# Sensitivity analysis of utility based prices and risk-tolerance wealth processes

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1 Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar, Standford, April 8 p. 1/38

2 Outline The prices of non replicable derivative securities depend on many factors: 1. risk-preferences of the investor: (a) reference probability measure P (b) utility function U = U(x) 2. current portfolio of the investor 3. trading volume in the derivatives Goal: study the dependence of prices on trading volume. Math Finance Seminar, Standford, April 8 p. 2/38

3 Model of a financial market There are d + 1 traded or liquid assets: 1. a savings account with zero interest rate. 2. d stocks. The price process S of the stocks is a semimartingale on (Ω, F, (F t ) 0 t T, P). Q : the family of local martingale measures for S. Assumption (No Arbitrage) Q = Math Finance Seminar, Standford, April 8 p. 3/38

4 Contingent claims Consider a family of m non-traded or illiquid European contingent claims with 1. maturity T 2. payment functions f = (f i ) 1 i m. Assumption No nonzero portfolio of f is replicable: q, f = m i=1 q i f i is replicable q = 0 Math Finance Seminar, Standford, April 8 p. 4/38

5 Pricing problem Question What is the (marginal) price p = (p i ) 1 i m of the contingent claims f? Intuitive Definition The marginal price p for the contingent claims f is the threshold such that given the chance to buy or sell at p trade an investor will 1. buy at p trade < p 2. sell at p trade > p 3. do nothing at p trade = p Math Finance Seminar, Standford, April 8 p. 5/38

6 Economic agent or investor Consider an investor with the portfolio (x, q), whose preferences are modeled by a utility function U : 1. U : (0, ) R, strictly increasing and strictly concave 2. The Inada conditions hold true: U (0) = U ( ) = 0 Math Finance Seminar, Standford, April 8 p. 6/38

7 Problem of optimal investment The goal of the investor is to maximize the expected utility of terminal wealth: u(x, q) = sup X X(x) E[U(X T + q, f )], where X(x) is the set of strategies with initial wealth x. Order structure: a portfolio (x, q) is better than a portfolio (x, q ) if u(x, q) u(x, q ). Math Finance Seminar, Standford, April 8 p. 7/38

8 Marginal utility based price Definition A marginal utility based price for the claims f given the portfolio (x, q) is a vector p(x, q) such that u(x, q) u(x, q ) for any pair (x, q ) satisfying x + q, p(x, q) = x + q, p(x, q). In other words, given the portfolio (x, q) the investor will not trade the options at p(x, q). Math Finance Seminar, Standford, April 8 p. 8/38

9 Computation of p(x) = p(x, 0) Define the conjugate function V (y) = max x>0 [U(x) xy], y > 0. and consider the following dual optimization problem: v(y) = inf Q Q E [ V ( y( dq )] dp ), y > 0 Q(y) : the minimal martingale measure for y. Math Finance Seminar, Standford, April 8 p. 9/38

10 Computation of p(x) = p(x, 0) Mark Davis gave heuristic arguments to show that if y corresponds to x in the sense that x = v (y) y = u (x) then p(x) = E Q(y) [f]. The precise mathematical results are given in a joint paper with Julien Hugonnier and Walter Schachermayer. Math Finance Seminar, Standford, April 8 p. 10/38

11 Computation of p(x) = p(x, 0) Theorem (Hugonnier,K.,Schachermayer) Let x > 0, y = u (x) and X be a non-negative wealth process. The following conditions are equivalent: 1. p(x) is unique for any f such that f K(1 + X T ) for some K > 0 2. Q(y) exists and X is a martingale under Q(y). Moreover, in this case p(x) = E Q(y) [f]. Math Finance Seminar, Standford, April 8 p. 11/38

12 Trading problem Assume that the investor can trade the claims at the initial time at the price p trade. Question What quantity q = q(p trade ) the investor should trade (buy or sell) at the price p trade? Using the marginal utility based prices p(x, q) we can compute the optimal quantity from the equilibrium condition: p trade = p(x qp trade, q) Math Finance Seminar, Standford, April 8 p. 12/38

13 Trading problem Qualitatively, there are two different cases: 1. If f is replicable, then q(p trade ) = any p trade = p(x), p trade p(x) 2. else if f is not replicable, then q(p trade ) = finite 0 p trade = p(x) p trade p(x) Math Finance Seminar, Standford, April 8 p. 13/38

