Risk Measures for Derivative Securities: From a Yin-Yang Approach to Aerospace Space

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1 Risk Measures for Derivative Securities: From a Yin-Yang Approach to Aerospace Space Tak Kuen Siu Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, AUSTRALIA

2 A Brief History of Binomial Tree Yin-Yang: I-Ching or Zhouyi (1,000 BC or before) and Taoism (the late 4th century BC) Origin in Probability Theory: Daniel Bernoulli (29 January March 1782); Coin tossing experiment {H, T } Discrete-time binomial tree in nance: Bill Sharpe? A beautiful paper by Cox, Ross and Rubinstein, CRR, (1979): Option valuation in a discrete-time binomial model

3 Behind the scene: Boyle, Siu and Yang (2002) Asian Financial Crisis in 1997: LTCM and derivative securities Reappraisal of Value at Risk (VaR): Non-Subadditivity Coherent risk measures by Artnzer, Delbean, Eber and Heath (1999) Tail risk, expected shortfall and a research report in Bank of Japan by Yamai and Yoshiba (2002)

4 The Challenge Traditional theories in nance: Linear risk Capital Asset Pricing Model and Arbitrage Pricing Theory Bigger universe of nonlinear risk: not well-explored! Examples: Derivative securities and hedged funds Current Practice: Traders use Greek Letters, such as Delta, Gamma, Rho,..., etc.

5 Main Idea: Boyle, Siu and Yang (2002) Consider a discrete-time nancial model consisting of a riskfree bond B and a stock S Deal with a European call option C written on S with strike price K and maturity T Build the two-level binomial model from the CRR binomial model Evaluate a coherent risk measure, namely Expected Shortfall (ES), for derivative securities

6 The Model Suppose {0, 1, 2,..., T } is the time parameter set in the rst level For each time point k in the rst level, [k, k + 1] is the time interval for risk measurement Divide [k, k + 1] into m equal sub-intervals Then {0, 1, 2,..., km, km + 1,..., T m} is the time parameter set in the second level

7 For each sub-interval [n, n + 1] in the second level, assume that, under a real-world probability measure P, B n+1 B n = ˆr S n+1 S n = { u with probability p d with probability 1 p Call price from the CRR binomial model: C km = 1 ˆr T m km T m km j=0 (S km u j d T m km j K) + ( T m km ) q j (1 q) T m km j j

8 Expected Shortfall (ES) for the Call C k,m : the discounted net loss C km ˆr m C (k+1)m of the call option C over [km, (k + 1)m] F km : the information generated by the values of S up to and including time km Under P, the distribution of C k,m F km : with probability ( m j C k,m = C km ˆr m C (k+1)m (S km u j d m j ) ) p j (1 p) m j, j = 0, 1,..., m.

9 ES for the call C: ES α ( C k,m F km ) = E P ( C k,m I { Ck,m V ar α }, F km ) = α 1 [E P ( C k,m I{ C k,m VaR α,p ( C k,m F km )} F km ) + VaR α,p ( C k,m F km )(α P ( C k,m VaR α,p ( C k,m F km ) F km )] Adjustment for the discrete loss distribution to ensure the coherent property for the ES

10 An Expression for the ES Dene j α = sup{j J C k,m (j) VaR α,p ( C k,m F km )}, where J represents the set {0, 1, 2,..., m}. Then ES α ( C k,m F km ) { jα [ 1 ( T m (k+1)m m ( T m (k + 1)m ) = )p j (1 p) m j ˆr T m km α j i j=0 i=0 q i (1 q) T m (k+1)m i (S km u j+i d T m km j i K) + ] q i (1 q) T m km i (S km u i d T m km i K) + } T m km i=0 ( T m km ) i + C k,m (u j α dm j α ) [ 1 α 1 j α j=0 ( m j )p j (1 p) m j ].

