Risk Minimizing Portfolio Optimization and Hedging with Conditional Value-at-Risk

Risk Minimizing Portfolio Optimization and Hedging with Conditional Value-at-Risk Jing Li, University of North Carolina at Charlotte, Department of Mathematics and Statistics, Charlotte, NC 28223, USA. Email: jli16@uncc.edu. Mingxin Xu 1, University of North Carolina at Charlotte, Department of Mathematics and Statistics, Charlotte, NC 28223, USA. Email: mxu2@uncc.edu. January 23, 2008 Abstract: We look at the problem of how to find a dynamic optimal portfolio so that the Conditional Value-at-Risk (CVaR) is minimized under the condition where the returns are bounded. CVaR is a risk measure based on the popular VaR that is coherent. In a complete market setting, we derive the exact optimal conditions. Then we provide applications in two classic complete market models: the Binomial model and the Black-Scholes model. In these cases, the procedures to find the optimal strategies are given with exact formulas. Numerical results show, as expected, dynamic portfolio provide much lower CVaR risk than static portfolios. 1. Introduction Portfolio optimization has been a long-standing classic topic in financial theory since Markowitz s ground-breaking work [12] in 1952. His choice of variance as the risk measure of a portfolio is no longer compatible with today s risk management practice. One risk measure, first proposed by J.P.Morgan in RiskMetrics, that the regulators (BASEL II) make extensive 1 The work of Mingxin Xu is supported by National Science Foundation under the grant SES-0518869 and John H. Biggs Faculty Fellowship. 1

2 use of is Value-at-Risk (VaR). VaR is defined as the loss level that will not be exceeded with a certain confidence level during a certain period of time. As a quantile of the portfolio, VaR is a threshold and does not give information about the loss size when the loss exceeds VaR. In addition, VaR is in general neither sub-additive or super-additive which either encourage or discourage actions of diversification or merger, and portfolio optimization with respect to VaR leads to non-smooth results. Recent research in the area of mathematical finance, following the economic theory of utility functions, developed into an axiomatic approach for risk measure: Coherent Risk Measure was first proposed by Artzner, et al. [3] and [4]. The axioms of coherence parallel the axioms of rational investors. Conditional Value-at- Risk (CVaR), sometimes called Shortfall Risk, defined as the expected loss during a certain period of time, conditional that the loss is greater than a loss threshold corresponding to a certain confidence level, studied by Rockafellar and Uryasev [13] and Acerbi and Tasche [1], turned out to be a distribution-based coherent risk measure. More importantly, CVaR is a vast improvement over VaR in producing smooth portfolio optimization results. The wide use of VaR and the advantage of CVaR has lead many financial institutions to consider using VaR for reporting purpose and CVaR for internal risk control. Numerical implementation of an optimization problem with quantile-based objective function does not have to be easy. The contribution of Rockafellar and Uryasev [13] is that they found an equivalent formula for CVaR as a convex function which opened the door for convex programming methods. Using Monte-Carlo simulation for a one-time step model with multiple assets, they formulated the portfolio optimization problem into a linear programming problem which can be efficiently implemented with standard programming software. The contribution of our paper is to work out the portfolio optimization problem with dynamic trading strategies (including multi-period models) in closed-form solutions for complete market models. In the classic setup of portfolio optimization, expected returns are maximized given limits on the risk of the portfolio. It can be made formal, but with some technical conditions, it is equivalent to minimize the risk of the portfolio given requirements for its

return. Our setup will focus on the later approach which takes focus on risk minimization as the objective. An attempt to employ the dynamic programming method for multi-period models was made by Ruszczyński and Shapiro [17]. They took the approach to modify the risk measure CVaR into a dynamic version conditional risk mappings for CVaR. In this paper, we will stick to the original CVaR and only let the portfolio composition adjusts over time because optimizing CVaR at each time period conditionally is very different than optimizing CVaR of the final wealth of a dynamic portfolio. In practice, portfolio rebalance occurs much more often than financial reports on return and risk. Our solution will anchor on duality methods based on risk neutral measures similar to those employed in option pricing and utility maximization problems. This martingale approach is well-studied in recent mathematical finance research partly because it allows for solutions to a wider ranges of problems which does not possess Markovian property and thus does not meet dynamic programming principles. Section 2 establishs the general risk minimization problem and provides the solution. Section 3 details the solutions to two popular complete market models: the Binomial model and the Black-Scholes model. Section 4 provide numerical examples and Section 5 concludes. 3 2. Dynamic Portfolio Optimization and Hedging with CVaR All results in this section work for multi-dimensional market models. In an attempt to keep the notations simple, we will present them in a one-dimensional case where there is a single risky asset. Let us consider a self-financing portfolio consisting of investments in a money market account and a stock account. Assume the money market account earns a risk-free rate r and the stock price S t is an adapted process on a filtered probability space (Ω, F, (F t ) t 0, P ) that satisfies the usual conditions. It is standard for continuous time models, as in the Black-Scholes model, to write the dynamics of this self-financing portfolio as dx t = ξ t ds t + r(x t ξ t S t )dt,

4 where ξ t is the number of shares of stocks held at time t. The riskiness of the return of this portfolio will be measured by CVaR. A good reference for CVaR is in the book written by Föllmer and Schied [9] where CVaR is given a third name Average VaR. We define VaR at level λ to be V ar λ (Z) = inf m P (Z + m < 0) λ }, λ (0, 1). From a risk management perspective, it can be viewed as the smallest capital reserve to add to a financial position Z for it to be admissible: the probability for the combined position to default is less than λ. Mathematically, it is the negative value of a λ-quantile of Z. CVaR is defined as a simple average of VaR (1) CV ar λ (Z) = 1 λ λ 0 V ar γ (Z)dγ. With this definition, it is easy to see why CVaR is different than VaR that it is smooth with respect to the change of the confidence level λ. To relate to the definition of expectation on the tail as stated in the Introduction, we have to be a little careful when we write down the equivalent form of (1) for the case when the probability space could be discrete CV ar λ (Z) = 1 λ ( ) E[Z1 Z<V arλ (Z)}] + V ar λ (Z)(1 P (Z < V ar λ (Z)). The question is how we should trade the shares throughout a finite holding period [0, T ] so that we can achieve minimal risk at time T, conditioned that the return is within acceptable range? Mathematically, we want to find the optimal investment strategy with initial capital X 0 = x: (2) ξ t = arg inf ξ t CV ar λ (X T ), subject to x d X t x u, for all t [0, T ].