14 Sensitivity analysis of utility based prices Main difficulty : p(x, q) is hard to compute except for the case q = 0. Linear approximation for small x and q : p(x + x, q) p(x) + p (x) x + D(x)q, where p (x) is the derivative of p(x) and D ij (x) = pi qj(x, 0), 1 i, j m. Math Finance Seminar, Standford, April 8 p. 14/38

15 Quantitative questions Question (Quantitative) How to compute p (x) and D(x)? Closely related publications: J. Kallsen (02) : formula for D(x) for general semimartingale model but in a different framework of local utility maximization. V. Henderson (02) : formula for D(x) in the case of a Black-Scholes type model with basis risk and for power utility functions. Math Finance Seminar, Standford, April 8 p. 15/38

16 Qualitative questions Question (Qualitative) When the following (desirable) properties hold true for any family of contingent claims f? 1. The marginal utility based price p(x) = p(x, 0) does not depend (locally) on x, that is, p (x) = 0 2. The sensitivity matrix D(x) has full rank 3. The sensitivity matrix D(x) is symmetric Math Finance Seminar, Standford, April 8 p. 16/38

17 Qualitative questions 4. The sensitivity matrix D(x) is negative semi-definite: q, D(x)q Stability of the linear approximation: for any p trade the linear approximation to the equilibrium equation: that is, p trade = p(x qp trade, q) p trade p(x) p (x)qp trade + D(x)q has the correct solution. Math Finance Seminar, Standford, April 8 p. 17/38

18 Risk-tolerance wealth process Definition (K., Sirbu) A maximal wealth process R(x) is called the risk-tolerance wealth process if R T (x) = U ( X T (x)) U ( X T (x)), where X(x) is the optimal solution of u(x) := u(x, 0) = sup X X(x) E[U(X T )]. Math Finance Seminar, Standford, April 8 p. 18/38

19 Risk-tolerance wealth process Some properties of R(x) (if it exists): 1. Initial value: R 0 (x) = u (x) u (x). 2. Derivative of optimal wealth strategy: R(x) R 0 (x) = X (x) := lim x 0 X(x + x) X(x). x Math Finance Seminar, Standford, April 8 p. 19/38

20 Main qualitative result Recall p(x + x, q) p(x) + p (x) x + D(x)q. Theorem (K., Sirbu) The following assertions are equivalent: 1. The risk-tolerance wealth process R(x) exists. 2. p (x) = 0 for any f. 3. D(x) is symmetric for any f. 4. D(x) has full rank for any (non-replicable) f. 5. D(x) is negative semidefinite for any f. Math Finance Seminar, Standford, April 8 p. 20/38

21 Existence of R(x) Recall that Q(y) is the solution of v(y) = inf Q Q E [ V ( y( dq )] dp ), y > 0 where V is the convex conjugate of U. Theorem (K., Sirbu) The following assertions are equivalent: 1. R(x) exists for any x > Q(y) = Q (same for all y!). Math Finance Seminar, Standford, April 8 p. 21/38

22 Second order stochastic dominance Definition then ξ 2 η if If ξ and η are nonnegative random variables, t 0 P(ξ x)dx t 0 P(η x)dx, t 0. We have that ξ 2 η iff E[W(ξ)] E[W(η)] for any convex and decreasing function W. Math Finance Seminar, Standford, April 8 p. 22/38

23 Existence of R(x) Case 1: a utility function U is arbitrary. Theorem (K., Sirbu) The following assertions are equivalent: 1. R(x) exists for any x > 0 and any utility function U. 2. There exists a unique Q Q such that d Q dp 2 dq dp Q Q. Math Finance Seminar, Standford, April 8 p. 23/38

24 Existence of R(x) Case 2: a financial model is arbitrary. Theorem (K., Sirbu) The following assertions are equivalent: 1. R(x) exists for any x > 0 and any financial model. 2. The utility function U is (a) the power utility: U(x) = (x α 1)/α, α < 1, if x (0, ) ; (b) the exponential utility: U(x) = exp( γx), γ > 0, if x (, ). Math Finance Seminar, Standford, April 8 p. 24/38

25 Computation of D(x) We choose R(x)/R 0 (x) = X (x) as a numéraire and denote f R = fr 0 (x)/r(x) : discounted contingent claims X R = XR 0 (x)/r(x) : discounted wealth processes Q R : the martingale measure for X R : dq R d Q = R T(x) R 0 (x) Math Finance Seminar, Standford, April 8 p. 25/38