11 Numerical Example Consider a European call with T = 2 months and K = 22. Suppose S 0 = 25 Assume that the time horizon for measuring the risk of the position is one month r = 0.7% per month and σ = 6% per month Two-level binomial model: m = 5, u = e , d = e and q =

12 The numerical values of ES and VaR for the call Table: ES and VaR for various values of p and α. p \ α ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

13 Yang-Yin Grows Everything Yang-Yin generates many patterns Think about the modern computing technologies Central Limit Theorem: Binomial => Normal CRR binomial model => Continous-time Black-Scholes-Merton model

14 Risk Measures for Derivatives in Continuous-Time Markets Literature: Siu and Yang (2000) and Yang and Siu (2001), Siu, Tong and Yang (2002) and Elliott, Siu and Chan (2008) Siu and Yang (2000) and Elliott, Siu and Chan (2008): Use of stochastic optimal control theory to evaluate risk measures for derivatives Bang-Bang type control: Use in Aerospace engineering Paul Wilmott's book on Quantitative Finance and uncertain volatility models widely used in the nance industry

15 Risk Measures in Elliott, Siu and Chan (2008) Consider a nancial model consisting of a bank account B and a share S A continuous-time, N-state observable Markov chain {X(t)} on (Ω, F, P) with state space {e 1, e 2,..., e N }. The price dynamics for B and S under P: db(t) = rb(t)dt, ds(t) = µ(t)s(t)dt + σ(t)s(t)dw (t), where µ(t) := µ, X(t) and σ(t) := σ, X(t) ; µ := (µ 1, µ 2,, µ N ) and σ := (σ 1, σ 2,, σ N ).

16 First Step: Valuation Esscher transform: Esscher (1932), Gerber and Shiu (1994), Siu, Tong and Yang (2004) and Elliott, Chan and Siu (2005) The regime-switching Esscher transform by Elliott, Chan and Siu (2005): 1. Dene a process θ := {θ(t)} by: θ(t) = θ, X(t), where θ = (θ 1, θ 2,..., θ N ). 2. The regime-switching Esscher transform Q θ P associated with θ := {θ(t)}: dq θ dp := G(t) exp( t 0 θ(u)dw (u)) E[exp( t 0 θ(u)dw (u)) F X (t)].

17 Consider a European-style option with payo V (S(T )) at maturity T Given S(t) = s and X(t) = x, a conditional price of the option is given by: V (t, s, x) = E θ [e r(t t) V (S(T )) S(t) = s, X(t) = x]. Proposition 1: Let V i := V (t, s, e i ), for each i = 1, 2,, N, and write V := (V 1, V 2,, V N ) R N. Write A(t) for the rate matrix of the chain at time t. Then, V i, i = 1, 2,, N, satisfy the following system of N-coupled P.D.E.s: rv i + V i t + rs V i s σ2 i s 2 V i s 2 + V, A(t)e i = 0, with terminal conditions V (T, s, e i ) = V (S(T )), i = 1, 2,, N.

18 Second Step: Risk Evaluation For each i = 1, 2,, N, let Λ i = [λ i, λ+ i ]. For example, when N = 2 (i.e. State 1 is Good Economy and State 2 is Bad Economy), λ 1 = 0.05; λ+ 1 = 0.10; λ 2 = 0.01; λ + 2 = Suppose λ(t) is the subjective appreciation rate of the share at time t. The chain modulates λ(t) as: λ(t) = λ, X(t), where λ := (λ 1, λ 2,, λ N ) R N with λ i Λ i, i = 1, 2,, N.

19 Consider, for each λ Θ, a process {θ λ (t)} dened by putting θ λ (t) = N i=1 ( µi λ i σ i ) X(t), e i. The regime-switching Esscher transform P θ λ with respect to {θ λ (t)}: dp θ λ dp := G(t) exp( t 0 θ λ (u)dw (u)) E[exp( t 0 θ λ (u)dw (u)) F X (t)]. P on G(t) Under P θ λ, ds(t) = λ(t)s(t)dt + σ(t)s(t)dw λ (t), where {W λ (t)} is a (G, P θ λ)-standard Brownian motion.