5 This is equivalent to the problem of minimizing CVaR on the return R T ξ t = arg inf ξ t CV ar λ (R T ), subject to r d R t r u, for all t [0, T ], whether it be percentage return R T = X T X 0 X 0 or log return R T = ln X T X 0 to identify the one-to-one correspondence between the quantiles of X T because we only need and R T. A more noted difference here to a classic setup is that we require the returns to be bounded by two constants r d and r u and we are not putting constraints on the expected returns. Even when the expected return is high enough, a particular realized return still have a chance to turn out to be very low. We are requiring the realized return to be above x d in all cases if we take x u =. Practically, this corresponds to a fund with a guaranteed return. The same question can be asked for a discrete time trading model where adjustments of the portfolio take place at 0 = t 0 < t 1 <... < t N = T, and the self-financing portfolio is X tn+1 = ξ tn S tn+1 + (X tn ξ tn S tn )(1 + r(t n+1 t n )). Since the discrete time trading model can be viewed as a special case of the continuous time trading model where the stock price and the investment strategy are held piece-wise constant over time intervals, we will derive all the theorems in the continuous time model, which can be readily applied to discrete time models. When we have an existing portfolio H t consisting of investments in derivatives, we can ask a second question as how to hedge our risk with a self-financing admissible portfolio. Let H be the random variable representing the final value of the existing portfolio. It is F T -measurable since H = H T. The optimal hedging problem is to solve (3) ξ t = arg inf ξ t CV ar λ (H + X T ), subject to x d X t x u, for all t [0, T ].

6 Thus when we combine the original portfolio and the hedging portfolio, the risk is minimized. It is straight-forward to see that if X T is the final wealth of the optimal portfolio for problem (2), by which we mean that min ξ t CV ar λ (X T ) = CV ar λ (X T ), then X T H is the optimal solution for problem (3) if it is the result of a self-financing strategy. In a complete market model, this is not an issue because all derivatives can be replicated. Therefore, we will focus on solving problem (2) in a complete market model in this paper. As mentioned in Section 1, problem (2) is complicated by the fact that the objective function involves quantile function and the corresponding numerical methods will have to involve ordering the returns. Rockafellar and Uryasev [13] and [14] found it to be the Fenchel- Legendre dual of expected shortfall where standard convex analysis applies. For λ (0, 1), definition (1) is equivalent to CV ar λ (Z) = 1 λ inf x R (E[(x Z)+ ] λx), when Z is integrable. Now we reformulate problem (2) into a more tractable convex optimization problem (4) ξt 1 = arg inf ξ t λ inf (E[(x X T ) + ] λx), x R subject to x d X t x u, for all t [0, T ]. If we let the stock price S t be as general as a semimartingale, then the market is complete when the risk neutral measure is unique. These fundamental theorems are developed in Delbaen and Schachermayer [5], El Karoui and Quenez [6], Kramkov [11] and Föllmer and Kabanov [7]. Define the set of risk neutral measures as M = Q P : S is a local martingale under Q}.

Now we make the main assumption of the section. 7 Assumption 2.1. S is a semimartingale on the filtered probability space (Ω, F, F t, P ). M has a unique element P. Under this assumption, the no-arbitrage price of any derivative security H is Ẽ[H] where the expectation is taken under the risk neutral measure P, and a corresponding hedging strategy can be found when the price is finite. Now we are ready to further simplify the main optimization problem (2). 2.1. Static Formulation of the Dynamic Problem. Let the initial wealth be X 0 = x r. Under Assumption 2.1, the space of final outcomes of self-financing strategies are those F T - measurable random variables X such that Ẽ[X] = x r. Therefore, the dynamic problem (4) is equivalent to the static problem (5) X = arg inf X 1 λ inf (E[(x x R X)+ ] λx), subject to Ẽ[X] = x r, x d X x u, where the constants satisfy x d < x r < x u. Let X T = X. Then X T is the final value of the optimal portfolio for problem (2): CV ar λ (X T ) = inf ξ t CV ar λ (X T ) = inf X 1 λ inf x R (E[(x X)+ ] λx) = 1 λ inf x R (E[(x X ) + ] λx). Martingale representation theorem applied to X t = Ẽ[X F t ] will produce the optimal hedging strategy ξ t for problem (2). Problem (5) is intrinsically much simpler than problem (2) because it looks for an optimal random variable X with convex objective function. The above simplification steps we have taken is based on classic duality theory (or sometimes called martingale approach) in mathematical finance. By duality, we mean there are two important spaces: the primal space consists of dynamics of self-financing portfolios and the dual space consists of risk neutral measures. The optimization problem in the primal space