26 Computation of D(x) Consider the Kunita-Watanabe decomposition: P R t = E Q R [ f R F t ] = Mt + N t, N 0 = 0, where 1. M is R(x)/R 0 (x) -discounted wealth process. Interpretation: hedging process. 2. N is a martingale under Q R which is orthogonal to all R(x)/R 0 (x) -discounted wealth processes. Interpretation: risk process. Math Finance Seminar, Standford, April 8 p. 26/38

27 Computation of D(x) Denote a(x) := xu (x)/u (x) the relative risk-aversion coefficient of u(x) = max X X(x) E[U(X T)]. Theorem (K., Sirbu) Assume that the risk-tolerance wealth process R(x) exists. Then D(x) = a(x) x E [ ] Q R NT N T Math Finance Seminar, Standford, April 8 p. 27/38

28 Computation of D(x) in practice Inputs: 1. Q. Already implemented! 2. R(x)/R 0 (x). Recall that R(x) R 0 (x) = lim x 0 X(x + x) X(x). x Decide what to do with one penny! 3. Relative risk-aversion coefficient α(x). Deduce from mean-variance preferences. In any case, this is just a number! Math Finance Seminar, Standford, April 8 p. 28/38

29 Model with basis risk Traded asset : ds t = S t (µdt + σdw t ). Non traded asset : d S = ( µdt + σd W t ) Denote by ρ = d WdW dt the correlation coefficient between S and S. In practice, we want to chose S so that ρ 1. Math Finance Seminar, Standford, April 8 p. 29/38

30 Model with basis risk Consider contingent claims f = f( S) whose payoffs are determined by S (maybe path dependent). To compute D(x) assume (as an example) the following choices: 1. Q is a martingale measure for S. 2. R(x)/R 0 (x) = 1 Then D ij (x) = α(x) x (1 ρ2 )Cov Q(f i, f j ). Math Finance Seminar, Standford, April 8 p. 30/38

31 Existence of u (x) Recall that u(x) = max X X(x) E[U(X T)]. 1. Does u (x) exist? Is u (x) strictly positive? 2. Does X (x) = lim x 0 X(x + x) X(x) x exist? Math Finance Seminar, Standford, April 8 p. 31/38

32 Assumption on the utility function Assumption There are strictly positive constants c 1 and c 2 such that c 1 < xu (x) U (x) < c 2, x > 0. Math Finance Seminar, Standford, April 8 p. 32/38

33 Sigma-bounded semimartingales Definition A semimartingale R is called sigma-bounded if there is H > 0 such that the stochastic integral is bounded. HdR Remark If R is locally bounded, then R is sigma-bounded. The reverse assertion does not hold true. Math Finance Seminar, Standford, April 8 p. 33/38

34 Assumption on the (jumps of) price processes Assumption The price process of the traded securities discounted by X(x), that is, is sigma-bounded. S X(x) = ( 1 X(x), S X(x) ) Remark The above assumption is useful only if we can explicitly compute the optimal investment strategy. Math Finance Seminar, Standford, April 8 p. 34/38

35 Assumption on the (jumps of) price processes The last assumption is satisfied in the following cases: 1. S is continuous 2. The financial model can be completed. 3. If purely discontinuous martingales have finite-dimensional basis (as stochastic integrals). Remark There are complete financial models, where the locally bounded analog of the assumption is not satisfied. Math Finance Seminar, Standford, April 8 p. 35/38

36 Assumption on the (jumps of) price processes Theorem (K.,Sirbu) Assume that there is a finite-dimensional local martingale M such that any bounded purely discontinuous martingale N is given by N = N 0 + HdM. Then every semimartingale on (Ω, F, (F t ) 0 t T, P) is sigma-bounded. Remark The condition of the theorem is invariant under an equivalent change of the reference probability measure. Math Finance Seminar, Standford, April 8 p. 36/38

37 Main result Theorem (K.,Sirbu) Let both assumptions above be satisfied. Then u (x) and X (x) exist for any x > 0 and c 1 < xu (x) u (x) < c 2, x > 0. where c 1 and c 2 are the same constants as in the assumption on U. Remark Both assumptions are essential for the assertion of the theorem to hold true. Math Finance Seminar, Standford, April 8 p. 37/38

38 Summary For non replicable contingent claims prices depend on the trading volume. The following conditions are equivalent: Approximate utility based prices have nice qualitative properties Risk-tolerance wealth processes exist. We need to solve the mean-variance hedging problem, where the risk-tolerance wealth process plays the role of the numéraire. Math Finance Seminar, Standford, April 8 p. 38/38

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