.8413 -2506.5706 -100.0000 100.0000
11 C1668 134.7972 0.3899 0.0006 -25.8791 968.7833 1702.1933 -2376.9177 -100.0000 100.0000
12 C1718 124.9809 0.3891 0.0006 -25.5083 957.2924 1624.8768 -2250.4661 -100.0000 100.0000
13 P1218 160.0896 0.2951 0.0005 -6.3805 888.6284 -2600.3408 1799.0157 -100.0000 100.0000
14 P1268 182.0700 -0.3216 0.0005 -5.5218 918.2728 -2871.9750 1960.6271 -100.0000 100.0000
15 P1318 205.3181 -0.3478 0.0006 -4.4863 942.0132 -3148.2922 2120.5767 -100.0000 100.0000
16 P1368 229.7699 -0.3737 0.0006 -3.2852 960.0800 -3428.1371 2278.0286 -100.0000 100.0000
17. P1418 255.3606 -0.3990 0.0006 -1.9306 972.7741 -3710.4821 2432.2800 -100.0000 100.0000
18 P1488 282.0253 -0.4236 0.0006 -0.4352 980.4463 -3994.4232 2582.7515 -100.0000 100.0000
19 P1518 309.6999 -0.4475 0.0036 1.1879 983.4800 -4279.1739 2728.9773 -100.0000 100.0000
20 P1568 338.3221 -0.4708 0.0006 2.9257 982.2762 -4564.0570 2870.5928 -100.0000 100.0000
21 P1618 367.8314 -0.4933 0.0006 4.7655 977.2426 -4848.4960 3007.3236 -100.0000 100.0000
22 P1668 398.1686 -0.5149 0.0006 6.6952 968.7833 -5132.0061 3138.9766 -100.0000 100.0000
23 P1718 429.2809 -0.5356 0.0006 8.7031 957.2924 -5414.1847 3265.4281 -100.0000 100.0000
(the position bounds usually implement trading constraints, especially on liquidity; here they are chosen arbitrarily) and a maximum value / maximum theta / minimum vega LRA strategy’ is
Model Parameters
a (Value) 1
b Theta 1
c (Delta) 0
d (Gamma) 0
e (Vega) 1 LRA Strategy f (Rho Rate) 0
g (Rho Yield) 0 Minimum Maximum Value 2299 6516 Number of Instruments 23 Delta 1.0000
Value Lower Bound 124.9809 1218.0000 Gamma 0.0005 Value Upper Bound Theta 38.1451 Delta Lower Bound -0.5356 -0.5356 1.0000 Vega 888.6284
Delta Upper Bound 1.0000 Rho Rate -5414.1847
Gamma Lower Bound 0.0005 0.0005 0.0006 Rho Yield -3716.8785 Gamma Upper Bound 0.0006
Theta Lower Bound -26.4751 8.7031 Result 1
Theta Upper Bound
Vega Lower Bound 888.6284 983.4800
Vega Upper Bound
Rho Rate Lower Bound -5414.1847 -5414.1847 2390.0997 Rho Rate Upper Bound 2390.0997
Rho Yield Lower Bound -3716.8785 -3716.8785 3265.4281 Rho Yield Upper Bound 3265.4281
One reason for minimizing the portfolio vega would be to keep the effects of model miss-specification with respect to the volatility of RUKN minimal. In general, however, limited risk arbitrage investment management allows the exact positioning of a securities, futures and options portfolio according to a trader’s or portfolio manager's beliefs and expectations about future market moves.
36
(result = 1 above means that the LRA strategy is not unique; furthermore, the portfolio value / portfolio theta / portfolio vega constraints are not considered as we maximize portfolio value / maximize portfolio theta / minimize portfolio vega). As interest rate risk and dividend yield risk are the dominating exposures if a purchase of the above European call or put options is considered, an interesting LRA strategy would be one that also minimizes this exposure - to 10%, say:
Model Parameters
a (Value) 1 b (Theta) 1
c (Delta) 0
d (Gamma) 0 e (Vega) -1 LRA Strategy f (Rho Rate) 0
g (Rho Yield) 0 Minimum Maximum Value 1326.1652 Number of Instruments 23 Delta 1.0000 Value Lower Bound 124.9809 1218.0000 Gamma 0.0005 Value upper Bound Theta 14.1605 Delta Lower Bound -0.5356 -0.5356 1.0000 Vega 888.6284 Delta Upper Bound 1.0000 Rho Rate -541.4185 Gamma Lower Bound 0.0005 0.0005 0.0006 Rho Yield 371.6879 Gamma Upper Bound 0.0006
Theta Lower Bound -26.4751 8.7031 Result 1 Theta Upper Bound
Vega Lower Bound 888.6284 983.4800 Vega Upper Bound Rho Rate Lower Bound -541.4185 -5414.1847 2390.0997 Rho Rate Upper Bound 239.0100 Rho Yield Lower Bound -371.6879 -3716.8785 3265.4281 Rho Yield Upper Bound 326.5428
37
On the other hand, a significant increase2 in the investor’s risk tolerances for portfolio delta (instantaneous investment risk) and the portfolio gamma (future dynamics) has the following effect on the corresponding LRA asset allocation:
the risk
A. Minimum Vega.
Model Parameters
a (Value) 1
b (Theta) 1
c (Delta) 0
d (Gamma) 0
e (Vega) -1 LRA Strategy
f (Rho Rate) 0
g (Rho Yield) 0 Minimum Maximum Value 120670.1326
Number of Instruments 23 Delta 99.4644
Value Lower Bound 124.9809 1218.0000 Gamma -0.2928
Value Upper Bound Theta 10962.7813 Delta Lower Bound -53.5608 -0.5356 1.0000 Vega -489195.4548
Delta Upper Bound 100.0000 Rho Rate 2390.0997 Gamma Lower Bound -53.1860 0.0005 0.0006 Rho Yield 3265.4281 Gamma Upper Bound 58.8630
Theta Lower Bound -26.4751 8.7031 Result Theta Upper Bound
Vega Lower Bound 888.6284 983.4800 Vega Upper Bound
Rho Rate Lower Bound -5414.1847 -5414.1847 2390.0997 Rho Rate Upper Bound 2390.0997 Rho Yield Lower Bound -3716.8785 -3716.8785 3265.4281 Rho Yield Upper Bound 3265.4281
² The new limits for delta and gamma are chosen such that the effects of a 1% change in interest rates
and dividend yield are of the same order of magnitude as a 1 CHF change in the value of RUKN.
38
1
The enormous vega (exposure with respect to volatility changes of RUKN) could be quite advantageous in the case where market analysts strongly believe in a decrease in RUKN volatility over the investment period considered: a 1% decrease in RUKN volatility would increase the LRA portfolio value by CHF 4892.00. Such volatility changes could be effects of changes in the underlying fundamentals of RUKN or be implications of changes in the dynamics of the futures and options markets (implied volatility).
5. Constrained Vega.
Model parameters
a (Value) 1
b (Theta) 1
c (Delta) 0
d (Gamma) 0
e (Vega) 0 LRA Strategy f (Rho Rate) 0
0 g (Rho Yield) Minimum Maximum Value 122880.6516
Number of Instruments 23 Delta 100.0000 Value Lower Bound 124.9809 1218.0000 Gamma -0.0005
Value Upper Bound Theta 78.0896 Delta Lower Bound -53.5608 -0.5356 1.0000 Vega -888.6284 Delta Upper Bound 100.0000
Rho Rate -5414.1847
Gamma Lower Bound -53.1860 0.0005 0.0006 Rho Yield -3716.8785 Gamma Upper Bound 58.8630
Theta Lower Bound -26.4751 8.7031 Result Theta Upper Bound
Vega Lower Bound -888.6284 888.6284 983.4800
Vega Upper Bound 983.4800 Rho Rate Lower Bound -5414.1847 -5414 1847 2390.0997
Rho Rate Upper Bound 2390.0997
Rho Yield Lower Bound -3716 8785 -3716.8785 3265.4281 Rho Yield Upper Bound 3265.4281
39
1
1
0
0
0
0
0
23
Model Parameters
a (Value)
b (Theta)
c (Delta)
d (Gamma)
e (Vega) LRA Strategy
f (Rho Rate)
g (Rho Yield) Minimum Maximum Value 122881 6516
Number of Instruments Delta 100 0000
Value Lower Bound 124.9809 1218.0000 Gamma 0.0000
Value Upper Bound Theta 58.1173 Delta Lower Bound -53.5608 -0.5356 1 0000 Vega 0.0000 Delta Upper Bound 100.0000 Rho Rate -5414 1847
Gamma Lower Bound -53.1860 0.0005 0 0006 Rho Yield -3716 8785
Gamma Upper Bound 58.8630
Theta Lower Bound -26.4751 87031 Result
Theta Upper Bound
Vega Lower Bound 0.0000 889.6284 983.4800
Vega Upper Bound 0 0000
Rho Rate Lower Bound 2390.0997
Rho Rate Upper Bound
-5414.1847 -5414.1847
2390.0997
Rho Yield Lower Bound -3716.8785 -3716.8785 3265.4281
Rho Yield Upper Bound 3265.4281
A constrained vega strategy would not try to position the LRA portfolio with respect to strong expectations about a decrease in RUKN volatility but rather strive to keep the effects of volatility changes (on LRA portfolio value) small: in the above example an adverse (i.e., upward) move in RUKN volatility of 1% over the next trading period would only cost CHF 8.90. Note in such a context also the following “zero exposure” (= “zero miss-specification error”) LRA strategies for both vega and rho:
C. Zero Vega.
40
Note the remarkable similarity of the positions held:
41
D. Zero Vega, Zero Rho (Rate).
Model Parameters
a (Value) 1
b (Theta) 1 c (Delta) 0
d (Gamma) 0 e (Vega) 0 LRA Strategy F (Rho Rate) 0
g (Rho yield)
Number of Instruments
0 Minimum Maximum Value 12.1800.0000
23 Delta 100_0000 Value Level Bound 124.9809 1218.0000 Gamma 0.0000 Value upper bound Theta 14.8512 Delta Lower Bound -53.5608 -0.5356 1_0000 Vega 0.0000 Delta Upper Bound 100.0000 Gamma Lower Bound
Rho Rate 0.0000 -53.1860 5.0005 0.0006
Gamma Upper Bound Rho Yield
58.8630 -3716.8785
Theta Lower Bound -26.4751 8.7031 Theta Upper Bound
Result
Vega Lower Bound 0.0000 888.6284 983 4800 Vega Upper Bound
Rho Rate Lower Bound 0.0000
0.0000 5414.1847 Rho Rate Upper Bound
2390.0997 0.0000
Rho Yield Bound -3716.8785 -3716 8785 3265.4281 Rho Yield Upper Bound 3265.4281
42
Portfolio rebalancing is now:
E. Zero Vega. Zero Rho (Rate), Zero Rho (Yield)
Model Parameters a (Value) 1
b (Theta) 1
c (Delta) 0
d (Gamma) 0
e (Vega) 0 LRA Strategy f(Rho Rate) 0
g (Rho Yield) 0 Minimum Maximum Value 121800.0000
Number of Instruments 23 Delta 100.0000 Value Lower Bound 124.9809 1218.0000 Gamma 0.0000 Value Upper Bound Theta 0.0000 Delta Lower Bound -53.5607 -0.5356 1.0000 Vega 0.0000 Delta Upper Bound 100.0000 Rho Rate 0.0000 Gamma Lower Bound t -53.1860 0.0005 0.0006 Rho Yield 0.0000 Gamma Upper Bound 58.8630 Theta Lower Bound -26.4751 8.7031 Result 1
Theta Upper Bound Vega Lower Bound 888.8284 983.4800 Vega Upper Bound
0.0000
0.0000 Rho Rate Lower Bound 0.0000 -5414.1847 2390.0997 Rho rate Upper Bound 0.0000 Rho Yield Lower Bound 0.0000 -3716.8785 3265.4231
Rho Yield Upper Bound 0.0000
43
Portfolio rebalancing is finally:
44
Minimum Premium Scenario. In the minimum premium scenario, the portfolio components’ parameters are
Instrument Parameters
Time Period Fixe Delta Gamma Theta Vega Am Rate Am Yield Low Bound Up Bound
??