8 is translated into an optimization problem in the dual space where it is always simpler in a complete market when the dual space consists of a singleton. 2.2. Solution to the Static Formulation. Rewrite the above static problem (5) by interchanging the order of infimum: (6) inf ξ t CV ar λ (X T ) = 1 ( ) λ inf inf E[(x x R X X)+ ] λx subject to Ẽ[X] = x r, x d X x u. we arrive to the final form of the main optimization problem (2) where we provide a direct solution in two steps: Step 1: Minimization of Expected Shortfall v(x) = inf X E[(x X)+ ] subject to Ẽ[X] = x r, x d X x u, Step 2: Minimization of CVaR inf ξ t CV ar λ (X T ) = 1 λ inf (v(x) λx). x R Schied [18] solved a general law invariant risk minimization problem of the type (6). We solve the CVaR minimization with the above two-step approach where we do not require the probability space to be atomless so the tree models are included. We give explicit computation methods for the Black-Scholes and Binomial model in Section 3. The solution for Expected Shortfall Minimization is studied in a semimartingale model in Föllmer and Leukert [8] and Xu [20]. Apply Proposition 4.1 in [8] to the above shortfall problem, we get the following result. Theorem 2.2 (Solution to Expected Shortfall Minimization Problem). Define a set A = ω Ω : }, d P (ω) > a set B = ω Ω : }, d P (ω) < a and set C = ω Ω : }, d P (ω) = a dp dp dp

where the constant a is to be determined on a case by case basis. X defined below is the optimal solution to the shortfall problem in Step 1 and the corresponding value function v(x) is given. 9 Case 1: x x d : X = any random variable X with values in [x d, x u ] where Ẽ[X] = x r, and v(x) = 0. Case 2: x d x x r < x u : X = any random variable X with values in [x, x u ] where Ẽ[X] = x r, and v(x) = 0. Case 3: x d < x r x x u : X = x d I Ax + k x I Cx + xi Bx, where A x, B x, C x are decided by level a x defined as } x r x d a x = sup a : P (B), x x d and k x is chosen so that the constraint x r = Ẽ[X ] = x d P (Ax ) + k x P (Cx ) + x P (B x ) is satisfied: k x = x r x d P (Ax ) x P (B x ) 1 P (Cx)>0} P (C x ). v(x) = (x x d )P (A x ) + (x k x )P (C x ). Case 4: x x u : X = x d I A + ki C + x u I B, where A, B, C are decided by level a defined as } x r x d a = sup a : P (B), x u x d and k is chosen so that the constraint x r = x d P (A) + k P (C) + xu P (B) is satisfied: k = x r x d P (A) xu P (B) 1 P (Cx)>0} P (C). v(x) = (x x d )P (A) + (x k)p (C) + (x x u )P (B).

10 Remark 2.3. The global minimum for function v(x) is 0. For the first two cases where x x r, the minimal value of 0 can be easily achieved by an admissible X x, including the special example of X x r that naturally satisfies the constraint of Ẽ[X ] = x r. For the latter two cases where x > x r, the solution is provided by Neyman-Pearson lemma. A part of the X should be as large as possible to minimize v(x) on the good set B, while the other part should be taken at the lower bound to offset this large number so that the risk-neutral expectation of X is guaranteed to stay at x r. In Case 3, we can equivalently define a x as a x = inf } x x r a : P (A), x x d because for fixed x level, A x is the largest set satisfying P (A x ) x xr, and B x is the largest set satisfying P (B x ) xr x d. When there is one or more point mass at a x, set C x is not empty and k x has to be chosen to satisfy the constraint of Ẽ[X ] = x r. When there is no point mass at a x, set C x is empty, and we have exact equality P (A x ) = x xr and P (B x ) = xr x d. Note that the sets A, B, C and the number k in Case 3 are functions of x, while in Case 4 they are not. Remark 2.4. Using Theorem 2.2 to solve Step 2, we need to find the global minimum among four cases: 1 λ 1 λ 1 λ inf x x d (v(x) λx) = 1 λ inf x x d (0 λx) = x d, inf x d x x r (v(x) λx) = 1 λ inf x r x x u (v(x) λx) = 1 λ inf (0 λx) = x r x d, x d x x r ( inf (x xd )P (A x ) + (x k x )P (C x ) λx ), x r x x u 1 λ inf x x u (v(x) λx) = 1 λ inf x x u ( (x xd )P (A) + (x k)p (C) + (x x u )P (B) λx ).

The solutions for the first two cases are simple where we observe the second case dominate the first case. We rewrite the thirds case as 1 λ inf x r x x u (v(x) λx) = 1 λ = x r + 1 λ = x r + 1 λ = x r + 1 λ inf ((x x d )P (A x ) + (x k x )P (C x ) λx) x r x x u inf ((x x d )P (A x ) + (x k x )P (C x ) λx + λx r ) x r x x u inf ((x x d )(P (A x ) λ P (A x )) + (x k x )(P (C x ) λ P (C x ))) x r x x u inf h(x) x r x x u where h(x) = (x x d )(P (A x ) λ P (A x )) + (x k x )(P (C x ) λ P (C x )), and solve the problem 11 inf h(x) x r x x u in Lemma 2.5. In case four, A, B, C and k are irrelevant to x. The minimization is simpler because it is linear in x with positive slope so the minimum is obtained when x = x u : 1 λ inf ( (x xd )P (A) + (x k)p (C) + (x x u )P (B) λx ) x>x u = 1 ( ) (xu x d )P (A) + (x u k)p (C) λx u λ 1 λ ( inf (x xd )P (A x ) + (x k x )P (C x ) λx ). x r<x x u We have shown here that the minimum obtained in the fourth case will not provide the global minimum because it is dominated by the results from the third case. It is easy to see that case two is also dominated by case three because it coincides with the result in case three when x = x r. Therefore, after Lemma 2.5, we arrive naturally to the result of Step 2 in Theorem 2.7. Lemma 2.5. Recall sets A, B, C are related to the number a in the same manner as in Theorem 2.2, namely A = ω Ω : }, d P (ω) > a B = ω Ω : }, d P (ω) < a and C = dp dp

