Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management

Transcription

1 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: Facsimile: Mark H.A. Davis Tokyo-Mitsubishi International plc 6 Broadgate, London EC2M 2AA Telephone: Facsimile: 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 risktransfer 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 singleinstrument 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 25

2 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 Appendix: References Base Value Scenario Minimum Premium Scenario Maximum Premium Scenario 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 (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 26

3 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 riskfree 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 Current Share Value (18 October 1995): CHF Recent Dividend Growth Rate Estimates (18 October 1995): UBS SBC Warburg James Capel Average 13.3% pa. 26.0% p.a 21.6% pa 20.3% p.a Dividend Yield: 27

4 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 yield variation range outlined below. for in the choice of the dividend Numerical Evaluation (Mathematics): With the initial values D0 = and S0 = 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: 28

5 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 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 29

6 The associated option risk parameters are 30

7 (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

8 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 Base Value: Maximum Premium: Minimum 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 32

9 (note that of course now the maximum and the minimum premium scenarios have the opposite meanings). The associated option risk parameters are 33

10 (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

11 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 35

12 Instrument Parameters Term Period 18, 10, Price 95 Delta Gamma Theta Vega Rho Rate Rho Yield???????? Instrument RUKN C C C C C C C C C C C P P P P P P P P P P P (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 Number of Instruments 23 Delta Value Lower Bound Gamma Value Upper Bound Theta Delta Lower Bound Vega Delta Upper Bound Rho Rate Gamma Lower Bound Rho Yield Gamma Upper Bound Theta Lower Bound Result 1 Theta Upper Bound Vega Lower Bound Vega Upper Bound Rho Rate Lower Bound Rho Rate Upper Bound Rho Yield Lower Bound Rho Yield Upper Bound 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

13 (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 Number of Instruments 23 Delta Value Lower Bound Gamma Value upper Bound Theta Delta Lower Bound Vega Delta Upper Bound Rho Rate Gamma Lower Bound Rho Yield Gamma Upper Bound Theta Lower Bound Result 1 Theta Upper Bound Vega Lower Bound Vega Upper Bound Rho Rate Lower Bound Rho Rate Upper Bound Rho Yield Lower Bound Rho Yield Upper Bound

14 On the other hand, a significant increase2 in the investor s risk tolerances for the portfolio delta (instantaneous investment risk) and the portfolio gamma (future risk dynamics) has the following effect on the corresponding LRA asset allocation: 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 Number of Instruments 23 Delta Value Lower Bound Gamma Value Upper Bound Theta Delta Lower Bound Vega Delta Upper Bound Rho Rate Gamma Lower Bound Rho Yield Gamma Upper Bound Theta Lower Bound Result 1 Theta Upper Bound Vega Lower Bound Vega Upper Bound Rho Rate Lower Bound Rho Rate Upper Bound Rho Yield Lower Bound Rho Yield Upper Bound 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

15 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 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 g (Rho Yield) 0 Minimum Maximum Value Number of Instruments 23 Delta Value Lower Bound Gamma Value Upper Bound Theta Delta Lower Bound Vega Delta Upper Bound Rho Rate Gamma Lower Bound Rho Yield Gamma Upper Bound Theta Lower Bound Result Theta Upper Bound Vega Lower Bound Vega Upper Bound Rho Rate Lower Bound Rho Rate Upper Bound Rho Yield Lower Bound Rho Yield Upper Bound

16 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 Note in such a context also the following zero exposure (= zero miss-specification error ) LRA strategies for both vega and rho: C. Zero Vega. Model Parameters a (Value) b (Theta) c (Delta) d (Gamma) e (Vega) f (Rho Rate) g (Rho Yield) Number of Instruments Minimum Maximum LRA Strategy Value Delta Value Lower Bound Gamma Value Upper Bound Theta Delta Lower Bound Vega Delta Upper Bound Rho Rate Gamma Lower Bound Rho Yield Gamma Upper Bound Theta Lower Bound Result Theta Upper Bound Vega Lower Bound Vega Upper Bound Rho Rate Lower Bound Rho Rate Upper Bound Rho Yield Lower Bound Rho Yield Upper Bound