18.10.95
RLKN
2 C1218
3 C1268
4 C1318
5 C1368
6 C1418
7 C1468 8 C1518
9 C1568
10 C1618
11 C1668
12 C1718
13 P1218
14 P1268
15 P1318
16 P1368
17 P1418
18 P1468
19 P1518
20 P1568
21 P1618
22 P1668
23 P1718
1218.0000
204.7840
186.7140
170.1043
154.8749
140.9314
128.1841
116.5454
105.9308
96.2598
87.4559
79.4474
172.8269
197.7817
224.2039
252.0029
281.0876
311.3687
342.7583
375.1721 408.5293
442.7538
477.7736
1.0000 0.0000
0.5449
0.5140 0.0006
0.4838 0.0006
0.4545 0.0006 0.4261 0.0006
0.3988 0.0006
0.3727 0.0006
0.3478 0.0006
0.3241 0.0006
0.3017
0.2806 0.0006
-0.3419 0.0006
-0.3728 0.0006
-0.4030 0.0006
-0.4323 0.0006
-0.4607 0.0006
-0.4880t 0.0006
-0.5141 0.0006
-0.5390 0.0006
-0.5626 0.0006
-0.5851 0.0006 -0.6062
0.0006
0.0006
16.3010
17.0304
-17.5671
-17.9259
18.1232
18.1761
18.1015
-17.9161
-17.6359
-17.2757
-16.8494
-10.7789 924.1420 -10.2175
-9.4633
-8.5313
-7.4377377 982.9147
-6.1998 -4.8343
-3.3589
-1.7870
-0.1360
0.0006 1.5812
924.1420
944.8005
957.7685
963.5962
962.9147
956.4002
944.7456
928.6381 908.7421
885.6873
860.0601
944.8005
957.7885
963.5962
956.4002 944.7456
928.6381 -5164.1503
908.7421 885.6873 -5783.2473
-6087.4798
2296.8321
2199-2079
2098.2657 -2949.7192 1995.5413
1892.3491
1789-7955 1688.7943
1590.0849
1494.2517
1401.7427
1312.8876
-2949.7613
-3262.7629
-3579.0824
3897.1843
-4215.7539
-4533.6849
-4850_0635
-5475_3608
-6087.4798
3321.8749
-3133.7832
-2770.7645
2597.7782
-2431.4184
2272.1599
2120.3195
-1976.0780 -1839.5015
1710.5598
2084.6796
2272.7705
2456.8345
2635.7892
2808.7755
2975.1353
3134.3938
3286.2342
3430.4757
3567.0522
3695.9939
-100.0000
-100.0000 -100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000
-100.0000 -100.0000
-100.0000
-100.0000
-100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000 100.0000
100.0000
100.0000 100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
and a maximum value / maximum theta / minimum vega LRA strategy is
Model Parameters
a (Value) 1
b (Theta)
c (Delta)
1
0
d (Gamma) 0
e (Vega) -1 LRA Strategy
f (Rho Rate) 0
g (Rho Yield) 0 Minimum Maximum Value 2434.1632
Number of Instruments 23 Delta 1.0000 Value Lower Bound 79.4474 1218.0000 Gamma 0.0006
Value Upper Bound Theta 35.2301 Delta Lower Bound -0.6062 -0.6062 1.0000 Vega 860.0601 Delta Upper Bound 1.0000 Rho Rate -6087.4798 Gamma Lower Bound 0.0006 0.0006 0.0006 Rho Yield -3321.8741 Gamma Upper Bound 0.0006 Theta Lower Bound -18.1761 1.5812 Result 1 Theta Upper Bound
Vega Lower Bound 860.0601 963.5962 Vega Upper Bound
Rho Rate Lower Bound -6087.4798 -6087.4798 2296.8321 Rho Rate Upper Bound 2298.8321 Rho Yield Lower Bound -3321.8741 -3321.8741 3695.9939 Rho Yield Upper Bound 3695.9939
45
0.0000 0.0000 0.0000 0.0000 -100.0000 100.0000
Portfolio rebalancing is substantial:
46
Maximum Premium Scenario. In the maximum premium scenario, the portfolio components’ parameters are
Instrument Parameters
Time Period 18.10.95 Price Delta Gamma Theta Vega Rho Rate Rho Yield Roh Low Bond Roh Upper Bound Instrument RKN 1218.0000 1.0000 0.0000 0.00000 0.0000 0.0000 0.0000 -100.0000 100.0000
2 C1218 326.4088 0.6649 0.00006 -32.3464 846.1670 2419.9967 -4053.8292 -100.0000 100.0000 3 C1268 307.1242 0.6421 0.0005 -33.2164 880.1838 2377.5786 -3914.8826 -100.0000 100.0000
4 C1318 288.9330 0.6194 0.0005 -33.9229 909.7256 2330.1434 3776.3918 -100.0000 100.0000
5 C1368 271.7880 0.5969 0.0005 -34.4748 934.8711 2278.5552 3638.9845 -100.0000 100.0000
6 C1418 255.6410 0.5746 0.0005 -34.8813 955.7531 2223.5998 -3503.2056 -100.0000 100.0000
7C1468 240.4437 0.5527 0.0005 -35.1525 972.5460 2165.9856 3369.5217 -100.0000 100.0000
8 C1518 226.1481 0.5312 0.0005 -35.2985 985.4555 2106.3470 -3238.3269 -100.0000 100.0000
9C1568 212.7071 0.5101 0.0005 -35.3296 994.7086 2045.2480 -3109.9489 -100.0000 100.0000 10 C1618 200.0744 0.4856 0.0005 -35.2560 1000.5468 1983.1867 -2984.6548 -100.0000 100.0000
11 C1668 188.2053 0.46695 0.0005 -35.0875 10003.2181 1920.5999 -2862.6575 -100.0000 100.0000 12 C1718 177.0566 0.4501 0.0005 -34.8337 1002.9743 1857.8883 -2744.1216 -100.0000 100.0000
13 P1218 150.4712 -0.2581 0.0005 -29186 846.1670 -2326.7971 1573.6168 -100.0000 100.0000
14 P1268 170.1159 -0.28090 0.0005 -1.8420 880.1838 -2564.0753 1712.5635 -100.0000 100.0000 15 P1318 190.8541 -0.3036 0.0005 -0.6021 909.7256 -2806.3708 1851.0543 -100.0000 100.0000
16 P1368 212.6385 -0.3262 0.0005 0.7925 934.8711 -3052.8191 1988.4615 -100.0000 100.0000 17 P1418 235.4209 -0.3484 0.0005 2.3324 955.7531 -3302.6347 2124.2404 -100.0000 100.0000 18 P1468 259.1529 -0.3704 0.0005 4.0077 972.5460 -3555.1091 2257.9243 -100.0000 100.0000
19 P1518 283.7867 -0.3919 0.0005 5.8081 985.4555 -3809.6078 2389.1191 -100.0000 100.0000t 20 P1569 309.2750 -0.4129 0.0005 7.7235 994.7086 -4065.5670 2517.4972 -100.0000 100.0000
21 P1618 335.5717 -0.4335 0.0005 9.7436 1000.5466 -4322.4885 2642.7912 -100.0000 100.0000 22 P1668 362.6320 -0.4535 0.0005t 11.8586 1003.2181 -4579.9354 2764.7885 -100.0000 100.0000
23 P1718 390.4127 -0.0005 0.0005 14.0589 1002.9743 -4837.5272 2883.3244 -100.0000 100.0000
and a maximum value / maximum theta / minimum vega LRA strategy is
Model Parameters
a (Value) 1 b (Theta) 1 c (Delta) 0 d (Gamma) 0
e (Vega) -1 LRA Strategy f (Rho Rate) 0
g (Rho Yield) 0 Minimum Maximum Value 2184 4463 Number of Instruments 23 Delta 1.0000 Value Lower Bound 150.4712 1218.0000 Gamma 0.0005 Value Upper Bound Theta 40.1494 Delta Lower Bound -0.4729 -0.4729 1.0000 Vega 846.1570
1.0000 Rho Rate -4837.5272 Gamma Lower Bound 0.0005 0.0005 0.0005 Rho Yield -4053.8292 Gamma Upper Bound 0.0005 Theta Lower Bound -35.3296 14.0589 Result Theta Upper Bound
Vega Lower Bound 846.1670 1003.2181 Vega Upper Bound
Rho Rate Lower Bound -4827.5272 -4837.5272 2419.9967 Rho Rate Upper Bound 2419.9967 Rho Yield Lower Bound -4053.8292 -4053.8292 2883.3244 Rho Yield Upper Bound
2883.3244
47
Delta Upper Bound
Portfolio rebalancing is in this case:
Note that quasi-LRA buy-and-hold strategies would be one efficient way to control the rebalancing of LRA portfolios over time. We shall now focus our attention on long- term Limited risk arbitrage strategies (that are of a stochastic nature, described by their expected value and their standard deviation). We still use the above simple Black & Scholes approximation of RUKN's value process but now assume American-style call and put options with strike prices ranging from CHF 1218.00 to 1668.00.