12 ω Ω : d P dp (ω) = a }. For any fixed x, define a x = inf } x x r a : P (A), k x = x r x d P (Ax ) x P (B x ) 1 x x P (Cx)>0} d P (C x ), where the sets are associated to x as A x = } ω Ω : d P dp x, etc. Denote the parameters a max, k max, A max, B max, C max corresponding to x = x u as the max-system; those parameters a min, k min, A min, B min, C min corresponding to x = x r as the min-system, those parameters a, k, A, B, C corresponding to x = x as the star-system. The solution to the minimization problem inf h(x), x r x x u where h(x) = (x x d )(P (A x ) λ P (A x )) + (x k x )(P (C x ) λ P (C x )), is: If d P dp 1 λ, P a.s., then inf x r x x u h(x) = h(x r ) = 0. Otherwise, if a max 1 P (A max) λ P (A max), then inf x r x x u h(x) = h(x u ). If a max < 1 P (A max), then inf λ P (A max) x r x x u h(x) = h(x ), where a = sup A = ω Ω : }, d P (ω) > dp a B = ω Ω : }, d P (ω) < dp a C = a : a < 1 P (A) λ P (A) ω Ω : d P }, dp (ω) = a }, x = xr x d( P (A )+ P (C )), are the parameters that form the star-system defined above. P (B ) Remark 2.6. In the case where the probability space is atomless and the Radon-Nikodým derivative d P dp (ω) has continuous distribution, we have C =, B = Ac, so the constraint x r = x d P (A) + x P (B) + k P (C) becomes xr = x d P (A) + x P (A c ), i.e., a x can be chosen for the precise equality P (A x ) = x xr to hold and k x = 0. The solution to the minimization problem inf xr x x u h(x) where h(x) = (x x d )(P (A x ) λ P (A x )) is: If d P dp 1 λ, P a.s., then inf x r x x u h(x) = h(x r ) = 0. Otherwise, if a max 1 P (A max) λ P (A max), then inf x r x x u h(x) = h(x u ). If a max < 1 P (A max), then inf λ P (A max) x r x x u h(x) = h(x ), where x = xr x d P (A ) 1 P (A ) } ω Ω : d P (ω) > dp a satisfies a = 1 P (A ). λ P (A ) and A =

Proof for Remark 2.6. Let us first prove 2.6 in the continuous distribution case. Suppose d P 1, P a.s. Then for any x [x dp λ r, x u ], d P (A x ) = P A x dp (ω)dp (ω) 1 λ P (A x). Thus h(x) = (x x d )(P (A x ) λ P (A x )) 0. When x = x r, P (A min ) = P (A min ) = 0 and h(x r ) = 0. Therefore, Now suppose P ( d P dp > 1 λ inf h(x) = h(x r ) = 0. x r x x u d P ) > 0. In this special case when has a continuous distribution, dp } ω : d P (ω) > a dp x, and we observe the we have the equality P (A x ) = x xr where A x = following: 13 A x increases as x increases; a x decreases as x and A x increases. Define function f(x) = x xr. We see that f(x) is an increasing function since f (x) = x r x d ( ) 2 > 0. Notice that the probability function P (A x ) is an increasing function of A x, so x f(x) P (A x ) A x. It s obvious from the definition that the set A x a x. P (A min ) = 0 and P (A max ) = xu xr x u x d. x ranges over the interval [x r, x u ], thus f(x) range over [ 0, xu xr x u x d ]. A min corresponds to the case x = x r. By the definition of function f(x), we know that f(x r ) = 0 = P (A min ). It is obvious that P (A max ) = f(x u ) = xu xr x u x d. d P (A x) dx = xr x d ( ) 2 ; D P (A x ) = 1 a x d P (A x) dx, D + P (A x ) = 1 a x+ d P (A x) dx. Use the definition of f(x), d P (A) dx = f (x) = xr x d ( ) 2. When the Radon-Nikodým derivative d P dp (ω) has continuous distribution, a x is a decreasing function of x where at times it can jump downward. Therefore, the left- and right-limit a x and a x+ exist. In fact, when a x > a x+, P (ax+ < d P dp (ω) < a x ) = 0. Since P and P are equivalent, P (a x+ < d P dp (ω) < a x ) = 0 and P (A x ) = P ( d P dp (ω) > a x ) = P ( d P dp (ω) > a x+). If we

14 denote the left- and right-derivatives as D P (A x+ɛ ) P (A x ) P (A x ) = lim, ɛ 0 ɛ D + P (A x+ɛ ) P (A x ) P (A x ) = lim, ɛ 0 ɛ we have D P (A x ) = lim ɛ 0 P (A x+ɛ ) P (A x ) ɛ Since and = lim ɛ 0 E[1 Ax+ɛ] E[1 Ax ] ɛ dp Ẽ[ d = lim P 1 dp d P ] Ẽ[ dp >a x+ɛ d P 1 d P ] dp >a x ɛ 0 ɛ Ẽ[ dp d P 1 a x< d P dp a x+ɛ ] ɛ Ẽ[ dp d P 1 a x < d P dp a x+ɛ ] ɛ 1 Ẽ[1 ax< d P a x+ɛ ɛ dp a x+ɛ ] = 1 P (Ax+ɛ ) P (A x ) a x+ɛ ɛ < 1 Ẽ[1 ax < a x ɛ d P ] dp a x+ɛ = 1 P (Ax+ɛ ) P (A x ) a x ɛ as ɛ 0. We conclude the left derivative is Similarly, the right derivative is D P (A x ) = 1 d P (A x ). a x dx D + P (A x ) = 1 d P (A x ). a x+ dx dp Ẽ[ d = lim P 1 a x < d P ] dp a x+ɛ ɛ 0 ɛ 1 d P (A x ), a x dx 1 d P (A x ), a x dx

