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Crash Hedging Strategies and Optimal Portfolios by Olaf Menkens School of Mathematical Sciences Dublin City University (DCU) St. Andrews, March 17th, 2009 Crash Hedging Strategies and Optimal Portfolios 1 1 Introduction 1.1 Optimal Investment in the Black–Scholes Setting Most basic setting: dP0,0(t) = P0,0(t) r0 dt , P0,0(0) = 1 , “bond” dP0,1(t) = P0,1(t) [µ0 dt + σ0 dW0(t)] , P0,1(0) = p1 , “stock” with constant market coeﬃcients µ0, r0, σ0 = 0 and where W0 is a Brownian motion on a complete probability space (Ω, F , P ). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 2 The optimal portfolio problem in this setting is to ﬁnd a solution of sup π E U X0 (T ) , π(·)∈A0(x) π where U is the utility function of the investor, and X0 denotes the wealth process of the investor given the portfolio strategy π. More speciﬁc, the wealth process satisﬁes π π dX0 (t) = X0 (t) [(r0 + π(t) [µ0 − r0]) dt + π(t)σ0 dW0(t)] , π X0 (0) = x. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 3 Classical solution methods are • the Martingale method (Pliska (1986), Karatzas et al. (1987), and Cox and Huang (1989)) and • the stochastic control method (Merton (1969 and 1971)). A solution of the optimal portfolio problem is called an optimal portfolio strategy and will be denoted by π ∗. The most used utility functions and the corresponding optimal portfolio strategies are i) Logarithmic utility: U (x) = ln(x) with π0 = µ0−r0 , ∗ 2 σ0 ii) HARA–utility: U (x) = 1 xγ with γ < 1, γ = 0 and with γ π0,γ = 1−γ µ0−r0 , ∗ 1 2 σ0 iii) Exponential utility: U (x) = − exp (−λx) with λ > 0 and with ∗ π0,λ(t) = π∗ 1 · µ0−r0 · exp (−r0 (T − t)). 2 σ0 λX 0,λ (t) c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 4 1.2 Traditional Crash Modelling Well–known empirical fact: Black–Scholes–price–model cannot explain large movements of real stock prices (the so–called “jumps” or “crashes”). Examples: Merton (1976) or Aase (1984). dP0,1(t) = P0,1(t) [µ0 dt + σ0 dW0(t) − k dN (t)] , P0,1(0) = p1 . where N is a Poisson process with intensity λ and k > 0 is the jump size. In the logarithmic–utility case U (x) = ln(x) the optimal ∗ portfolio strategy πp calculates to 2 2 2 ∗ = σ 0 + k (µ 0 − r 0 ) − πp kλ − (µ0 − r0) + σ0 + k (µ0 − r0) . 2kσ0 2 2 kσ0 2kσ02 c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 5 ∗ Level Lines of the Optimal Portfolio Strategie πP 1 1 1 1 1 0 0 0.9 1 0 Probability of no crash occurring within the next year The market coefficients are r = 0.05, µ = 0.1, σ = 0.2, π = 1.25, Ψ = 0.08125. −1 0.8 −1 0 −1 1 ∗ 0.7 −1 −2 −2 0 0.6 −2 −1 −2 1 0.5 −3 −3 0 −3 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 k −− Crash height The variables are the crash height k and the probability that no crash occurs within the next year. The market coeﬃcients are assumed to be r0 = 0.05, µ0 = 0.1 and σ0 = 0.2. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 6 ∗ The Dependence of the Optimal Portfolio Strategie πP on the Crash Height and the Crash Intensity The market coefficients are r = 0.05, µ = 0.1, σ = 0.2, π = 1.25, Ψ = 0.08125. 2 π −− Wealth fraction invested in the risky asset 1 ∗ 0 −1 −2 −3 −4 0 0.2 0.4 1 0.9 0.6 0.8 0.7 0.6 0.5 k −− Crash height 0.8 0.4 0.3 0.2 λ −− Crash intensity 1 0.1 0 ∗ This graphic shows the dependence of πP from the crash height k and the crash intensity λ. The market coeﬃcients are assumed to be r0 = 0.05, µ0 = 0.1 and σ0 = 0.2. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 7 Remarks: • Although these processes deliver a better ﬁt, they do not help to determine optimal portfolio strategies under the threat of a crash! • The time to maturity is very important in crash modelling. However, this variable is neglected in traditional crash modelling. • Asymmetry of risk is not taken into consideration: – For π ∈ [0, 1], you can loose at most your investment in the stock. – For π < 0 or π > 1, however, you can loose much more than just your investment in the stock! c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 8 1.3 Alternative Crash Modelling 1. Hua and Wilmott (1997) → Number and size of crashes in a given time interval are bounded. ⇒ No probabilistic assumptions on height, number and times of occurence of crashes. 2. Korn and Wilmott (2002) → Determine worst case bounds for the perfor- mance of optimal investment. For simplicity: One bond, one stock, at most one crash in [0, T ] with a crash height of k with 0 < k∗ ≤ k ≤ k∗ < 1. Security prices in “normal times”: dP0,0 (t) = P0,0 (t) r0 dt , P0,0 (0) = 1 , “bond” dP0,1 (t) = P0,1 (t) [µ0 dt + σ0 dW0 (t)] , P0,1 (0) = p1 , “stock” At crash time: stock price falls by a factor of k ∈ [k∗ , k∗ ]. π Consequence: The wealth process X0 (t) at crash time t satisﬁes: π π π X0 (t−) = (1 − π(t)) X0 (t−) + π(t)X0 (t−) π π =⇒ (1 − π(t)) X0 (t−) + π(t)X0 (t−) (1 − k) π π = (1 − π(t)k) · X0 (t−) = X0 (t). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 9 Thus: Following the portfolio process π(·) if a crash of size k happens at time t leads to a ﬁnal wealth of π X π (T ) = (1 − π(t)k) · X0 (T ) , π if X0 (·) denotes the wealth process in the model without any crash. Hence: • “High” values of π(·) lead to a high ﬁnal wealth if no crash occurs at all, but to a high loss at the crash time. • “Low” values of π(·) lead to a low ﬁnal wealth if no crash occurs at all, but to a small loss (or even no loss at all!) at the crash time. Moral: We have two competing aspects (“Hedging vs. Return”) for two diﬀerent scenarios (“Crash or not”) and are therefore faced with a balanced problem between crash impact minimization and return maximization. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 10 Aim: Find the best uniform worst case bound, e.g. solve ' $ sup inf E [U (X π (T ))] , π(·)∈A0(x) 0≤τ ≤T k∈K & % where the ﬁnal wealth satisﬁes X π (T ) = (1 − π(τ )k) X0 (T ) in the π case of a crash of size k at stopping time τ . Moreover, K = {0} ∪ [k∗, k∗]. 1 Note: To avoid bankruptcy we require π(t) < k∗ for all t ∈ [0, T ]. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 11 Important Remarks: (!) • We do not (!) compare two diﬀerent strategies scenario-wise. Typically, two diﬀerent strategies have two diﬀerent worst case scenarios! • The worst case bounds do not depend on the probability of the worst case! • Assuming µ0 > r0, we do not have to consider portfolio pro- cesses π(t) that can attain negative values since the utility func- tion is increasing in x. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 12 Two extreme strategies (in the logarithmic utility case): 1. “Playing safe”: π(t) ≡ 0 =⇒ worst case scenario: no crash (!), leading to the following worst case bound of W CB0 = E ln X 0 (T ) = ln(x) + r0T. 2. “Optimal investment in the crash–free world”: π(t) ≡ π0 = µ0−r0 =⇒ worst case scenario: a crash of maximum ∗ 2 σ0 size k∗ (at any arbitrary time (!)), leading to the following worst case bound of 2 ∗ 1 µ0 − r 0 W CBπ∗ = E ln X π0 (T ) = ln(x) + r0T + T 0 2 σ0 + ln 1 − π0k∗ . ∗ c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 13 Insights: • it depends on the time to maturity which one of the above strategies is better. • strategy 1 takes too few risk to be good if no crash occurs while strategy 2 is too risky to perform well if a crash occurs =⇒ the optimal strategy should balance this out! • a constant portfolio process cannot be the optimal one. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 14 2 Optimal Investment under the Threat of a Crash This chapter is based on Korn and Wilmott (2002), Korn and M. (2005), and M. (2006). 2.1 The Set up U (x) = ln(x), the price of the bond and the risky asset are assumed to be given by dP1,0(t) = P1,0(t) r1 dt , P1,0(τ ) = P0,0(τ ) , dP1,1(t) = P1,1(t) [µ1 dt + σ1 dW1(t)] , P1,1(τ ) = (1 − k) P0,1(τ ) , respectively, with constant market coeﬃcients r1, µ1 and σ1 = 0 after a possible crash of size k at time τ . For simplicity, the initial market will also be called market 0, while the market after a crash will be called market 1. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 15 Definition 2.1 1. The wealth process X π (t) in the crash model is deﬁned as π X0 (t) for 0 ≤ t < τ X π (t) = π π,τ,X0 (τ ) (1) [1 − π(τ )k] X 1 (t) for t ≥ τ ≥ 0 , given the occurrence of a jump of height k at time τ , is strictly positive. 2. The problem to solve sup inf E [ln (X π (T ))] , (2) π(·)∈A(x) 0≤τ ≤T, k∈K where the ﬁnal wealth X π (T ) in the case of a crash of size k at time τ is given by π π (T ) = 1 − π(τ )k X π,τ,X0 (τ ) T , (3) X [ ] 1 ( ) π π,τ,X0 (τ ) with X1 (t) as above, is called the worst case scenario portfolio problem. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 16 Definition 2.2 1. The value function to the above problem is deﬁned via νc(t, x) = sup inf E ln X π,t,x(T ) . (4) π(·)∈A(t,x) t≤τ ≤T, k∈K 2. The value function in the crash–free setting of market i will be denoted by π,t,x νi(t, x) = sup E ln Xi (T ) π(·)∈Ai(t,x) for i = 0, 1. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 17 Definition 2.3 For i = 0, 1 let us name 1. the optimal portfolio strategy in market i, assuming that no crash will happen, by ∗ := µi − ri . πi 2 σi 2. Moreover, 2 2 1 µi − r i σi Ψi := ri + = ri + πi 2 ∗ 2 σi 2 will be called the utility growth potential or earning potential of market i. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 18 Deﬁne for an arbitrary admissible portfolio strategy π(t) π,t,x νπ (t, x) := E ln X0 (T ) T 1 = ln (x) + E 2 π(s) (µ0 − r0) + r0 − π 2(s)σ0 ds 2 t 2 T σ0 2 = ln (x) − E π(s) − π0 2 − 2 Ψ0 ds ∗ 2 σ0 t T 2 σ0 = ln (x) + E Ψ0 − π(s) − π0 2 ds . ∗ 2 t In particular, the value function of market i given that no crash occurs is νi(t, x) = ln (x) + Ψi (T − t) . c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 19 2.2 A Main Result Definition 2.4 ˆ 1. A portfolio strategy π determined via the equation ν1 (t, x (1 − π (t)k∗)) ˆ ˆ for π (t) ≥ 0 ˆ ν (t, x) = for all t ∈ [0, T ] ˆ ν1 (t, x (1 − π (t)k∗)) ˆ for π (t) < 0 will be called a crash hedging strategy. ˜ 2. A portfolio strategy π is a partial crash hedging strategy, if ˜ there exists an S ∈ (0, T ) such that π is a crash hedging strategy on [0, S] and is a solution to the worst case scenario portfolio problem on [S, T ]. ˆ Hereby, the convention ν (t, x) := νπ (t, x) is used. ˆ c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 20 Rewriting the determining equation for the non–negative crash hedg- ˆ ing strategy π gives ν (t, x) = ν1 t, x 1 − π (t)k∗ ˆ ˆ T 2 σ0 ⇐⇒ ln (x) + Ψ0 − π (s) − π0 2 ds ˆ ∗ 2 t = ln (x) + ln 1 − π (t)k∗ + Ψ1 (T − t) ˆ T 2 σ0 ⇐⇒ ln 1 − π (t)k∗ ˆ = Ψ0 − Ψ 1 − π (s) − π0 2 ds . (5) ˆ ∗ 2 t ˆ Assuming that π is diﬀerentiable, diﬀerentiating with respect to t yields −ˆ′(t)k∗ π 2 σ0 = π (t) − π0 2 + Ψ1 − Ψ0 ˆ ∗ 1 − π (t)k∗ ˆ 2 1 2 σ0 ⇐⇒ ˆ π ′(t) = π (t) − ∗ ˆ π (t) − π0 2 + Ψ1 − Ψ0 . ˆ ∗ k 2 c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 21 Theorem 2.5 ˆ 1. If Ψ1 ≥ r0, then there exists a unique crash hedging strategy π , which is given by the solution of the diﬀerential equation 1 2 σ0 ′ ∗ 2 ˆ π (t) = ˆ π (t) − ∗ (ˆ(t) − π0 ) + Ψ1 − Ψ0 , π (6) k 2 and ˆ π (T ) = 0 . (7) 1 Moreover, this crash hedging strategy is bounded by 0 ≤ π ≤ k∗ , if Ψ1 > Ψ0 . ˆ In the case of Ψ1 ≤ Ψ0 , the crash hedging strategy is bounded by 0 ≤ π ≤ˆ ∗ 2 π0 − σ2 (Ψ0 − Ψ1 ). 0 ˆ 2. If Ψ1 < r0, then there exists a unique crash hedging strategy π , which is given by the solution of the diﬀerential equation 1 2 σ0 ′ ∗ 2 ˆ π (t) = ˆ π (t) − (ˆ(t) − π0 ) + Ψ1 − Ψ0 , π (8) k∗ 2 and ˆ π (T ) = 0 . (9) ∗ 2 Furthermore, this crash hedging strategy is bounded by π0 − 2 σ0 (Ψ0 − Ψ1 ) ≤ ˆ π (t) < 0 for t ∈ [0, T ). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 22 ∗ 3. If Ψ1 < Ψ0 and π0 < 0, there exists a partial crash hedging strategy π (which ˜ ˆ is diﬀerent from π ), if ∗ ln 1 − π0 k∗ S := T − > 0. (10) Ψ0 − Ψ1 ˜ With this, π is on [0, S] given by the unique solution of the diﬀerential equation 1 2 σ0 ′ ∗ 2 ˜ π (t) = ˜ π (t) − (˜(t) − π0 ) + Ψ1 − Ψ0 , π (11) k∗ 2 ∗ and ˜ π (S) = π0 . (12) ˜ ∗ On [S, T ] set π (t) := π0 . This partial crash hedging strategy is bounded by ∗ 2 ∗ π0 − 2 σ0 ˜ (Ψ0 − Ψ1 ) ≤ π ≤ π0 < 0. The optimal portfolio strategy for an investor, who wants to maximize her worst case scenario portfolio problem, is given by ∗ ¯ ˜ π (t) := min {ˆ(t), π (t), π0 } π for all t ∈ [0, T ], (13) ˜ ¯ where π (t) is only taken into account if it exists. π will be named the optimal crash hedging strategy. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 23 Geometric Interpretation of the Crash Hedging Strategy 2.5 The initial market coefficients are given by r = 0.05, µ = 0.1, σ = 0.2, π = 1.25, Ψ = 0.08125. π hat The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2, 1 0 2 iii) vi) ν < ν hat − B ν <ν 1 π π π 0 π −− Wealth fraction invested in the risky asset 0 ∗ i) ν > ν hat − B 1 π π 0 1.5 π∗ = 1.75, Ψ = 0.09125. Moreover, k = 0.05 and k∗ = 0.2. π∗ 0 0 ii) crash is negative 0 1 ν > ν hat − B π π ∗ v) crash is positive iv) crash is positive 0.5 νπ < ν0 νπ < ν hat − Bπ 1 1 0 0 5 10 15 20 25 30 35 40 Time in years c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 24 ∗ Example for Ψ1 = Ψ0 and π0 ≥ 0 1.4 The initial market coefficients are given by r0 = 0.05, µ0 = 0.1, σ0 = 0.2, π0 = 1.25, Ψ0 = 0.08125. The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.15, σ1 = 0.4, 1.2 π −− Wealth fraction invested in the risky asset 1 ∗ 0.8 π1 = 0.625, Ψ1 = 0.08125. Moreover, k∗ = 0.05 and k = 0.2. ∗ 0.6 0.4 0.2 ∗ 0 0 10 20 30 40 50 60 70 80 90 100 Time in years ˆ ¯ ˆ This graphic shows π = π = φ0 (blue dash–dotted line with black background), ϕ = π0 ˆ ∗ ¯ ˆ ∗ (blue dotted line), ϕ (green line), φ1 (red dash–dotted line), and π1 (red dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 25 ∗ Example for Ψ1 > Ψ0 and π0 ≥ 0 6 The initial market coefficients are given by r0 = 0.05, µ0 = 0.1, σ0 = 0.2, π0 = 1.25, Ψ0 = 0.08125. The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2, 1 5 π1 = 1.75, Ψ1 = 0.09125. Moreover, k∗ = 0.05 and k = 0.2. Observe that t0 = 89.5149. 1 π −− Wealth fraction invested in the risky asset ∗ 4 1 3 ∗ 2 1 ∗ 0 0 10 20 30 40 50 60 70 80 90 100 Time in years ˆ ¯ ¯ ˆ This graphic shows π (black dashed line), π (black line), ϕ (green line), φ0 (blue dash– ˆ ∗ ∗ dotted line), φ1 (red dash–dotted line), π0 (blue dotted line), and π1 (red dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 26 ∗ Example for r0 ≤ Ψ1 ≤ Ψ0 and π0 ≥ 0 2 1.8 The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.1, σ1 = 0.2, The initial market coefficients are given by r0 = 0.02, µ0 = 0.1, σ0 = 0.2, π0 = 2, Ψ0 = 0.1. 1.6 π −− Wealth fraction invested in the risky asset ∗ 1.4 1.2 π1 = 1.25, Ψ1 = 0.08125. Moreover, k∗ = 0.05 and k = 0.2. 1 ∗ 0.8 0.6 0.4 0.2 ∗ 0 0 2 4 6 8 10 12 14 16 18 20 Time in years ˆ ¯ ˆ ¯ ˆ This graphic shows π = π (black line), ϕ (cyan dotted line), ϕ (green line), φ0 (blue ˆ ∗ ∗ dash–dotted line), φ1 (red dash–dotted line), π0 = 2 (blue dotted line), and π1 (red dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 27 ∗ Example for r0 ≤ Ψ1 ≤ Ψ0 and π0 ≥ 0, the long term behaviour 2 1.8 The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.1, σ1 = 0.2, The initial market coefficients are given by r0 = 0.02, µ0 = 0.1, σ0 = 0.2, π0 = 2, Ψ0 = 0.1. 1.6 π −− Wealth fraction invested in the risky asset ∗ 1.4 1.2 π1 = 1.25, Ψ1 = 0.08125. Moreover, k∗ = 0.05 and k = 0.2. 1 ∗ 0.8 0.6 0.4 0.2 ∗ 0 0 10 20 30 40 50 60 70 80 90 100 Time in years ˆ ¯ ˆ This graphic shows the long term behaviour of π = π (black line), ϕ (cyan dotted line), ¯ ˆ ˆ0 (blue dash–dotted line), φ1 (red dash–dotted line), π0 = 2 (blue dotted ϕ (green line), φ ∗ ∗ line), and π1 (red dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 28 ∗ Example for Ψ1 < r0 and π0 ≥ 0 1.5 The initial market coefficients are given by r0 = 0.1, µ0 = 0.15, σ0 = 0.2, π∗ = 1.25, Ψ0 = 0.13125. The market coefficients after a possible crash are assumed to be r1 = 0.03, µ1 = 0.05, σ1 = 0.2, 1 π −− Wealth fraction invested in the risky asset 0 0.5 π1 = 0.5, Ψ1 = 0.035. Moreover, k∗ = 0.05 and k = 0.2. ∗ 0 −0.5 ∗ −1 0 2 4 6 8 10 12 14 16 18 20 Time in years This graphic shows π = π (black line), ϕ = ϕ (green line with cyan dotted points), φ0 ˆ ¯ ¯ ˆ ˆ ˆ ∗ (blue dotted line), and π ∗ (red (blue dash–dotted line), φ1 (red dash–dotted line), π0 1 dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 29 ∗ Example for Ψ1 > Ψ0 and π0 < 0 5 The initial market coefficients are given by r0 = 0.05, µ0 = 0, σ0 = 0.2, π0 = −1.25, Ψ0 = 0.08125. The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2, 4 1 1 π −− Wealth fraction invested in the risky asset 3 ∗ 1 2 π∗ = 1.75, Ψ = 0.09125. Moreover, k = 0.05 and k∗ = 0.2. 1 ∗ 0 −1 1 1 −2 0 1 2 3 4 5 6 7 8 9 10 Time in years ˆ ¯ ¯ ∗ This graphic shows π (black dashed line), π = ϕ = π0 (blue dotted line with black ˆ ˆ ∗ background), φ0 (blue dash–dotted line), φ1 (red dash–dotted line), and π1 (red dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 30 ∗ Example for r0 ≤ Ψ1 ≤ Ψ0 and π0 < 0 5 The initial market coefficients are given by r = 0.05, µ = −0.05, σ = 0.2, π = −2.5, Ψ = 0.175. 4 The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2, 0 1 π = 1.75, Ψ = 0.09125. Moreover, k = 0.05 and k = 0.2. Observe that S = 8.5936. 3 1 0 ∗ π −− Wealth fraction invested in the risky asset 2 1 0 1 0 0 ∗ 0 −1 ∗ −2 −3 1 −4 1 ∗ −5 0 1 2 3 4 5 6 7 8 9 10 Time in years ˆ ¯ ˜ ˆ This graphic shows π (black dashed line), π = π (black line), ϕ (cyan dotted line), ϕ ¯ ˆ ˆ0 (blue dash–dotted line), φ1 (red dash–dotted line), π0 (blue dotted line), (green line), φ ∗ ∗ and π1 (red dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 31 ∗ Example for Ψ1 < r0 and π0 < 0, the long term behaviour 5 The initial market coefficients are given by r = 0.1, µ = 0.05, σ = 0.2, π = −1.25, Ψ = 0.13125. The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.05, σ = 0.2, 4 1 0 π = 0.5, Ψ = 0.035. Moreover, k = 0.05 and k = 0.2. Observe that S = 19.3701. 3 1 π −− Wealth fraction invested in the risky asset 0 ∗ 2 1 0 1 0 ∗ 0 0 −1 ∗ −2 −3 1 1 ∗ −4 0 2 4 6 8 10 12 14 16 18 20 Time in years ˆ ¯ ˜ ˆ This graphic shows π (black dashed line), π = π (black line), ϕ (cyan dotted line), ϕ ¯ ˆ ˆ0 (blue dash–dotted line), φ1 (red dash–dotted line), π0 (blue dotted line), (green line), φ ∗ ∗ and π1 (red dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 32 ∗ Example for Ψ1 < r0 and π0 < 0 0.5 The initial market coefficients are given by r0 = 0.1, µ0 = 0.05, σ0 = 0.2, π∗ = −1.25, Ψ0 = 0.13125. The market coefficients after a possible crash are assumed to be r1 = 0.03, µ1 = 0.05, σ1 = 0.2, 0 π1 = 0.5, Ψ1 = 0.035. Moreover, k∗ = 0.05 and k = 0.2. Observe that S = 1.3701. −0.5 π −− Wealth fraction invested in the risky asset 0 −1 −1.5 ∗ −2 −2.5 −3 ∗ −3.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time in years ˆ ¯ ˜ ˆ This graphic shows π (black dashed line), π = π (black line), ϕ (cyan dotted line), ϕ ¯ ˆ ˆ0 (blue dash–dotted line), φ1 (red dash–dotted line), π0 (blue dotted line), (green line), φ ∗ ∗ and π1 (red dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 33 3.2 Optimal Portfolios Given the Probability of a Crash In this section, let us suppose that the investor knows the probability of a crash occurring. Let p, with p ∈ [0, 1], be the probability of a crash happening. In this situation, the optimization problem writes to sup inf Ep ln X π,t,x (T ) t≤τ ≤T, π(·)∈A(t,x) k∈K π,t,x := sup p· inf E ln X π,t,x (T ) + (1 − p) E ln X0 (T ) t≤τ ≤T, π(·)∈A(t,x) k∈K π,t,x = sup p· inf E ν1 τ, X0 (τ ) (1 − π(τ )k) + (1 − p) E [νπ (t, x)] . t≤τ ≤T, π(·)∈A(t,x) k∈K c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 34 Observe that the two extremes, p ∈ {0, 1} are straightforward to solve: • p = 1: sup inf E1 ln X π,t,x (T ) = sup inf E ln X π,t,x (T ) . t≤τ ≤T, t≤τ ≤T, π(·)∈A(t,x) π(·)∈A(t,x) k∈K k∈K Thus, this is the original worst case scenario portfolio problem. The solution is already known. π,t,x • p = 0: sup inf E0 ln X π,t,x (T ) = sup E ln X0 (T ) , t≤τ ≤T, π(·)∈A(t,x) π(·)∈A(t,x) k∈K which is the classical optimal portfolio problem. The solution is well–known and is given in our notation (see Deﬁnition 1) by ∗ π0 . c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 35 Let us now consider the case p ∈ (0, 1). Denoting the crash hedging ˆ strategy in this situation by πp and the corresponding utility function ˆ by νp (t, x) := νπp (t, x), the deﬁning equilibrium equation for the ˆ crash hedging strategy can be written as νp (t, x) = p · ν1 t, x 1 − πp(t)k∗ ˆ ˆ + (1 − p) νπp (t, x) ˆ ⇐⇒ νp (t, x) = p · ν1 t, x 1 − πp(t)k∗ ˆ ˆ ˆ + (1 − p) νp(t, x) ⇐⇒ νp (t, x) = ν1 t, x 1 − πp(t)k∗ , ˆ ˆ ˆ ˆ hence πp ≡ π . This result shows that the crash hedging strategy remains the same even if the probability of a crash is known. Thus, this result justiﬁes the wording worst case scenario of the above developed concept. This is due to the fact that the worst case scenario should be independent of the probability of the worst case and which has been shown above. Let us summarize this result in a proposition. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 36 Proposition 3.1 Given that the probability of a crash is positive, the worst case scenario portfolio problem as it has been deﬁned in Deﬁnition 2.1 is independent of the probability of the worst case. If the probability of a crash is zero, the worst case scenario portfolio problem reduces to the classical crash–free portfolio problem. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 37 3.3 The q –quantile crash hedging strategy Obviously, the concept of the worst case scenario has the disadvan- tage that additional information (namely the given probability of a crash) is not used. However, if the probability of a crash and the probability of the crash size is known, it is possible to construct the (lower) q –quantile crash hedging strategy. Assume that pc(t) ∈ [0, 1] is the probability of a crash at time t ∈ [0, T ] and let p(k, t) ∈ [0, 1] be the density of the distribution function for a crash of size k ∈ [k∗, k∗] at time t. Moreover, suppose that a function q : [0, T ] −→ [0, 1] is given. With this deﬁne c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 38 0 if 1 − pc (t) ≥ q(t) kq inf kq : 1 − pc (t) + pc (t) p(k, t) dk ≥ q(t) if 1 − pc (t) < q(t) kq (t; π) := k∗ and π ≥ 0 k∗ sup k : 1 − p (t) + p (t) p(k, t) dk ≥ q(t) else q c c kq for any given portfolio strategy π. This has the following interpre- tation. The probability that at most a crash of size kq (t) at time t happens is q(t). Equivalently, the probability that a crash higher than kq (t) will happen at time t is less than 1 − q(t). Obviously, this is a Value at Risk approach. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 39 Notice that the worst case of a nonnegative portfolio strategy is either a crash of size k∗ or no crash. On the other hand, the worst case of a negative portfolio strategy is either a crash of size k∗ or no crash. Correspondingly, the q–quantile calculates diﬀerently for negative portfolio strategies (see the third row) than for the nonnegative portfolio strategies (see the second row). Furthermore, denote by {0} if kq (t) = 0 Kq (t) := {0} ∪ [k∗, kq (t)] if kq (t) = 0 and π ≥ 0 . {0} ∪ k (t), k ∗ else [ q ] c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 40 Definition 3.2 1. The problem to solve sup inf E [ln (X π (T ))] , (14) π(·)∈A(x) 0≤τ ≤T, k∈Kq (t) where the ﬁnal wealth X π (T ) in the case of a crash of size k at time s is given by π π (T ) = 1 − π(τ )k X π,τ,X0 (τ ) T , (15) X [ ] 1 ( ) π π,τ,X0 (τ ) with X1 (t) as above, is called the (lower) q –quantile scenario portfolio problem. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 41 2. The value function to the above problem is deﬁned via νq (t, x) = sup inf E ln X π,t,x(T ) . (16) π(·)∈A(t,x) t≤τ ≤T, k∈Kq (t) ˆ 3. A portfolio strategy πq determined via the equation ˆ νπq (t, x) = ν1 (t, x (1 − πq (t)kq (t))) ˆ for all t ∈ [0, T ] with kq (t) > 0 will be called a (lower) q –quantile crash hedging strategy. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 42 Remark 3.3 1. It is straightforward to see that the 1–quantile scenario portfolio problem is equivalent to the worst case scenario portfolio prob- lem in Deﬁnition 2.1. Moreover, the 1–quantile crash hedging strategy is equivalent to the crash hedging strategy in Deﬁnition 3.1 in M. (2006), p. 602. 2. Notice that the q–quantile scenario portfolio problem is only a q–quantile concerning the crash. The randomness of the market movement represented in the model by a geometric Brownian motion has been averaged out, namely by taking the expectation – and not the q–quantile. Deﬁne the support of kq to be supp (kq ) := {t ∈ [0, T ] : kq (t) > 0} . c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 43 Theorem 3.4 Let us suppose that kq is continuously diﬀerentiable on supp (kq ) with respect to t. 1. Then there exists a unique (lower) q-quantile crash hedging ˆ strategy πq , which is on supp (kq ) given by the solution of the diﬀerential equation 1 2 σ0 ˆ′ πq (t) = πq (t) − ˆ πq (t) − π0 2 + Ψ1 − Ψ0 − πq (t)kq (t), ˆ ∗ ˆ ′ kq (t) 2 ˆ πq (T ) = 0 . ∗ For t ∈ [0, T ] \ supp (kq ) set πq (t) := π0. ˆ Moreover, the q–quantile crash hedging strategy is for t ∈ supp (kq ) bounded by 1 1 ˆ 0 ≤ πq (t) < ≤ if Ψ1 ≥ r0. kq (t) k∗ c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 44 ∗ Additionally, if Ψ1 ≤ Ψ0 and π0 ≥ 0, the q–quantile crash hedg- ing strategy has another upper bound with πq < π0− 2 (Ψ0 − Ψ1). ˆ ∗ σ2 0 On the other side, if Ψ1 < r0 the q–quantile crash hedging strategy is bounded by ∗ 2 π0 − ˆ Ψ − Ψ1) < πq (t) < 0 2( 0 for t ∈ [0, T ). σ0 ∗ 2. If Ψ1 < Ψ0 and π0 < 0, there exists a partial q–quantile crash ˜ ˆ hedging strategy πq at time t (which is diﬀerent from πq ), if ∗ ln 1 − π0kq (t) Sq (t) := T − >0 for t ∈ supp (kq ) . (17) Ψ 0 − Ψ1 ˜ With this, πq (t) is given by the unique solution of the diﬀerential equation c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 45 1 2 σ0 ˜′ πq (t) = πq (t) − ˜ πq (t) − π0 2 + Ψ1 − Ψ0 − πq (t)kq (t), ˜ ∗ ˜ ′ kq (t) 2 ∗ πq (Sq (t)) = π0 . ˜ ∗ For Sq (t) ≤ 0 set πq (t) := π0. This partial crash hedging strategy ˜ is bounded by ∗ 2 ∗ π0 − 2 (Ψ0 − Ψ1) < πq ≤ π0 < 0. ˜ σ0 If kq is independent of the time t, the optimal portfolio strategy for an investor, who wants to maximize her q–quantile scenario portfolio problem, is given by ¯ ˜ ∗ πq (t) := min {ˆq (t), πq (t), π0} π for all t ∈ [0, T ], (18) ˜ ¯ where πq will be taken into account, if it exists. πq will also be called the optimal q –quantile crash hedging strategy. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 46 Remark 3.5 1. It is also possible to solve the above problem if kq is not contin- ˆ uously diﬀerentiable. In order to verify this deﬁne πk to be the unique solution of 1 2 σ0 ˆ πk′ (t) = ˆ πk (t) − πk (t) − π0 2 + Ψ1 − Ψ0 ˆ ∗ and (19) k 2 ˆ πk (T ) = 0, (20) ˆ ˆ for k > 0. Set then πq (t) := πkq (t)(t) where the convention ∗ π0(t) := π0 is used in order to include the case kq (t) = 0. Note ˆ that this procedure is also possible for continuously diﬀeren- tiable kq . However, only if kq is continuously diﬀerentiable, it is ˆ possible that πq is also continuously diﬀerentiable. ˆ′ ˆ′ 2. Notice that πk < πk for k1 < k2. Hence, πk1 ≥ πk2 with strict ˆ ˆ 1 2 ˆ ˆ inequality applying on [0, T ). Thus, in particular, πq (t) > π (t) for t ∈ [0, T ) for any q which satisﬁes q(t) < 1 for t ∈ [0, T ). Moreover, πq1 (t) ≤ πq2 (t), if q1 > q2. ˆ ˆ c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 47 3. For this remark, let us suppose that the market conditions do not change, hence Ψ1 = Ψ0. Moreover, keep in mind that any ∗ πk is bounded by π0 from above. Thus, it is clear that ˆ 0 for t = T ψ(t) := ∗ π0 else ˆ is an upper bound for any πk with k > 0. Unfortunately, it is not possible to show that ˆ πk∗ −→ ψ for k∗ ↓ 0 with k∗ = 0, since πk∗ is only known implicitly and ˆ not explicitly. However, this is exactly what can be observed in practice. ∗ Moreover, keep in mind that the case k = 0 yields π0 as the ∗ optimal portfolio with π0 ≡ ψ. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 48 ∗ Example of k −→ 0 for Ψ1 = Ψ0 and π0 ≥ 0 0.7 The initial market coefficients are given by r0 = 0.05, µ0 = 0.15, σ0 = 0.4, π0 = 0.625, Ψ0 = 0.08125. The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.1, σ1 = 0.2, 0.6 π −− Wealth fraction invested in the risky asset ∗ 0.5 0.4 π1 = 1.25, Ψ1 = 0.08125. Moreover, k∗ = 0.05 and k = 0.2. ∗ 0.3 0.2 0.1 ∗ 0 0 1 2 3 4 5 6 7 8 9 10 Time in years This graphic shows π = πk∗ (black dashed line), π k∗ (red dashed line), π k∗ (blue dash– ˆ ˆ ˆ ˆ 2 10 dotted line), π k∗ (cyan dotted line), π ˆ ˆ ∗ (green solid line), and π0 (black dotted line). k∗ 100 1000 c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 49 The Range of (Optimal) q –Quantile Crash Hedging ∗ Strategies for Ψ1 = Ψ0 and π0 ≥ 0 0.8 The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.1, σ1 = 0.25, The initial market coefficients are given by r0 = 0.05, µ0 = 0.1, σ0 = 0.25, π∗ = 0.8, Ψ0 = 0.07. 0.7 0.6 π −− Wealth fraction invested in the risky asset 0 0.5 π∗ = 0.8, Ψ1 = 0.07. Moreover, k∗ = 0.05 and k∗ = 0.2. 0.4 0.3 0.2 0.1 1 0 0 1 2 3 4 5 6 7 8 9 10 Time in years This graphic shows πk∗ (black solid line), πk∗ (black dashed line), the range of possible ˆ ˆ q–quantile crash hedging strategies (light grey and dark grey area), the range of possible ∗ optimal q–quantile crash hedging strategies (dark grey area), and π0 (black dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 50 The Range of (Optimal) q –Quantile Crash Hedging ∗ Strategies for Ψ1 > Ψ0 and π0 ≥ 0 25 The initial market coefficients are given by r = 0.05, µ = 0.1, σ = 0.2, π = 1.25, Ψ = 0.08125. The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2, 1 0 20 1 π −− Wealth fraction invested in the risky asset 0 ∗ 1 0 15 π = 1.75, Ψ = 0.09125. Moreover, k = 0.05 and k = 0.2. 0 ∗ 0 10 ∗ 5 1 1 ∗ 0 0 5 10 15 20 25 30 35 40 Time in years ˆ This graphic shows πk∗ (black solid line), πk∗ (black dashed line), the range of possible ˆ q–quantile crash hedging strategies (light grey and dark grey area), the range of possible ∗ optimal q–quantile crash hedging strategies (dark grey area), and π0 (black dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 51 The Range of (Optimal) q –Quantile Crash Hedging ∗ Strategies for r0 < Ψ1 < Ψ0 and π0 < 0 25 The initial market coefficients are given by r = 0.05, µ = −0.05, σ = 0.2, π∗ = −2.5, Ψ = 0.175. The market coefficients after a possible crash are assumed to be r1 = 0.03, µ1 = 0.1, σ1 = 0.2, 0 20 0 π −− Wealth fraction invested in the risky asset 15 0 π1 = 1.75, Ψ1 = 0.09125. Moreover, k∗ = 0.05 and k = 0.2. 0 10 ∗ Observe that S = 8.5936 and that S = 5.1586. 0 ∗ 5 0 ∗ −5 0 1 2 3 4 5 6 7 8 9 10 Time in years ˆ This graphic shows πk∗ and πk∗ (black solid line), πk∗ and πk∗ (black dashed line), the range ¯ ˆ ¯ of possible q–quantile crash hedging strategies (light grey area), the range of possible ∗ optimal q–quantile crash hedging strategies (dark grey area), and π0 (black dotted line). c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 52 4 Extensions Possible extensions are • More crashes (see Korn and Wilmott (2002), Korn and M. (2005), M. (2004)). =⇒ System of diﬀerential equations. • More stocks (see e. g. Hua and Wilmott (1997)). =⇒ Numerical methods and crash coeﬃcients. • General utility functions (see Korn and M. (2005), M. (2004)). =⇒ Stochastic control approach. c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 53 • Connection to problems in actuarial mathematics (see Korn (2005)). =⇒ Investing in the presence of additional risk processes. • Worst case scenario optimization for reinsurance (see Korn, M. , Steﬀensen, work in progress) • Costs and beneﬁts of crash hedging (see M. (2004)). =⇒ Calculating the costs and the potential beneﬁts of crash hedging. • Diﬀerential games (see Korn and Steﬀensen (2005)) • Market coeﬃcients after a crash depend on the crash size k (see M. (2004)). =⇒ Diﬀerential equations for π and ˆ, the worst case crash size. ˆ k c Olaf Menkens School of Mathematical Sciences, DCU Crash Hedging Strategies and Optimal Portfolios 54 4 References 1. Ralf Korn and Paul Wilmott, Optimal Portfolios under the Threat of a Crash, International Journal of Theoretical and Applied Fi- nance, 5(2):171 – 187, 2002. 2. Ralf Korn, Worst–Case Scenario Investment for Insurers, In- surance: Mathematics and Economics, 36(1):1 – 11, February 2005. 3. Ralf Korn and Olaf Menkens, Worst–Case Scenario Portfolio Optimization: A New Stochastic Control Approach, Journal of Mathematical Methods of Operations Research, 62(1):123 – 140, 2005. 4. Ralf Korn and Mogens Steﬀensen, On Worst–Case Portfolio Optimization, SIAM J. Control Optim., 46(6):2013 - 2030, 2007. 5. Olaf Menkens, Crash Hedging Strategies and Worst–Case Sce- nario Portfolio Optimization, International Journal of Theorec- tical and Applied Finance, June, 9(4):597 – 618, 2006. c Olaf Menkens School of Mathematical Sciences, DCU

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