48
2. LRA Option Strategies
Limited risk arbitrage (LRA) option strategies (see also Part I: Securities Markets, Part II: Securities and Derivatives Markets and Part V: A Guide to Efficient Numerical Implementations) involve shares x(t) of common stock [for example,
Swiss Re registered shares S(t) as considered earlier in this paper] as well as corresponding futures contracts and European call and put options [on the stocks themselves or on a stock market index I(t)] and thus generalize the more traditional stock option trading strategies (i.e., covered call strategies, protective put strategies. spreads and combinations).
Specifically, in the context of this paper, we assume the canonical Black & Scholes securities market setting
risk - averse evolution
risk -neutral evolution
and consider trading and portfolio management strategies
position in common stock
positions in futures F1 (t),..,FK (t)
positions in call options c1 (t),..,cL (t)
positions in put options P1 (t),..,PM (t)
with associated value function
that are the solutions of the linear program
where RA denotes the limited risk arbitrage objectives and AC additional linear constraints.
Since the return Rn on the above risk/arbitrage portfolio n = n(t) has the (conditional) variance
49
LRA investment management clearly minimizes both instantaneous investment risk³
etc.] and future portfolio risk dynamics to a degree that is
consistent with the investor’s stated objectives (i.e, the risk tolerances
and ).
If we now introduce the instantaneous value appreciation rate
(lambda)
of a contingent claim, then the limited risk arbitrage optimization program
RA:
maximizes the value appreciation rate of an investor’s option portfolio while keeping its derivatives risk exposure within the specified tolerance band.
Furthermore, we have
which shows that this optimization program at the same time maximizes the (conditionally) expected return of the investor's securities and derivatives portfolio to an extent that is consistent with the stated investment management objectives (RA and AC).
Moreover, we can write the moments of the return on the risk/arbitrage portfolio over a given investment period [0, H] in the form
³ In a securities (i.e., stocks and bonds) portfolio context, instantaneous investment risk is usually defined in terms of the variance /standard deviation of return, whereas in a derivatives portfolio context the contingent claim sensitivities (of the first order: delta, etc.) are used.
50
and thus limited risk arbitrage investment management is a generalization to the derivatives markets of the myopic portfolio optimization techniques that extend Markowitz portfolio selection in the traditional financial markets.
Recall also that in a dynamically complete securities market setting (such as the simple canonical one considered here) contingent claims are redundant and therefore (after appropriate identifications) general limited risk arbitrage investment management only involves solving a linear program in strategy space (see Part I: Securities Markets, Part II: Securities and Derivatives Markets and Part V: A Guide to Efficient Numerical Implementations).
In the case of American options the Black & Scholes partial deferential equation
can be solved with numerical methods (Brennan and Schwartz [4, 5]). The implicit finite difference method approximates the partial differential operators in this equation by the finite differences
that are defined on a two dimensional rectangular grid in time t and state x (see Fig. 1 below) while the corresponding explicit finite difference approximation is
A discretization of the above Black & Scholes partial differential equation with the implicit finite difference operators then leads to the (tridiagonal) system
of linear equations with (state dependent) coefficients
that can easily be solved backwards in time by using the boundary conditions and early exercise criteria which characterize the given contingent claim v.
51
and
A discretization with the explicit instead of the implicit finite difference operators substantially simplifies these calculations. The corresponding linear equation system is in this case
and the (state dependent) coefficients are
[where the local consistency conditions
have to be satisfied, see also Part III A Risk/Arbitrage Pricing Theory and Part IV An Impulse Control Approach to Limited Risk Arbitrage].
Fig 1: Finite Difference Method
With the contingent claim sensitivities
in an implicit and
52
and
and
in an explicit finite difference approximation for the market variable x, risk/arbitrage strategies
position in common stock
positions in futures
positions in European call options
positions in European put options
positions in American call options
positions in American put options
involving shares of common stock as well as futures contracts and European and American call and put options on the stocks themselves or on a stock market index [where
is the associated value function] are the solutions of the state dependent linear programs
(variance of return minimization) or
53
[expected return maximization, where
is the associated option value appreciation rate].
In this way, risk/arbitrage trading and portfolio management systems (see Fig. 3 below) that operate on the basis of the finite difference method support the design of longer-term limited risk arbitrage investment management strategies by determining at any time before or during the relevant investment period the current and all future optimal (state dependent linear optimization) portfolio positions that over the entire investment horizon reduce both the instantaneous investment risk and the future portfolio risk dynamics to values within a specified tolerance band and at the same time achieve a maximum rate of portfolio value appreciation over each single trading period.
As noted earlier, the simple canonical (linear optimization in strategy space) setting that we consider here readily extends to any dynamically complete (diffusion type) securities market and any given set of portfolio management objectives of an investor in the form of von Neumann-Morgenstern utility functions.
The necessary identifications are (see Part I: Securities Markers and Part II: Securities and Derivatives Markets):
Lattice approaches on the other hand work with a discrete representation of the market variable
(risk -neutral state evolution) in the form of a binomial lattice (Cox and Ross [6] and Cox, Ross and Rubinstein [7]).
54
Fig 2: Lattice Approach
The parameters
risk -averse risk-neutral P, u state evolution state evolution
of such a lattice describing the securities market dynamics in a risk-averse and in a risk-neutral financial economy can be calculated by noting that
and with
therefore
holds.
55
The current sensitivities of a contingent claim and their future evolution over the claim’s entire lifetime can then be determined together with its current and all future prices by using a dynamic programming procedure that operates on the underlying recombining lattice with root
and branching process
risk-averse state evolution
risk-neutral state evolution
Any claim v contingent on the market variable x is uniquely characterized by a function F(j), 0 ≤ j ≤ m, representing the payments to its holder at maturity (terminal
condition), a function X(i,j), 1 ≤ i ≤ m and 0 ≤ j ≤ i, representing intertemporal cashflows to which its holder is entitled (payoff function) and boundary conditions L(i,j) ≤ vij ≤ U(i, j), 0 ≤ i ≤ m – 1 and 0 ≤ j ≤ i, for its value process. In an arbitrage
pricing theory framework the claim’s value function consequently satisfies the equation
(risk-neutral pricing formula) which provides the basic algorithm for the above mentioned dynamic programming procedure.
The claim’s sensitivities can now (similar to the explicit finite difference approach) he approximated by
where
or alternatively by
where
56
[note that these conditionally expected rates of change of the option value with respect to time t and the market variable x are defined only in the context of a discrete-time lattice approximation of the state dynamics; we have however
as
Based on this information the current and all future optimal asset positions (that over the entire investment horizon reduce both instantaneous investment risk and future portfolio risk dynamics to values within a given tolerance band and at the same time achieve a maximum rate of portfolio value appreciation over each single trading period) can then be determined by solving (as part of the above mentioned dynamic programming procedure) the state dependent linear programs
(variance of return minimization) or
(expected return maximization) if the sensitivities of the contingent claims are defined as in an explicit finite difference approximation and
or
57
in the case where the sensitivity approximations of the contingent claims are defined as conditionally expected rates of change with respect to time and the market variable.
Note finally that the futures price
of the tradable asset represented by the market variable x satisfies the stochastic differential equation
and can therefore be approximated by a binomial lattice with parameters
The limited risk arbitrage (LRA) techniques briefly outlined above and developed in detail in the publication series
• Risk/Arbitrage Strategies.. A New Concept for Asset/Liability Management, Optimal Fund Design and Optimal Portfolio Selection in a Dynamic, Continuous-Time Framework
have been implemented in the form of a financial / (re)insurance techniques toolbox (FRT), see Fig. 3 below. The toolbox runs under Windows 3.1, 3.11 and 95 as well as under Windows NT 3.51 and NT 4.0.
58
Fig.3: Financial / (Re)insurance Techniques Toolbox (FRT)
FRT can be used in asset /liability management applications as well as for the rapid development of advanced risk transfer solutions for Fortune 500 companies. An extreme value techniques toolbox (EVT) handels the liability side while on the asset side multivariate stochastic models of the (jump) diffusion type are used for the evolution of the main financial markets variables like interest rates, stocks, stock indices and foreign currencies (for details, see Part V: A Guide to Efficient Numerical Implementations).
3. “Mitarbeiter-Option” Trading
In the final section of our paper we shall now outline how the LRA (lattice) techniques described above can be used to implement a “Mitarbeiter-Option” trading system along the lines of Bühlmann, Davis and List [3].
We consider an American-style call and put “Mitarbeiter-Option” schedule with strike prices ranging from CHF 1218.00 to 1668.00. The options are of the forward- start variety, starting in 3.5 years and maturing in 5 years time. As in the introduction,
59
we distinguish between the base value, the minimum premium and the maximum premium scenario (defined as in the European call option case).
The LRA strategy considered is of the maximum value / maximum time value maximum theta type with constrained instantaneous investment risk, i.e.,
and constrained future portfolio risk dynamics, i.e.,
[where the option sensitivities are defined as conditional (monthly) motes of change of the option value with respect to time t and the market variable x ].
The results
• model parameters • option values and sensitivities • positions held • investment portfolio characteristics
conclude the paper.
60
Base Value Scenario.
61
62
63
64
LRA Investment Portfolio (Expectations) Time Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta
0 261'107.8437 12'896.6271 1.0000 0.1000 7'996.3040 1 274'263.5644 2'137.2363 1.0000 0.1000 8'477.9165 2 271'210.2526 8'234.5989 1.0000 0.1000 6'327.3782 3 275'879.5549 4'805.8299 0.8618 0.1000 8456.6532 4 273'746.0089 8'541.5509 0.9433 0.1000 6353.8147 5 276'043.3745 7'367.7707 0.8096 0.1000 8'390.2826 6 275'962.6457 8'368.6040 0.9062 0.1000 6369.1485 7 276'554.6563 8'850.4326 0.7668 0.1000 8'339.9977 8 278'995.1083 6'923.5195 0.8870 0.1000 8'416.3746 9 280'120.1071 6'736.1518 0.7774 0.1000 8'406.7416
10 283'836.0525 3'230.1986 0.8743 0.0999 8533.3100 11 281'218.0214 6'750.6784 0.7749 0.0999 6'380.3000 12 284'506.6144 3'412.3623 0.8670 0.0996 8'486.4288 13 282'001.9699 6'560.5712 0.7763 0.0995 8'336.0997 14 284'847.8332 3'396.4925 0.8632 0.0990 8'425.1039 15 282'454.8485 6'225.3991 0.7800 0.0989 8'281.1847 16 284'845.8893 3'212.8509 0.8618 0.0981 8'348.5767 17 282'772.9682 5'495.3234 0.7851 0.0979 8'217.0168 18 284'551.7527 2'879.9441 0.6889 0.0969 8'230.9346 19 282'641.6929 4'763.5202 0.7910 0.0966 8'133.1958 20 283'899.1052 2'468.6362 0.7026 0.0954 8'126.9128 21 282'042.2071 4'140.2039 0.7974 0.0950 8'028.6782 22 282'971.5963 1'993.0534 0.7157 0.0935 8'009.7353 23 281'178.8634 3'488.2940 0.8040 0.0931 7'911.9333 24 281'813.0062 1'482.1999 0.7282 0.0913 7'881.0443 25 280'102.0210 2'832.0636 0.8107 0.0909 7784.2376 26 280'473.9094 962.4967 0.7402 0.0689 7'742.7472 27 278'864.1401 2'192.7385 0.8174 0.0885 7'647.7543 28 279'005.8899 455.7133 0.7516 0.0862 7'596.7897 29 277'518.6831 1'588.3852 0.8240 0.0859 7'504.4876 30 277'447.6350 -30.3537 0.7625 0.0833 7'444.6132 31 276'016.3805 1'002.4898 0.8305 0.0829 7'352.4393 32 275'867.1548 -465.6082 0.7728 0.0803 7'288.7997 33 274'434.1244 458.3910 0.8369 0.0795 7'194.5274 34 274'309.3615 -839.2558 0.7827 0.0770 7'131.0288 35 272'915.9626 -18.3302 0.7187 0.0760 7'014.9836 36 272'823.8848 -1'137.4704 0.7920 0.0737 6'973.1611 37 271'454.4356 -402.8851 0.7319 0.0722 6'857.2699 38 271'444.3131 -1'345.5442 0.8010 0.0703 6'816.4297 39 270'094.4437 -694.5320 0.7443 0.0683 6'700.4266 40 270'173.0072 -1'416.5567 0.8095 0.0669 6'660.8478 41 268'798.2315 -734.6300 0.8604 0.0641 6'560.8063 42 269'266.1502 -1'222.6948 0.8175 0.0632 4'621.5643 43 268'018.6405 -706.0681 0.7670 0.0599 4'442.8759 44 268'504.6548 -1'277.1122 0.8253 0.0594 4'464.6479 45 267'591.6359 -1'062.3180 0.7775 0.0556 4'233.0308 46 267'626.7441 -1'343.1223 0.8328 0.0541 4'333.0537 47 267'044.4472 -1'041.3581 0.7673 0.0514 4'045.0943 48 266'955.4256 -1'322.2977 0.8396 0.0466 4'106.4621 49 266'673.6896 -962.0612 0.7967 0.0472 3'889.7303 50 266'529.8213 -1'222.0965 0.6463 0.0434 3'887.5183
65
LRA Investment Portfolio (Standard Deviations) Time Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta
0 0.0000 1 1'385.0581 2 10'509.1906 3 3'526.2555 4 9'219.1152 5 3'548.4724 6 10'375.1068 7 5'738.6584 8 12'353.7551 9 9'743.8322
10 8'886.3867 11 10'776.5728 12 9'322.4916 13 11'294.8043 14 9'397.1587 15 11'409.2520 16 9'243.3545 17 11'187.8611 16 8'959.2330 19 10'909.2275 20 8'748.9540 21 10'709.4107 22 8'642.2849 23 10'580.1814 24 8'667.7373 25 10'539.9062 26 8'812.0506 27 10'580.0351 26 9'040.2720 29 10'678.3868 30 9'315.6246 31 10'659.8293 32 9'614.4971 33 10'598.5657 34 9'932.2912 35 10'649.2676 36 10'316.0832 37 10'866.7023 30 10'872.5293 39 11'449.9681 40 12'031.4680 41 11'859.6396 42 11'230.9521 43 10'583.9843 44 10'208.2454 45 9'297.4089 46 8'917.9883 47 8'237.2376 48 7'630.5666 49 7'140.7175 50 6'401.7339
Minimum Premium Scenario.
0.0000 0.0000 0.0000 0.0000 12'470.3890 0.0000 0.0000 197.7285 11'248.4652 0.0000 0.0000 331.6423 14'555.7429 0.5072 0.0000 178.0203 13'016.3898 0.3319 0.0000 302.8477 15'899.6341 0.5870 0.0000 251.2916 12'443.4769 0.4186 0.0000 282.5420 16'946.6760 0.6172 0.0000 298.2902 10'841.0436 0.4618 0.0000 258.1688 12'905.6693 0.6290 0.0000 191.0520 10'507.4166 0.4853 0.0013 165.0662 11'999.4626 0.6321 0.0023 219.5343 9'644.8124 0.4983 0.0034 233.0341
11'105.4860 0.6304 0.0048 284.5659 8'770.9671 0.5048 0.0061 333.5087
10'249.3581 0.6258 0.0078 383.2139 7'923.7646 0.5072 0.0091 456.5513 9'158.8867 0.6194 0.0109 503.6405 7'154.2826 0.7248 0.0124 572.5534 8'233.3884 0.6118 0.0143 639.4848 8'423.0944 0.7116 0.0157 717.0289 7'574.5477 0.6035 0.0176 781.2345 5'764.2106 0.6984 0.0190 862.8081 6'963.1513 0.5947 0.0210 923.6828 5'176.5719 0.6853 0.0223 1'006.2951 6'445.5550 0.5855 0.0243 1'063.4067 4'654.5488 0.6724 0.0256 1'144.8735 5'947.2021 0.5761 0.0275 1'197.9670 4'191.7094 0.6596 0.0286 1'276.6817 5'475.0688 0.5666 0.0305 1'325.7448 3777.9852 0.6470 0.0316 1'400.2909 5'022.1525 0.5570 0.0333 1'445.6294 3'435.8377 0.6346 0.0343 1'515.2439 4'593.4418 0.5474 0.0358 1'567.0611 3'190.8786 0.6224 0.0369 1'621.4480 4'997.5436 0.6953 0.0380 1'640.8327 3'096.2366 0.6105 0.0392 1'719.4716 3'887.4264 0.6814 0.0399 1'737.1743 3'224.2193 0.5987 0.0413 1'810.4041 3'809.6334 0.6679 0.0416 1'828.0779 3'632.8613 0.5872 0.0431 1'897.7645 4'008.0957 0.5096 0.0433 1'922.4203 3'820.6833 0.5759 0.0449 4'346.5522 4'040.6486 0.6416 0.0444 4'317.7122 3'519.9260 0.5648 0.0462 4'302.3029 3'725.2163 0.6289 0.0455 4'379.2340 3'230.4591 0.5539 0.0460 4'225.7469 3'422.6685 0.6165 0.0466 4'336.1075 2'939.3307 0.5432 0.0458 4'236.8055 3'079.5772 0.6043 0.0473 4'260.4418 2'620.9703 0.5327 0.0459 4'194.7047
66
LRA Investment Portfolio (Expectations) Time Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta
0 282'804.4064 1 283'233.3392 2 283'703.7071 3 283'809.7034 4 282'087.8194 5 283'025.0578 6 283'226.0470 7 283'300.5037 8 284'291.9034 9 283'560.3413
10 286'492.9705 11 283'959.2491 12 287'022.9204 13 284'383.9075 14 287'360.2107 15 284'565.5467 16 287'446.0573 17 284'616.6378 18 287'252.2478 19 284'203.0497 20 286'777.7346 21 285'997.2426 22 286'042.4417 23 285'060.0010 24 285'073.2141 25 283'912.0405 26 283'759.0218 27 282'603.2632 28 282'270.7153 29 281'185.8655 30 280'688.1052 31 279'714.4352 32 279'066.8085 33 278'239.3943 34 277'462.8645 35 276'803.7319 36 275'924.7635 37 275'417.1243 38 274'483.5395 39 274'119.8960 40 273'165.4932 41 272'854.2919 42 272'450.0348 43 271'874.0684 44 271'505.5176 45 270.814.9571 48 270'648.0262 47 269'872.5438 48 269'703.6815 49 269'083.2536 50 288'808.9665
-136.2816 1'923.2645 4'182.1480 5'538.0041 8'553.4359 8778.7049 9'165.3529
10'131.4924 9'350.9618
11'042.9894 8'067.7445
11'446.5808 8'182.3642
11'482.2184 8'118.1463
11'256.0376 7'903.6144
10'838.8598 7'565.9898
10'280.8599 7'132.3417 7'189.5546 6'629.0119 6'606.2650 6'074.3483 5'992.5801 5'446.3266 5'370.5979 4'808.3429 4'759.4667 4'186.5726 4'175.7323 3'597.6734 3'633.1582 3'057.2527 3'142.9446 2'578.6242 2'702.7041 2'172.9219 2'334.3571 1'851.9752 2'044.2773 1'732.7788 1'729.6484 1'365.9902 1'340.6058 1'046.2648 1'005.5176
725.5044 731.1762 455.7379
1.0000 0.1000 5'506.1821 1.0000 0.1000 5'480.8482 0.6865 0.1000 5'400.4403 0.8759 0.1000 5'399.6674 0.6510 0.1000 5'270.4463 0.8321 0.1000 5'290.8831 0.6528 0.1000 5'223.9416 0.8153 0.1000 5'214.8627 0.6652 0.1000 5'174.6897 0.8103 0.1000 5'137.9624 0.6810 0.1000 5'156.2242 0.8111 0.0999 5'063.5820 0.6978 0.0999 5'087.2213 0.5609 0.0997 4'939.9397 0.7145 0.0996 5'010.0545 0.5929 0.0992 4'859.9348 0.7306 0.0990 4'922.8988 0.6215 0.0983 4'769.9912 0.7460 0.0980 4'824.8477 0.6475 0.0970 4'669.1508 0.7606 0.0967 4'715.7925 0.6710 0.0953 4'625.7954 0.7743 0.0949 4'596.2347 0.6925 0.0931 4'500.4836 0.7872 0.0928 4'466.8764 0.7122 0.0905 4'365.8131 0.7994 0.0899 4'324.4460 0.7304 0.0876 4'223.1811 0.8107 0.0866 4'173.7914 0.7471 0.0843 4'074.0410 0.8214 0.0828 4'017.1345 0.7626 0.0807 3'919.9064 0.8315 0.0786 3'855.9970 0.7769 0.0768 3'762.1637 0.8409 0.0742 3'691.9096 0.7902 0.0727 3'603.9787 0.8497 0.0694 3'526.1818 0.8026 0.0684 3'439.5517 0.8580 0.0643 3'359.6171 0.8142 0.0637 3'275.9608 0.8658 0.0588 3'192.8788 0.8249 0.0573 3'109.4532 0.8732 0.0545 1'313.8255 0.8350 0.0524 1'241.1501 0.8801 0.0505 1'285.3010 0.8444 0.0473 1'112.2268 0.8866 0.0467 1'150.8329 0.8531 0.0425 1'045.8115 0.8927 0.0414 1'046.1941 0.8614 0.0381 1'076.3193 0.8985 0.0351 984.0491
67
LRA Investment Portfolio (Standard Deviations) Time Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta
0 0.0000 0.0000 0.0000 0.0000 0.0000 1 2'382.1806 5'563.7579 0.0000 0.0000 86.1962 2 3'236.0401 8'850.7233 0.7271 0.0000 87.1471 3 3'737.5785 10'323.4845 0.4825 0.0000 142.0069 4 4'145.0338 14'227.6831 0.7591 0.0000 285.3858 5 3'986.9702 13'458.9694 0.5546 0.0000 290.8588 8 4'867.0304 12'677.2780 0.7575 0.0000 261.5276 7 5'108.4193 13'306.7726 0.5791 0.0000 324.6937 8 6'491.4730 11'487.0699 0.7467 0.0000 268.8721 9 6'851.4184 13'087.6702 0.5861 0.0000 364.6436
10 8'119.4702 9'057.8300 0.7323 0.0000 263.8537 11 8'352.3102 12'530.2459 0.5848 0.0010 400.1922 12 8'936.5842 8'439.4254 0.7163 0.0020 319.2577 13 9'372.0593 11'839.9812 0.8279 0.0029 403.8837 14 9'429.5330 7'777.4986 0.6996 0.0044 392.3357 15 9'999.4117 11'071.8295 0.8053 0.0056 489.0480 18 9'641.3122 7'105.6394 0.6828 0.0073 483.3698 17 10'279.7281 10'307.1639 0.7834 0.0088 553.6255 18 9'657.7568 6'447.1045 0.6659 0.0106 590.1967 19 10'316.6586 9'572.8619 0.7621 0.0122 653.7264 20 9'566.4553 5'816.4095 0.6492 0.0142 706.3603 21 9'010.2227 5'949.3610 0.7414 0.0159 756.0966 22 9'434.4860 5'220.8300 0.6328 0.0180 827.7552 23 8'761.2062 5'329.5563 0.7214 0.0197 879.8106 24 9'301.7475 4'657.2979 0.6167 0.0217 950.3385 25 8'522.1714 4'749.2110 0.7019 0.0235 1'003.0519 26 8'860.8082 4'092.3328 0.6009 0.0253 1'073.7136 27 8'293.8361 4'204.7849 0.6831 0.0272 1'123.4197 28 8'398.5043 3'553.6650 0.5854 0.0288 1'193.0069 29 8'057.6869 3'693.4535 0.6647 0.0307 1'239.2518 30 7'883.7386 3'045.1172 0.5703 0.0321 1'308.8063 31 7'763.2297 3'214.4745 0.6469 0.0341 1'349.6855 32 7'319.5295 2'571.1419 0.5556 0.0352 1'414.3921 33 7'440.6351 2'773.4223 0.6296 0.0373 1'454.3882 34 6'687.4657 2'139.4088 0.5412 0.0380 1'515.7983 35 7'003.9133 2'387.7131 0.6128 0.0403 1’553.4777 34 5'983.9840 1'769.9898 0.5272 0.0407 1'611.5843 37 6'420.2472 2'080.6245 0.5965 0.0432 1'647.4400 38 5'249.1035 1'502.6666 0.5136 0.0433 1'702.5189 39 5'756.4314 1'905.5908 0.5807 0.0461 1'737.6514 40 4'754.6726 1'372.4542 0.5033 0.0462 1'790.2894 41 4'827.1779 1'698.2864 0.5653 0.0472 1'823.9705 42 4'284.9939 1'269.0227 0.4874 0.0471 3'109.9938 43 4'007.4889 1'510.3163 0.5503 0.0466 3'052.1051 44 3'939.8945 1'088.8863 0.4748 0.0480 2'911.5827 45 3'407.6402 1'346.7334 0.5358 0.0462 2'896.6694 48 3'615.7216 928.0570 0.4626 0.0485 2'741.1530 47 2'996.7739 1'193.4007 0.5217 0.0462 2'691.5581 48 2'947.4135 716.2703 0.4506 0.0469 2'605.9843 49 2'676.1787 1'081.9585 0.5080 0.0463 2'430.6620 50 2'347.1628 573.2094 0.4390 0.0442 2'390.1926
Maximum Premium Scenario
68
LRA Investment Portfolio (Expectations) lime Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta
0 246'675.3502 1 268'036.7063 2 262'469.9556 3 270'204.4086 4 267'719.3584 5 272'513.4237 6 276'043.4554 7 274'674.3544 8 277'312.3444 9 276'533.3251
10 279'111.3047 11 278'806.6201 12 280'415.7504 13 281'503.7716 14 281'235.9080 15 281'952.0994 16 281'707.7543 17 282'004.4854 18 281'629.9808 19 281'697.4378 20 283'409.5460 21 281'044.0160 22 282'577.0425 23 280'132.0525 24 281'497.4015 25 279'148.3419 26 280'284.9258 27 277'895.0671 28 278'912.7446 28 276'532.4559 30 277'466.3119 31 275'108.4181 32 275'989.6859 33 273'672.4741 34 274'510.6473 35 272'276.4430 36 273'030.1064 37 270'966.7603 38 271'446.1036 39 269'219.2721 40 269'966.8096 41 268'008.9209 42 268'688.8969 43 267'080.4063 44 267'608.6606 45 266'835.8242 46 266'739.3477 47 266'221.3601 48 266'091.6557 49 265'569.4798 50 265'701.1319
22'904.5494 1'844.9317
11'652.8913 3'968.8185 9'256.8549 4'770.0559 4'129.5309 5'166.7994 4'084.9098 5'308.3268 3'976.1751 4'377.3704 3'742.4075 2'430.4032 3'377.9646 2'197.7362 2'797.6707 1'827.3592 2'285.7273 1'359.6915 -597.8331 857.7763
-1'048.7405 315.8261
-1'513.9253 -418.2924
-2'002.9068 -1'021.1268 -2'458.8380 -1'588.1231 -2881.4111 -2'104.3951 -3'256.4364 -2'557.5911 -3'584.1558 -2'937.8852 -3'861.3708 -3'240.8962 -4'116.5037 -2'865.2121 -4'175.7289 -2'715.5746 -3'805.5504 -2'518.9198 -3'758.9828 -2'978.3249 -3'600.3258 -2758.7463 -3341.9415 -2'523.8385 -2969.2771
1.0000 1.0000 1.0000 1.0000
0.9363 0.9731 0.8955 0.9493 0.8697 0.9310 0.8534 0.9174 0.8430 0.7398 0.8367 0.7412 0.8331 0.7444 0.8315 0.7487 0.8313 0.7538 0.8322 0.7594 0.8338 0.7652 0.6827 0.7712 0.6935 0.7773 0.7040 0.7834 0.7141 0.7894 0.7238 0.7954 0.7331 0.8013 0.7421 0.8070 0.7507 0.8127 0.7591 0.8182 0.7670 0.7080 0.7747 0.7183 0.7821 0.7282 0.7892