If a x is continuous at x, i.e., a x+ = a x = a x, then the derivative exists 15 dp (A x ) dx = 1 a x d P (A x ) dx = x r x d a x (x x d ) 2. Now let s turn to the first derivative and second derivative of the function we would like to minimize: h(x) = (x x d )(P (A x ) λ P (A x )), and show it to be a convex function. On x (x r, x u ), when a x is continuous at x, we have h (x) = (P (A x ) λ P (A x )) + (x x d ) = ( P (A x ) λ x x ) r + (x x d ) x x d = P (A x ) λ x x r x x d + ( 1 a x λ = P (A x ) λ + 1 a x (1 P (A x )). ( dp (A x ) dx λ d P ) (A x ) dx ( xr x d a x (x x d ) λ x ) r x d 2 (x x d ) 2 ) xr x d x x d When a x is discontinuous at x, we can define the left- and right-derivatives Similarly to the above calculation, we get D h(x + ɛ) h(x) h(x) = lim, ɛ 0 ɛ D + h(x + ɛ) h(x) h(x) = lim. ɛ 0 ɛ D h(x) = P (A x ) λ + 1 a x (1 P (A x )), D + h(x) = P (A x ) λ + 1 a x+ (1 P (A x )). When a x is continuous at x, P (Ax ) = P ( d P dp (ω) > a x) = 1 P ( d P dp (ω) a x) = 1 F (a x ), where F ( ) is the cumulative distribution function of the Radon-Nikodým derivative d P dp. Since P (A x ) = x xr, F (a x ) = xr x d. We have also started by assuming d P dp has a continuous

16 distribution, therefore the derivative of F ( ) exists and is the probability density function f( ). When a x = a x+, P (Ax ) is strictly increasing as x increases, thus f(a x ) > 0. By Implicit Differentiation Theorem, the derivative of a x exists and can be computed as With Chain Rule, we know (a x ) = ( 1 a x x r x d f(a x )(x x d ) 2 < 0. ) = a x a 2 x Now we can compute the second derivative of h(x): > 0. h (x) = dp (A ( ) x) 1 + (1 dx a P (A x )) 1 d P (A x ) x a x dx ( ) 1 = (1 P (A x )) > 0. a x Here 1 P (A x ) is positive since the maximum value of P (A x ) is P (A max ) = xu xr x u x d < 1 by definition. Clearly, the second derivative indicates that h (x) is strictly decreasing at those points x (x r, x u ) where a x is continuous. When a x is discontinuous, we have D h(x) = P (A x ) λ + 1 a x (1 P (A x )) < D + h(x) = P (A x ) λ + 1 a x+ (1 P (A x )). We recognize that this is a kink point for h(x). Finally, we conclude h(x) is convex on (x r, x u ). It is easy to see that h(x) is continuous at both left and right end points. Therefore, it is convex on the closed interval [x r, x u ]. If we can find x [x r, x u ], where 0 [D h(x ), D + h(x )], then it is the minimum. Otherwise if D + h(x r ) 0 the infimum is obtained at x = x r ; if D h(x u ) 0 the infimum is obtained at x = x u. If the derivative of h(x) exists at x = x, then the above condition collapse to h (x ) = 0, or equivalently, a = 1 P (A ). When the λ P (A ) derivative does not exist, it is obvious that 0 [D h(x ), D + h(x )] is still equivalent to the existence of a [a x, a x +] where a = 1 P (A ) λ P (A ), and the corresponding x can be computed from the equation P (A ) = x x r x x d.

Recall that P ( d P dp > 1 λ ) > 0. Since P (A min ) = P ( d P dp > a min) = 0, a min > 1 λ. So D h(x r ) = P (A min ) λ + 1 a min (1 P (A min )) = λ + 1 a min < 0. On the other end, D + h(x u ) = P (A max ) λ + 1 a max (1 P (A max )). We have D + h(x u ) > 0 if and only if a max < 1 P (A max) λ P (A max). In this case, the minimum occurs at x (x r, x u ) where a = 1 P (A ) λ P (A ). If a max 1 P (A max) λ P (A max), the minimum is achieved at the right end point with the value h(x u ). 17 Proof for Lemma 2.5. As in the proof for Remark 2.6, let F ( ) be the cumulative distribution function of the Radon-Nikodým derivative d P dp. Then for fixed x, we have F (a x ) = 1 P (A x ). In the proof for Remark 2.6, we have assumed that d P dp has a continuous distribution. This essentially dealt with case where F ( ) is continuous: it could either be strictly increasing or flat. Now to deal with the general case for the distribution of d P, we only need to discuss dp the remaining case where F ( ) has a jump, i.e., there is a point mass at d P dp = a x. See Fig. 1: Figure 1. F (a) is the cumulative distribution function of the Radon- Nikodým derivative d P dp.

18 Recall the definitions a x = inf A x = } x x r a : P (A), x x d ω Ω : d P dp (ω) > a x k x = x r x d P (Ax ) x P (B x ) 1 P (Cx)>0} P (C x ), }, C x = ω Ω : d P } dp (ω) = a x, B x = h(x) = (x x d )(P (A x ) λ P (A x )) + (x k x )(P (C x ) λ P (C x )), ω Ω : d P dp (ω) < a x }, and we would like to find When d P dp inf h(x). x r x x u has a point mass at a x, i.e., P ( d P dp = a x) = P (C x ) > 0, the distribution function F has a jump at a x : F (ax ) F (a x ) = P (C x ). As in the proof for Remark 2.6, we first discuss the case d P dp 1 λ, P a.s. Similarly we can show P (A x ) 1 λ P (A x) and P (C x ) 1 λ P (C x). It is easy to check that x k x 0 when x r x x u. We conclude, inf h(x) = h(x r ) = 0. x r x x u Now suppose P ( d P dp > 1 λ ) > 0. As soon as the equality P (A) = x xr, we have P (B) = xr x d and P (C) = 0. Therefore, we need only discuss two cases: P (A x ) = x xr, P (Bx ) = xr x d and P (C x ) = 0, P (A x ) < x xr, P (Bx ) < xr x d and P (C x ) > 0. In the case when P (A) < x xr, P (B) < x r x d and P (C) > 0, the sets A, B, C as well as the point a x do not change with a small change in x. However, in this case where x increases from xr x d P (A) P (B)+ P to xr x d( P (A)+ P (C)) (C) P (B), k x decreases from xr x d P (A) P (B)+ P (C) to r d. And the derivative of