17 Note the remarkable similarity of the positions held: 41

18 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) 0 Minimum Maximum Value Number of Instruments 23 Delta 100_0000 Value Level Bound Gamma Value upper bound Theta Delta Lower Bound _0000 Vega Delta Upper Bound Rho Rate Gamma Lower Bound Rho Yield Gamma Upper Bound Theta Lower Bound Result Theta Upper Bound Vega Lower Bound Vega Upper Bound Rho Rate Lower Bound Rho Rate Upper Bound Rho Yield Bound Rho Yield Upper Bound

19 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 Number of Instruments 23 Delta Value Lower Bound Gamma Value Upper Bound Theta Delta Lower Bound Vega Delta Upper Bound Rho Rate Gamma Lower Bound t Rho Yield Gamma Upper Bound Theta Lower Bound Result 1 Theta Upper Bound Vega Lower Bound Vega Upper Bound Rho Rate Lower Bound Rho rate Upper Bound Rho Yield Lower Bound Rho Yield Upper Bound

20 Portfolio rebalancing is finally: 44

21 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?? RLKN C C C C C C C C C C C P P P P P P P t _ P P P _ P 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 Number of Instruments 23 Delta Value Lower Bound Gamma Value Upper Bound Theta Delta Lower Bound Vega Delta Upper Bound Rho Rate Gamma Lower Bound Rho Yield Gamma Upper Bound Theta Lower Bound Result 1 Theta Upper Bound Vega Lower Bound Vega Upper Bound Rho Rate Lower Bound Rho Rate Upper Bound Rho Yield Lower Bound Rho Yield Upper Bound

22 Portfolio rebalancing is substantial: 46

23 Maximum Premium Scenario. In the maximum premium scenario, the portfolio components parameters are Instrument Parameters Time Period Price Delta Gamma Theta Vega Rho Rate Rho Yield Roh Low Bond Roh Upper Bound Instrument RKN C C C C C C C C C C C P P P P P P P t 20 P P P t P 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 Number of Instruments 23 Delta Value Lower Bound Gamma Value Upper Bound Theta Delta Lower Bound Vega Delta Upper Bound Rho Rate Gamma Lower Bound Rho Yield Gamma Upper Bound Theta Lower Bound Result Theta Upper Bound Vega Lower Bound Vega Upper Bound Rho Rate Lower Bound Rho Rate Upper Bound Rho Yield Lower Bound Rho Yield Upper Bound

24 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 longterm 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 to

25 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 with associated value function 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) 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

26 LRA investment management clearly minimizes both instantaneous investment risk³ etc.] future and portfolio risk dynamics consistent with the investor s stated objectives (i.e, the risk tolerances and ). If we now introduce the instantaneous value appreciation rate to a degree that is (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

27 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 and 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

28 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 and have to be satisfied, see also Part III A Risk/Arbitrage Pricing Theory and Part IV An Impulse Control Approach to Limited Risk Arbitrage]. and Fig 1: Finite Difference Method With the contingent claim sensitivities in an implicit and 52

29 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

30 [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

31 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

32 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

33 [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

34 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

35 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 to The options are of the forwardstart variety, starting in 3.5 years and maturing in 5 years time. As in the introduction, 59

36 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

37 Base Value Scenario. 61

38 62

39 63

40 64

41 LRA Investment Portfolio (Expectations) Time Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta 0 261' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

42 LRA Investment Portfolio (Standard Deviations) Time Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' Minimum Premium Scenario. 66

43 LRA Investment Portfolio (Expectations) Time Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta 0 282' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

44 LRA Investment Portfolio (Standard Deviations) Time Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' Maximum Premium Scenario 68

45 LRA Investment Portfolio (Expectations) lime Period Portfolio Value Portfolio Time Value Portfolio Delta Portfolio Gamma Portfolio Theta 0 246' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '