On optimal portfolios with derivatives in a regime-switching market

Transcription

1 On optimal portfolios with derivatives in a regime-switching market Department of Statistics and Actuarial Science The University of Hong Kong Hong Kong MARC, June 13, 2011 Based on a paper with Jun Fu and Hans U. Gerber

2 Introduction Portfolio selection problem is one of the key topics in finance. This is recent work This topic is one of Marc s research areas

3 Literature review de Finetti (1940), in the context of choosing optimum reinsurance levels, Bruno de Finetti essentially proposed mean-variance analysis with correlated risks. Markowitz s single-period mean-variance model (1952, 1959) tradeoff between return (mean) and risk (variance). Tobin (1958) extends Markowitz s model to include a risk-free asset market portfolio, separation theorem. Samuelson (1969) extended the work of Markowitz to a multi-period setting. Dynamic programming method was employed to find the optimal consumption strategy so as to maximize the overall utility of consumption.

4 Literature review Merton (1969, 1971) extended these work to continuous-time setting. Ito calculus and the methods of continuous-time stochastic optimal control were introduced. Cox and Huang (1989, 1991), and Pliska (1986) introduced the martingale technique to deal with the continuous-time optimal consumption and investment problem. The problem, which is dynamic in nature, can be reduced to a static one by using the martingale representation theorem. Van Weert, Dhaene and Goovaerts, (2010) Optimal portfolio selection for general provisioning and terminal wealth problems, Insurance: Mathematics and Economics, vol. 47, no. 1, pp

5 Literature review In the literature, most models contain risky and risk free assets Kraft (2003) considers the optimal portfolio problem with wealth consisting of stock, option and bond. By introducing the elasticity of the portfolio with respect to the stock price, the paper shows that we can use the elasticity as the single control variable.

6 Literature review Options: Di Masi et al. (1994), Buffington and Elliott (2001), Guo (2001) Optimal Trading Rules, Optimal Portfolio: Zariphopoulou (1992), Zhang (2001), Zhou and Yin (2003), Cheung and Yang (2004) Risk Theory: Asmussen (1989)

7 Portfolio with derivatives

8 Notation W t : standard Brownian motion α t : continuous-time stationary Markov chain which takes values in the regime space M = {1,..., d} and has a transition rate matrix Q = (q ij ) R d d. B(t): S t : risk free bond price stock price

9 Price dynamics db B ds S = r(t, α t )dt, = µ(t, α t )dt + σ(t, α t )dw t. (1)

10 Dynamic for derivatives do(t, α t, S t ) = O t (t, α t, S t )dt + O S (t, α t, S t )ds t + 1 d 2 O SS(t, α t, S t )(ds t ) 2 + q αtko(t, k, S t )dt, k=1 (2)

11 The discounted price of derivative V (t, α t, S t ) t V (t, α t, S t ) = exp( V (t, α t, S t ) is a martingale. 0 r(u, α u )du) O(t, α t, S t ). (3) V t (t, α t, S t ) + V S (t, α t, S t )r(t, α t )S t + 1 d 2 V SS(t, α t, S t )σ 2 (t, α t )St 2 + q αtkv (t, k, S t ) = 0. (4) From this we have O t + O S rs O SSσ 2 S 2 + k=1 d q αtko(t, k, S t ) = ro (5) k=1

12 Here we are not pricing the regime switching risk. That is we do not change the transaction probability of the Markovian chain when we obtain the risk neutral probability.

13 We construct a self-financing portfolio which contains short one option, long O S the underlying stock, the portfolio value is Π = O + O S S, (6) from the self-financing property and (2), we have dπ = do + O S ds and (5) leads to = O t dt 1 2 O SSσ 2 S 2 dt d q αtko(t, k, S t )dt, (7) k=1 dπ = rodt + O S rsdt = r( O + O S S)dt. (8) Therefore, Π = O + O S S is risk free, hence O + O S S is a delta-neutral portfolio.