h(x) is easily calculated as 19 h (x) = (P (A) λ P (A)) + (1 dk x dx )(P (C) λ P (C)) = P (A) λ + 1 a x (1 P (A)) Since a x is not changing with respect to x in this case, h (x) is constant on x ( x r x d P (A) P (B) + P (C), x r x d ( P (A) + P ) (C)). P (B) This formula is exactly the same as the one in the continuous case. Therefore, the convexity of h(x) is preserved on the interval x [ x r x d P (A) P (B) + P (C), x r x d ( P (A) + P ) (C)), P (B) i.e., D h( x r x d P (A) P (B) + P (C) ) h (x), for x ( x r x d P (A) P (B) + P (C), x r x d ( P (A) + P ) (C)). P (B) At the other end point x = xr x d( P (A)+ P (C)), k P (B) x dropped to r d and the old sets A and C combine to produce the new set A x = A C and we encounter the other case where P (A x ) = x xr, P (B x ) = xr x d above and P (C x ) = 0. The left derivative at this point is computed D h( x r x d ( P (A) + P (C)) ) = P (A) λ + 1 (1 P (B) a P (A)), x where A is the old set before x reach the right end point. The right derivative is computed in the proof for Remark 2.6: D + h( x r x d ( P (A) + P (C)) ) = P (A x ) λ + 1 (1 P (B) a P (A x )). x+

20 The difference P (A x ) λ + 1 a x (1 P (A x )) ( P (A) λ + 1 ) (1 a P (A)) x = P (C) 1 a x P (C) = 0, by the definition of set C. Since 1 a x+ 1 a x, we keep the convexity of h(x) at x = xr x d( P (A)+ P (C)) P (B) because D h( x r x d ( P (A) + P (C)) P (B) ) D + h( x r x d ( P (A) + P (C)) ). P (B) We also conclude after this discussion that when there is a point mass at d P dp = a x, the convex function h(x) becomes linear. In any case, combining the results we have just shown and those in the proof of Remark 2.6, we know that h(x) is convex all the time on x [x r, x u ]. As shown in Fig. 2, in a case like the Binomial model where there are only point masses, h(x) is a piecewise constant convex function; in a case like the Black-Scholes model where the distribution is continuous and spans the whole positive part of the real line, h(x) is a continuously differentiable convex function. Figure 2. The left picture is how h(x) look like in the Binomial model; the right pictures is for the Black-Scholes model. The discussion about minimizing h(x) in the proof of Remark 2.6 is also valid here and we conclude our proof by verifying the star-system when it happens at the right end point

0 (D h( xr x d( P (A)+ P (C)) P (B) ), D + h( xr x d( P (A)+ P (C)) )). This means a P (B) x is the largest value where 21 P (A) λ + 1 a x (1 P (A)) < 0. Therefore, we get the desired condition a = sup a : a < 1 P (A) λ P (A) }. Theorem 2.7 (Solution to CVaR Minimization Problem (6)). Define the sets A, B, C and the numbers a x, k x the same way as in Lemma 2.5 and Theorem 2.2. Denote the min-system, max-system, star-system as in Lemma 2.5. The solution to problem (6) is as follows: If d P dp 1 λ, P a.s., then X = x r is optimal, and the minimal risk is CV ar λ (X ) = x r. Otherwise, if a max 1 P (A max) λ P (A max), then the optimal risk is achieved by the max-system. X = x d I Amax + k max I Cmax + x u I Bmax is optimal, and the associated minimal risk is CV ar λ (X ) = x r + 1 λ [(x u x d )(P (A max ) λ P (A max ))+(x u k max )(P (C max ) λ P (C max ))]. If a max < 1 P (A max), then the optimal risk is achieved by the star-system. Set λ P (A a max) = } sup a : a < 1 P (A) λ P, A = ω Ω : }, d P (ω) > (A) dp a B = ω Ω : }, d P (ω) < dp } a C = ω Ω : d P (ω) = dp a, x = xr x d( P (A )+ P (C )). Then X = x d I A + k I C + x I B P (B ) is optimal, and the associated minimal risk is CV ar λ (X ) = x r + 1 λ [(x x d )(P (A ) λ P (A )) + (x k )(P (C ) λ P (C ))]. Remark 2.8. For the continuous case as in Remark 2.6 where C =, B = A c, and the constraint x r = x d I A + xi A c, we can simplify the result as follows:

22 If d P dp 1 λ, P a.s., then X = x r is optimal, and the minimal risk is CV ar λ (X ) = x r. Otherwise, if a max 1 P (A max) λ P (A max), then set x = x u. The optimal portfolio is X = x d I Amax + x I A c max, and the optimal risk is CV ar λ (X ) = x r + 1 λ (x u x r )(P (A max ) λ P (A max )). If a max < 1 P (A max) λ P (A max), then x = xr x d P (A ) 1 P (A ) and A = ω Ω : d P dp (ω) > a } satisfies a = 1 P (A ) λ P (A ). The optimal portfolio is X = x d I A + x I A c, and the optimal risk is CV ar λ (X ) = x r + (x r x d )(1 1 λa ). Proof. Theorem 2.2 gives the solution to the shortfall problem in Step 1. Leveraging these results, we have discussed in Remark 2.4 that the solution for step 2 is achieved by finding the solution to the third case of 1 λ inf x r<x x u (v(x) λx) = x r + 1 λ inf x r<x x u h(x), where h(x) = (x x d )(P (A x ) λ P (A x )) + (x k x )(P (C x ) λ P (C x )). Now combine the solution to inf h(x), x r<x x u found in Lemma 2.5, we quickly arrive to the conclusion. 3. Some Complete Market Examples