20 Future net loss of the option position over [t, t + h]: V (t, h) := e rh V (t, S(t), X(t)) V (t + h, S(t + h), X(t + h)) Given S(u) = s and X(u) = x, u [t, t + h], the generalized scenario expectation for the option position V over [t, t + h]: ρ(u, s, x) := sup E θλ [exp( r(t + h u)) V (t, h) S(u) = s, X(u) = x], λ Θ where E θλ [ ] is an expectation under P θ λ. Write ρ i := ρ(u, s, e i ), i = 1, 2,, N, and ρ := (ρ 1, ρ 2,, ρ N ).

21 Proposition 2. For each i = 1, 2,, N, let R i := ρ i s { and λ( R λ + i ) = i if R i > 0 λ i if R Then ρ i, i = 1, 2,, N, i < 0. satisfy the following system of N-coupled P.D.E.s: ρ i u + 1 ρ i 2 σ2 i s2 2 s 2 + λ( R i )s ρ i s rρ i + ρ, A(t)e i = 0, with the following terminal conditions: ρ(t+h, S(t+h), e i ) = e rh V (t, S(t), X(t)) V (t+h, S(t+h), e i ). For the case of an American-style option, a system of coupled variational inequalities for the risk measures was obtained.

22 What Next? Incorporate credit risk and counterparty risk in the OTC markets Liquidity risk due to large trading positions Applications to modern insurance products with embedded options Non-Markovian situation: Use of functional Itô± calculus for nonlinear evaluation of dynamic convex risk measures in Siu (2011)

23 References 1. Artzner, P., Delbaen, F., Eber, J. and Heath, D Coherent measures of risk. Mathematical Finance 9 (3), Boyle, P.P., Siu, T.K. and Yang, H Risk and probability measures. Risk 15 (7), Cox, J.C., Ross, S.A. and Rubinstein, M Option pricing: a simplied approach. Journal of Financial Economics 7, Elliott, R.J., Chan, L.L. and Siu, T.K Option pricing and Esscher transform under regime switching. Annals of Finance 1(4), Elliott, R.J., Siu, T.K. and Chan, L.L A P.D.E. Approach for Risk Measures for Derivatives With Regime Switching. Annals of Finance, 4(1), Esscher, F On the probability function in the collective theory of risk, Skandinavisk Aktuarietidskrift, 15,

24 7. Gerber, H.U. and Shiu, E.S.W Option pricing by Esscher transforms (with discussions). Transactions of the Society of Actuaries 46, Siu, T.K. and Yang, H A P.D.E. approach for measuring risk of derivatives. Applied Mathematical Finance 7(3), Siu, T.K., Tong, H. and Yang, H Bayesian risk measures for derivatives via random Esscher transform. North American Actuarial Journal 5(3), Siu, T.K., Tong, H. and Yang, H On pricing derivatives under GARCH models: a dynamic Gerber-Shiu's approach. North American Actuarial Journal 8(3), Siu, T.K Functional Itô's calculus and dynamic convex risk measures for derivative securities. Preprint. 12. Yamai, Y. and Yoshiba, T Comparative analyses of Expected Shortfall and Value-at-Risk: Expected Utility Maximization and Tail Risk.

25 Institute for Monetary and Economic Studies. Bank of Japan. Vol.20, No.2 / April Yang, H. and Siu, T.K Coherent risk measures for derivatives under Black-Scholes economy. International Journal of Theoretical and Applied Finance 4(5),

Pricing Exotic Options Under a Higher-order Hidden Markov Model

Pricing Exotic Options Under a Higher-order Hidden Markov Model Wai-Ki Ching Tak-Kuen Siu Li-min Li 26 Jan. 2007 Abstract In this paper, we consider the pricing of exotic options when the price dynamic