14 Substituting dπ = r( O + O S S)dt into do = dπ + O S ds results in do = r( O + O S S)dt + O S ds = rodt + O S (µ r)sdt + O S σsdw (9)

15 where do O = [r + ε O(µ r)]dt + ε O σdw, (10) ε O (t, α t, S t ) = O S(t, α t, S t )S t O(t, α t, S t ) is the elasticity of the option price with respect to the stock price

16 For a portfolio of option, stock, and bond of the form the dynamics of X is X = χ O O + χ S S + χ B B, dx = χ O do + χ S ds + χ B db, where χ O, χ S, and χ B denote the number of shares of options, stocks and bonds respectively and db = rbdt

17 dx X = χ OdO X = x O do O + x S + χ SdS X + χ BdB X ds S + x B db B, (11) where x O = χ OO X, x S = χ SS X, x B = χ BB X, denote the percentages of wealth invested in the three assets and we have x O + x S + x B = 1.

18 dx X = [x O (r + ε O (µ r)) + x S µ + x B r]dt + [x O ε O σ + x S σ]dw = [x O ε O (µ r) + x S (µ r) + r]dt + [x O ε O σ + x S σ]dw = [r + ε(µ r)]dt + εσdw (12) where, as in Kraft (2003), ε denotes the elasticity of the whole portfolio with respect to the stock price, i.e. ε = x O ε O + x S, (13) and note that the elasticities of the stock price and bank account value with respect to the stock price are ε S = 1, ε B = 0.

19 Optimization problem J(t, α t,, X t ) = max ε {E[U(X T ) F t ]}, (14) s.c. dx X = [r + ε(µ r)]dt + εσdw, X 0 > 0, X T 0.

20 Hamilton-Jacobi-Bellman (HJB) optimality condition max DJ = 0, (15) where D is the Dynkin operator and DJ is given by DJ = J t + J X [r + ε(µ r)]x J XX ε 2 σ 2 X 2 + d q αt,kj(t, k, X t ), k=1 where the subscripts denote the partial derivatives of J with respect to t and x. (16)

21 Optimal elasticity by setting DJ ε DJ ε = J X (µ r)x + J XX εσ 2 X 2, (17) = 0, we can obtain the optimal elasticity as ε = J X (µ r) J XX X σ 2, (18)

22 HJB equation J t + J X X [r J X (µ r) 2 J XX X σ 2 ] J XX X 2 [ J X (µ r) J XX X σ ]2 + d q αt,kj(t, k, X t ) = 0, k=1

23 HJB equation 1 2 J2 X (µ r)2 = J XX σ 2 [J t + J X Xr + with the terminal condition d q αt,kj(t, k, X t )], (19) k=1 J(T, α T, X T ) = U(X T ). (20)

24 CRRA utility Assuming that the agent has constant relative risk aversion (CRRA) with utility function given by where γ < 1 and γ 0 U(x) = x γ γ,

25 Solution J(t, i, x) = a(t, i) x γ γ, (21) where a(, i) is a continuous function with a(t, i) = 1 for each i M. Then, immediately it follows that J t (t, i, x) = a (t, i) x γ γ, J X (t, i, x) = a(t, i)x γ 1, J XX (t, i, x) = a(t, i)(γ 1)x γ 2.

26 where λ(t, i)a(t, i) = a (t, i) + λ(t, i) = 1 2 d q i,k a(t, k), (22) k=1 i) r(t, i) [µ(t, ] 2 γ r(t, i)γ. σ(t, i) γ 1

27 Matrix form (Λ(t) Q)a(t) = a (t), (23) where λ(t, 1) 0 0. Λ(t) = , 0 0 λ(t, d) a(t) = (a(t, 1), a(t, 2), a(t, d)), a (t) = (a (t, 1), a (t, 2), a (t, d)),

28 Solution J(t, i, x) = a(t, i) x γ, i M, γ where a(t, i) is uniquely determined by (23). Furthermore, due to (18), we have ε (t, i) = µ(t, i) r(t, i) (1 γ)σ 2 (t, i). (24)

29 Solution For a portfolio of only stock and option with x S + x O = 1, by tracking this optimal elasticity, the optimal portfolio policy (xs and xo ) can be obtained through (13). But if an agent also invests in the bank account beyond the stock and option, the processes (xs, xo and x B ) cannot be determined uniquely.

30 According to (24) where γ < 1 and γ 0, we find that the smaller is γ, the smaller is the elasticity of the portfolio. This is consistent with the definition of Arrow-Pratt index of risk aversion xu (x) U (x) = 1 γ, (25) which indicates higher level of risk aversion for smaller value of γ.

31 CARA utility U(x) = 1 γ e γx, where γ > 0. Similarly, we conjecture that the solution to (19) and (20) is given by J(t, i, x) = 1 exp[ γg(t, i)x]a(t, i), (26) γ where a(, i) and g(, i) are continuous functions and a(t, i) = g(t, i) = 1 for each i M.

32 CARA utility g (t, i) + g(t, i)r(t, i) = 0, (27) a (t, i) d a(t, i) + J(t, k, x) q i,k J(t, i, x) k=1 = 1 2 and the terminal conditions imply that i) r(t, i) (µ(t, ) 2, (28) σ(t, i) T g(t, i) = exp( r(s, i)ds). (29) t

33 CARA utility If the interest rate is independent of the state of the Markovian chain, a (t, i) d a(t, i) + k=1 q i,k a(t, k) a(t, i) = 1 2 i) r(t) (µ(t, ) 2, σ(t, i) and similar to what we have done for CRRA utility, it follows that (Λ(t) Q)a(t) = a (t) (30)

34 CARA utility λ(t, i) = 1 i) r(t) [µ(t, ] 2, 2 σ(t, i) λ(t, 1) 0 0. Λ(t) = λ(t, d) a(t) = (a(t, 1), a(t, 2), a(t, d), a(t) = (a(t, 1), a(t, 2), a(t, d)),, and a(t ) = 1.

35 Solution J(t, i, x) = 1 exp[ γg(t)x]a(t, i), i M. γ By substituting this result into (18), we have ε (t, i) = µ(t, i) r(t) γg(t)xσ 2 (t, i). (31)

36 In contrast to CRRA utility, for smaller value of γ, the Arrow-Pratt index of risk aversion for CARA utility given by xu (x) U (x) = γx (32) indicates lower level of risk aversion, and (31) implies higher elasticity of portfolio which is more sensitive to the changes in the stock price.

37 Analyzing the optimal portfolio: Without risk free asset For the portfolio containing only stock and option, we have x S + x O = 1, x S + x O ε O = ε, and these two equations imply that x S = J X (µ r) J XX X σ ε O ε O 1 ε O. (33)

38 Analyzing the optimal portfolio: Without risk free asset J X (µ r) J XX X σ 2 : J X (µ r) J XX X σ ε O : ε O 1 ε O : optimal strategy in Merton s model the modified term of speculation the pure delta neutral hedging term

39 Analyzing the optimal portfolio: Without risk free asset X = χ O O + χ S S, to make it delta-neutral, we require that and equivalently, χ O O S + χ S = 0, x S = x O O S S O = x Oε O, (34) which, combined with x S + x O = 1, results in ε O x S =. 1 ε O

40 Effect of γ We use the CRRA utility function, which, due to (21), admits J X (µ r) J XX X σ 2 = x S = µ r (1 γ)σ 2, (35) µ r 1 (1 γ)σ 2 ε O, 1 ε O 1 ε O (36) where γ < 1, γ 0.

41 Observations Same as the optimal solution of the Merton s problem, for the reduced portfolio optimization problem, the smaller is γ, the more risk-averse is the agent as indicated by the Arrow-Pratt index of risk aversion, so the less is its wealth invested in stock and the more is invested in the bank account.

42 Observations As γ decreases, the modified term of speculation in xs approaches zero and xs converges to the term of pure delta neutral hedging.

43 Discrete-Time Model with Regime Switching Based on a paper with K.C. Cheung: K. C. Cheung and H. Yang, Asset Allocation with Regime-Switching: Discrete-Time Case, ASTIN Bulletin, Vol. 34, No. 1, , 2004.