3.1. Binomial Model. In the recombining tree shown below, we have the following dynamics for the stock and the self-financing portfolio: 23 S n+1 (H) = us n, with P (ω n = H) = p and P (ω n = H) = p S n+1 (T ) = ds n, with P (ω n = T ) = q and P (ω n = T ) = q X n+1 = ξ n S n+1 + (X n ξ n S n )(1 + r), where p, q are risk-neutral probability, p, q are physical probability, u, d are the step size for up move and down move respectively, and r is the risk-free interest earned for one time step. Our main goal is first to find inf ξ n CV ar λ (X N ) s.t. Ẽ[X N ] = x r, x d X N x u, then to find the corresponding dynamic hedging ξ n. The optimal final portfolio X N and the minimal risk inf ξ n CV ar λ (X N ) can be found easily according to Theorem 2.7. To find the corresponding optimal hedging ξ n, we first calculate X n = ( 1 1+r )(N n) Ẽ[XN F n] for n 1, 2,..., N 1}, then compute ξ n = X n+1(h) X n+1 (T ) S n+1 (H) S n+1 (T ) martingale representation theorem. by Algorithm 3.1. CVaR Minimization for Binomial Model (1) We first arrange the states ω Ω such that P P (ω) is descending: P P (ω 1) P P (ω 2)... (2) If P P (ω 1) 1 λ, then the optimal risk is inf ξ n CV ar λ (X N ) = x r, and the optimal hedge is ξ n = 0 for all n. Stop. (3) If there is some ω such that P (ω) > 1, then we need to find the max-system. P λ (4) If a max > 1 P (A max) λ P (A max), then the optimal portfolio is X N = x di Amax +k max I Cmax +x u I Bmax, and the optimal risk is inf ξn CV ar λ (X N ) = x r + 1 λ [(x u x d )(P (A max ) λ P (A max ))+ (x u k max )(P (C max ) λ P (C max ))]. Go to (6).

24 (5) If a max < 1 P (A max) λ P (A max), then find the star-system where A is the smallest set satisfying a < 1 P (A) λ P (A). Compute x from the equation P (A ) = x xr, and set the rest elements of the star-system. The optimal portfolio is X N = x di A + k I C + x I B, and the optimal risk is inf ξn CV ar λ (X N ) = x r + 1 λ [(x x d )(P (A ) λ P (A )) + (x k )(P (C ) λ P (C ))]. (6) Calculate X n = ( 1 1+r )(N n) Ẽ[X N F n] for all n. (7) Calculate ξ n = X n+1(h) X n+1 (T ) S n+1 (H) S n+1 (T ). 3.2. Black-Scholes Model. Let s turn our attention to the Black-Scholes model, a complete market model with continuously distributed stock price. The dynamics of the stock price and the self-financing portfolio are as follows: ds t = S t (µdt + σdw t ) dx t = ξ t ds t + (X t ξ t S t )rdt. Our main goal is first to find inf ξ t CV ar λ (X T ) s.t. Ẽ[X T ] = x r, x d X T x u, then to find the corresponding dynamic hedging ξ t. According to Theorem 2.2, and 2.7 of the general complete market, we need to check whether d P (ω) dp T 1 for all ω Ω, and if not, then whether a λ max > 1 P (A max). λ P (A max) The Radon- Nikodým derivative for geometric Brownian motion is Z t := d P dp t = e θwt θ 2 2 t, where θ = µ r. σ Obviously, d P dp (ω) T > 1 λ for some ω Ω since ess sup Z T =. To check the second inequality, and possibly to find solution to the equation a = 1 P (A), we need to find the λ P (A) explicit relation among the three elements P (A), P (A) and a. Notice that A = d P } W T dp T > a = < θ T T 2 } ln a WT θ = < θ T T T 2 } ln a θ, T

we have 25 P (A) = N( θ T 2 P (A) = N( θ T 2 ln a θ T ), ln a θ T ), where N( ) is the cumulative distribution function for normal distribution. Link the above two equations with the definition P (A) = x xr, we can solve for P (A max ), a max, P (A max ) sequentially: P (A max ) = x u x r x u x d a max = θ [ θ ] T T N 1 ( 2 P (A max )) P (A max ) = N( θ T 2 ln a max θ T ), and solve the equation a = 1 P (A) λ P (A) through its explicit form: a = 1 T N(θ ln a 2 θ ) T λ N( θ T ln a 2 θ ). T A direct application of Theorem 2.7 gives the optimal final portfolio X T and the minimal CV ar. To find the corresponding optimal hedging ξ t, we apply the martingale representation theorem. X t = e r(t t) Ẽ[X T F t ] = e r(t t) x Pt (A c ) + x d Pt (A )} = e r(t t) x + (x d x ) P t (A )}, where P t (A ) is the conditional probability under risk-neutral measure. Since A := ω : Z T > a}, where Z t is the Radon-Nikodým derivative process, we have then Z T = Z t e θ(w T W t) θ2 2 (T t) >

26 a on A. So P t (A ) = P t (Z t e θ(w T W t) θ2 2 (T t) > a ) ( = P WT W ) a t ln Z t < t T t θ T t + θ T t 2 a ln Z = N( t θ T t + θ T t). 2 And Z t can be represented by the stock price S t : Z t = e θwt 1 2 θ2t W t = ln Z t + 1 2 θ2 t θ St ln S t = S 0 e σwt+(µ 1 2 σ2 )t S W t = 0 (µ 1 2 σ2 )t σ So, Z t = e θ σ [ µ+r σ 2 t ln S t 2 S ] 0, Give the stock price S t at time t, the optimal strategy ξ t is given by: ξ t = dx t ds t = [ dx t dz t ][ dz t ds t ] = [e r(t t) (x d x 1 )( e 1 2 ( ln a 2π = e r(t t) (x d x 1 )( e 1 2 ( ln a 2π = e r(t t) (x d x 1 )( e 1 2 1 2π Z t θ T t + θ Z t θ T t + θ T t 1 Z t ) 2 2 )( θ T t )][Z t( θ )] σs t ) 2 2 1 )( ) σs t T t T t θ T t [ ln a + θ σ ( µ+r σ2 t ln S t )+ θ2 2 S 0 2 (T t)]}2 )( Algorithm 3.2. CVaR Minimization for Black-Scholes Model 1 ). σs t T t (1) Calculate P (A max ), P (A max ), a max. If a max > 1 P (A max) λ P (A max), then set x = x u, a = a max, and inf ξt CV ar λ (X T ) = x r + 1 (x λ u x r )(P (A max ) λ P (A max )). Otherwise, find the unique solution x, a, A to the equation a = 1 P (A) λ P (A), and inf ξ t CV ar λ (X T ) = x r + (x r x d )(1 1 λa ).