0.1000 10'422.4201 1.0000 11'448.8872 0.1000 11'150.2288 0.1000 11'505.5933 0.1000 11'349.9805 0.1000 11'564.7371 0.1000 11'653.3470 0.1000 11'615.5453 0.1000 11'708.4380 0.0999 11'650.6536 0.0998 11'740.0267 0.0995 11'705.1373 0.0994 11'746.4255 0.0989 11'757.0109 0.0987 11'727.8390 0.0980 11'720.3176 0.0976 11'690.7550 0.0968 11'662.6321 0.0963 11'625.1229 0.0953 11'585.6258 0.0948 11'649.2422 0.0936 11'489.8436 0.0931 11'543.1368 0.0917 11'379.5246 0.0911 11'423.0699 0.0896 11'263.7851 0.0890 11'272.6628 0.0873 11'132.8253 0.0867 11'135.5604 0.0848 10'994.4625 0.0842 10'992.6854 0.0821 10'850.9897 0.0817 10'846.1503 0.0793 10'704.7796 0.0790 10'697.2652 0.0763 10'558.3176 0.0761 10'546.0165 0.0732 10'413.8235 0.0727 10'387.3278 0.0700 10'245.4447 0.0690 10'231.3779 0.0666 10'101.0781 0.0651 7'430.6940 0.0633 7'151.0473 0.0610 7'322.1735 0.0597 6'998.0887 0.0568 7'157.7199 0.0562 6'881.5153 0.0522 6'944.1637 0.0512 6'792.6827 0.0479 6'672.3876
69
LRA Investment Portfolio (Standard Deviations) Time Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta
0 0.0000 1 3'185.6427 2 16'508.9246 3 5'747.7921 4 17'659.8372 5 9'084.6105 6 13'378.9051 7 11'631.8553 8 14'521.8159 9 13'262.5825
10 15'169.8833 11 14'172.9729 12 15'323.2573 13 12'557.3795 14 15'131.3460 15 12'436.5431 16 14'607.4143 17 12'196.0373 18 14'263.4010 19 11'969.5821 20 11'414.1326
21 11'675.2373 22 11'492.6407 23 11'910.8877 24 11'735.3833 25 11'942.8773 26 12'104.5683 27 12'203.4717 28 12'576.8131 29 12'559.6873 30 13'100.7074 31 12'992.6716 32 13'654.4895 33 13'503.4627 34 14'246.2990 35 14'129.8480 36 14'878.7546 37 14'981.7972 38 15'629.5847 39 16'895.0995 40 17'683.3984 41 18'093.5992 42 16'660.8066 43 16'446.1601 44 14'940.8757 45 14'354.5383 46 13'164.5275 47 12'619.7318 48 11'370.6078 49 10'685.9684 50 9'616.5936
Appendix: References
0.0000 0.0000 19'301.1090 0.0000 17'733.9193 0.0000 20'042.6628 0.0000 18'034.1290 0.3511 19'378.7031 0.2303 14'666.9502 0.4451 18'372.9783 0.3144 13'995.6848 0.4935 17'256.0059 0.3650 13'180.6481 0.5213 15'459.5351 0.3979 12'342.6880 0.5379 12'969.5785 0.6728 11'556.4393 0.5477 12'021.0358 0.6712 10'779.1733 0.5531 11'162.5232 0.6677 10'228.2947 0.5555 10'465.2432 0.6629 8'694.2431 0.5558 4'896.8094 0.6571 8'227.0158 0.5545 9'431.1195 0.6507 7'687.2362 0.5521 8'812.0225 0.6438 7'242.1782 0.7307 8'395.1661 0.6366 6'862.6757 0.7204 8'026.6554 0.6292 6'555.3575 0.7102 7'666.3944 0.6216 6'322.6679 0.7000 7'400.3579 0.6139 6'178.1747 0.6900 7'146.5223 0.6061 6'172.7903 0.6801 6'969.9941 0.5983 6'407.6613 0.6703 7'584.3333 0.5905 7'060.0952 0.6606 7'826.1336 0.5827 7'344.2633 0.6510 7'746.3943 0.5750 6'700.5461 0.6416 6'686.2261 0.7062 6'036.7353 0.6323 6'061.3195 0.6957 5'363.2072 0.6231 5'431.4840 0.6853 4'651.7244 0.6141
0.0000 0.0000 0.0000 282.2917 0.0000 734.4458 0.0000 281.8281 0.0000 731.8672 0.0000 328.5573 0.0000 428.9128 0.0000 369.9700 0.0000 424.6868 0.0015 389.8213 0.0027 416.5145 0.0038 396.1969 0.0053 422.2974 0.0065 314.7490 0.0082 462.4128 0.0094 408.8745 0.0113 537.0520 0.0125 530.6511 0.0144 652.8039 0.0156 672.3045 0.0175 766.7754 0.0186 822.3090 0.0205 924.2090 0.0215 973.7410 0.0234 1'079.1514 0.0244 1'123.0361 0.0262 1'215.4065 0.0271 1'267.2167 0.0289 1'357.1925 0.0297 1'404.4615 0.0314 1'490.6790 0.0321 1'533.7602 0.0339 1'615.3016 0.0344 1'654.9806 0.0361 1'731.0646 0.0365 1'769.0814 0.0382 1'838.5143 0.0385 1'878.4837 0.0398 1'939.2657 0.0403 1'983.9156 0.0412 2'054.5048 0.0421 2'076.0187 0.0424 8'264.5200 0.0437 6'514.3428 0.0436 6'270.4572 0.0452 6'486.1787 0.0446 6'293.6778 0.0466 6'396.7187 0.0455 6'388.1758 0.0464 6'212.4620 0.0466 6'480.1606
70
[1] N. Bühlmann, M. H. A. Davis and H.-F. List, New Tax-efficient, Option-based Compensation Packages - Part I: Compound Option Structures, AFIR 1997, Vol. I, p. 197-226
[2] N. Bühlmann, M. H. A. Davis and H.-F. List, New Tax-efficient, Option-based Compensation Packages - Part II: Investment Protection, , AFIR 1997, Vol. I, p. 227- 255
[3] N. Bühlmann, M. H. A. Davis and H.-F. List, New Tax-efficient, Option-based Compensation Packages - Part III: A Note on the Implementation, , AFIR 1997, Vol. I, p. 257-282
[4] M. J. Brennan and E. S. Schwartz, The Valuation of the American Put Option, Journal of Finance 32, 449-462 (1977)
[5] M. J. Brennan and E. S. Schwartz, Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis, Journal of Financial and Quantitative Analysis 13, 461-474 (1978)
[6] J. C. Cox and S. A. Ross, The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics 3, 229-263 (1976)
[7] J. C. Cox, S. A. Ross and M. Rubinstein, Option Pricing: A Simplified Approach, Journal of Financial Economics 7, 229-263-(1979)
71