44 Discrete-Time Model with Regime Switching Discrete-time setting: investor can decide the level of consumption, c n at time n = 0, 1, 2,..., T 1 After consumption, all the remaining money will be invested in a risky asset The random return of the risky asset in different time periods will depend on the credit ranking which is modeled by a time-homogeneous Markov chain {ξ n } 0 n T with state space M = {1, 2,..., M} and transition probability matrix P = (p ij )

45 Absorption State Default Risk Assume that state M of the Markov Chain is an absorbing state: p Mj = 0 j = 1, 2,..., M 1, p MM = 1. Default occurs at time n if ξ n = M. In this case, the investor can only receive a fraction, δ, of the amount that he/she should have received. The recovery rate δ, is a random variable, valued in [0, 1]

46 Wealth Process {W n } 0 n T : wealth process of the investor W n+1 = { (Wn c n )Rn ξn (1 {ξn+1 M} + δ1 {ξn+1 =M}) if ξ n M, W n c n if ξ n = M, n = 0, 1,..., T 1, where 1 { } is the indicator function. R i n is the return of the risky asset in the time period [n, n + 1], given that the Markov chain is at regime i at time n.

47 Assumptions The random returns R0 i, Ri 1,..., Ri T 1 are i.i.d. with distribution F i ; they are assumed to be strictly positive and integrable R i n is independent of R j m, for all m n The Markov chain {ξ} is stochastically independent of the random returns in the following sense: P(ξ n+1 = i n+1, R in n B ξ 0 = i 0,..., ξ n = i n ) = p ini n+1 P(R in n B) for all i 0,..., i n, i n+1 S, B B(R) and n = 0, 1,..., T 1

48 Assumptions 0 c n W n (Budget constraint) The recovery rate δ is stochastically independent of all other random variables

49 Given that the initial wealth is W 0 and the initial regime is i 0 M := M \ {M}, the objective of the investor is to max {c 0,...,c T } E 0 [ T n=0 ] 1 γ (c n) γ over all admissible consumption strategies. Here 0 < γ < 1. Admissible consumption strategy: a feedback law c n = c n (ξ n, W n ) satisfying the budget constraint Optimal Consumption Strategy: Ĉ = {ĉ 0,..., cˆ T }

50 For n = 0, 1,..., T, the value function V n (ξ n, W n ) is defined as [ T ] V n (ξ n, W n ) = max E 1 n {c n,c n+1,...,c T } γ (c k) γ. Bellman s Equation: V n (ξ n, W n ) = max 0 cn Wn E n [U(c n ) + V n+1 (ξ n+1, W n+1 )] n = 0, 1,..., T V T (ξ T, W T ) = 1 γ W γ T k=n

51 Suppose λ > 0, w > 0, and 0 < γ < 1 are fixed constants. The function f : [ 0, w] R defined by will achieve its unique maximum at f (c) = c γ + λ(w c) γ (37) ĉ = w 1 + λ 1 1 γ (38) and the maximum value is given by f (ĉ) = w γ (1 + λ 1 1 γ ) 1 γ. (39)

52 Define some symbols recursively: H (i) = {E[(R i ) γ ]} 1 1 γ, i M, L (i) 0 = 0, i M, L (i) n = H (i) K (i) n 1 {i M} + n1 {i=m}, i M, n = 1, 2,..., T, K (i) 1 = [1 p im + p im E(δ γ )] 1 1 γ, i M, K n (i) M 1 = p ij (1 + L (j) n 1 )1 γ + p im E(δ γ )(1 + L (M) j=1 i M, n = 2,..., T. n 1 )1 γ Note that K (M) s are not defined. M (i) is well-defined since we have assumed that R i is integrable. 1 1 γ,

53 For n = 0, 1,..., T, the value function is given by V T n (i, w) = 1 γ w γ (1 + L (i) n ) 1 γ, (40) and the optimal consumption strategy is given by ĉ T n (i, w) = w(1 + L (i) n ) 1. (41)

54 From this result, we see that if we are now at time T n, and in regime i, then we should consume a fraction of our wealth which is equal to L (n). i Thus our optimal consumption strategy depends heavily on the current regime and the remaining investment time through the function L.

55 For each i M, L ( ) i is increasing in n: 0 = L (i) 0 L (i) 1... L (i) T. (42) For each i M, K ( ) i is increasing in n: 0 K (i) 1 K (i) 2... K (i) T. (43)

56 The monotonicity of L implies at the same regime, we should consume a larger fraction of our wealth when we are closer to the maturity. This strategy is quite reasonable. If we are closer to the maturity, a short-term fluctuation in the return of the risky asset will bring a loss to us that we may not have enough time to cover. Therefore, we should consume more and invest less.