(2) Compute optimal hedge 27 ξt = e r(t t) (x d x 1 )( e 1 2 1 2π θ T t [ ln a + θ σ ( µ+r σ2 t ln S t )+ θ2 2 S 0 2 (T t)]}2 )( 1 ). σs t T t 4. Results: Dynamic & Static Hedging 4.1. Binomial Model. Dynamic Hedging Take a 2-step binomial model as an example, where the sampling times are t = t 0, t 1, t 2. Also assume p = 7, q = 1, u = 2, d = 1, r = 1, S 8 8 2 4 0 = 4, X 0 = 1, x u = 2, x d = 1, λ = 0.1. We calculate the risk-neutral probability p = 1+r d u d = 1 u 1 r, q = = 1, and x 2 u d 2 r = (1 + r) 2 = 25. 16 P (T T ) = 1 4, P (T T ) = 1 64, P (T T ) = 16 P P (T H) = 1 4, P (T H) = 7 64, P 16 (T H) = P 7 P (HT ) = 1 4, P (HT ) = 7 64, P 16 (HT ) = P 7 P (HH) = 1 49, P (HH) = 4 64, P 16 (HH) = P 49. By Algorithm 3.1, we first put the states in descending order: (TT), (TH,HT), (HH)}. Since P P (T T ) > 1 λ, we jump to step 3 and calculate the max-system. A max is the largest set such that P (A) < xu xr x u x d = 0.4375, so A max = (T T )}. Similarly, B max is the largest set such that P (B) < xr x d x u x d = 0.5625, so B max = (HH)}. And consequently C max = (HT, T H)}, a max = P P (C) = 16 7, k max = xr x d P (A max) x u P (Bmax) P (C max) = 13 8. With the above calculated max-system, we see that a max < 1 P (A max) λ P (A max), so we proceed with step 5. The only set A that s smaller than A max is A =. Its corresponding a = 16, and we have the inequality a > 1 P (A) λ P (A). Therefore, the star-system includes A =, a = 16 in this simple 2-step recombining binomial tree. We calculate the corresponding x = 7 4, k = 7 4, and the optimal risk is inf ξ t CV ar λ (X t ) = x r + 1 λ [(x x d )(P (A ) λ P (A ))+(x k )(P (C ) λ P (C ))] = 1.6328. Now we calculate the portfolio value X 1 (H), X 1 (T ). We know that X 2 (T T ) = x d, X 2 (T H) = X 2 (HT ) = k,

28 and X 2 (HH) = x, so X 1 (H) = 1 1 + r ( px + qk ) = 1.4, X 1 (T ) = 1 1 + r ( pk + qx d ) = 1.1. And the hedging ξ 1 (H) = X 2(HH) X 2 (HT ) S 2 (HH) S 2 (HT ) = 0 ξ 1 (T ) = X 2(T H) X 2 (T T ) S 2 (T H) S 2 (T T ) = 0.25 ξ 0 = X 1(H) X 1 (T ) S 1 (H) S 1 (T ) = 0.05. Static Hedging If we can only determine the hedging at the beginning, i.e., ξ 0, then the portfolio values along the binomial tree are X 2 = ξ 0 S 2 + (1 + r) 2 (X 0 ξ 0 S 0 ). To constrain X 2 on the interval [x d, x u ], we require ξ 0 [ 0.0577, 0.0449], and the resulting CV ar λ (X 2 ) with λ = 0.1, is tabulated as follows: P (ω) X 2 (ω) ξ 0 = -0.0577 ξ 0 ( 0.0577, 0) ξ 0 = 0 ξ 0 (0, 0.0449) ξ 0 = 0.0449 49 X 64 2 (HH) 1 1.5625 2 7 X 64 2 (HT ) 1.6923 1.5625 1.4615 7 X 64 2 (T H) 1.6923 1.5625 1.4615 1 X 64 2 (T T ) 1.8654 1.5625 1.3269 CV ar 0.1 (X 2 ) -1-1 -1.5625-1.4405 Obviously the static hedging is not as good as the dynamic hedging since the optimal risk is inf xi0 CV ar λ (X 2 ) = 1.5625, achieved at ξ 0 = 0. 4.2. Black-Scholes Model. Assume λ = 0.1, T = 0.5, r = 0.05, S 0 = 10, X 0 = 10, x d = 5, x u = 30. Let s compare the optimal risk from dynamic hedge and static hedge for three examples:

29 Example 1 Example 2 Example 3 µ 0.1 0.3 0.4 σ 0.2 0.1 0.08 Static CV ar -0.2532-1.2131-4.5710 Dynamic CV ar -10.2531-11.8619-27.2300 5. Conclusion We have so far found exact solutions for CVaR minimization in complete market models like the Binomial model and the Black-Scholes model. When we compare the minimal risk resulting from dynamic portfolio management to a static portfolio management, we get very different numbers in magnitude. This suggests that the procedure provided in this paper help to find a dynamic solution that is far better than a static solution. The procedure

30 also works for higher dimensional case as long as the market is complete. However, the incomplete market is still an open question.

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