57 For any fixed i M and w > 0, ĉ 1 (i, w) ĉ 2 (i, w)... ĉ T (i, w).

58 Next, we may guess that at any time period, say T n, if we are at a better regime, then we should consume less and invest more. We need two ingredients: 1. A criterion to compare the distributions of the returns in different regimes = second order stochastic dominance 2. Market has to regular enough = stochastically monotone transition matrix

59 Insurance: Mathematics and Economics 31 (2002) 3 33 Review The concept of comonotonicity in actuarial science and finance: theory J. Dhaene, M. Denuit, M.J. Goovaerts, R. Kaas, D. Vyncke DTEW, K.U. Leuven, Naamsestraat 69, 3000 Leuven, Belgium Received December 2001; accepted 6 June 2002 Abstract In an insurance context, one is often interested in the distribution function of a sum of random variables. Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not realistic. We will determine approximations for sums of random variables, when the distributions of the terms are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. In this paper, the theoretical aspects are considered. Applications of this theory are considered in a subsequent paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented Elsevier Science B.V. All rights reserved. Keywords: Comonotonicity; Actuarial science and finance; Sums of random variables 1. Introduction In traditional risk theory, the individual risks of a portfolio are usually assumed to be mutually independent. Standard techniques for determining the distribution function of aggregate claims, such as Panjer s recursion, De Pril s recursion, convolution or moment-based approximations, are based on the independence assumption. Insurance is based on the fact that by increasing the number of insured risks, which are assumed to be mutually independent and identically distributed, the average risk gets more and more predictable because of the Law of Large Numbers. This is because a loss on one policy might be compensated by more favorable results on others. The other well-known fundamental law of statistics, the Central Limit Theorem, states that under the assumption of mutual independence, the aggregate claims of the portfolio will be approximately normally distributed, provided the number of insured risks is large enough. Assuming independence is very convenient since the mathematics for dependent risks are less tractable, and also because, in general, the statistics gathered by the insurer only give Corresponding author. address: david.vyncke@econ.kuleuven.ac.be (D. Vyncke) /02/$ see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

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63 Compound binomial model Suppose X and Y are two random variables. If E[g(X )] E[g(Y )] for any increasing and concave function g such that the expectations exist, then we say X is dominated by Y in the sense of second order stochastic dominance and it is denoted by X SSD Y. Suppose P = (p ij ) is an m m stochastic matrix. It is called stochastically monotone if m p il l=k m l=k p jl for all 1 i < j m and k = 1, 2,..., m.

64 Suppose P is a M M matrix. Let e k = (1,..., 1, 0,..., 0) (i.e. first k coordinates are 1, the rest are 0) for k = 1, 2,..., M. Let D M = {(x 1,..., x M ) R M x 1 x M } and P D = {y D M Py D M }. Suppose that P is an M M stochastic matrix. The following statements are equivalent: 1. P is stochastically monotone; 2. P D = D M ; 3. e k P D for all k = 1, 2,..., M.

65 Suppose that the transition probability matrix P is stochastically monotone. Assume that and Then we have for n = 1, 2,..., T, R 1 SSD R 2 SSD R M 1, (44) H (i) K (i) 1 1 i M. (45) L (1) n L (2) n L (M 1) n L (M) n, (46) and K n (1) K n (2) K n (M 1). (47)

66 Meaning of R 1 SSD SSD R M 1 Preference of investor: increasing and concave utility function + Return of the risky asset in regime i: R i + Definition of SSD order The M 1 regimes are ranked according to their favorability to the risk-averse investor: regime 1 is the most favorable, regime M 1 is the most unfavorable

67 Meaning of P being stochastically monotone: For 1 i < j M 1 (regime i is more favorable to regime j) M l=k p il is the probability of switching to the worst M k + 1 regimes from regime i M l=k p jl is the probability of switching to the worst M k + 1 regimes from regime j Intuitively, if the market is regular enough, we should have M p il l=k for all possible k. This precisely means that P is stochastically monotone. M l=k p jl