Drawdowns, Drawups, their joint distributions, detection and financial risk management

Transcription

1 Drawdowns, Drawups, their joint distributions, detection and financial risk management June 2, 2010

2 The cases a = b The cases a > b The cases a < b Insuring against drawing down before drawing up Robust replication Semi-robust hedges

3 Motivation The price of a stock or index is fluctuate, and may have a big drop or a big rally over a period [0,T]. The present decrease from the historical high The present increase over the historical low S&P500, 2007 Now Historical High SP500 Historical Low Jan Dec Dec Nov 2009

4 Mathematical Definitions A stochastic process {X t ;t 0}. Its drawdown and drawup processes. DD t = supx s X t, DU t = X t inf X s. s t s t Drawdowns and drawups. T D (a) = inf{t 0 DD t a}, T U (b) = inf{t 0 DU t b}. The goal: characterize the probability P x (T D (a) T U (b) T).

5 The cases a = b The cases a > b The cases a < b The first range time ρ(a) = T D (a) T U (a) sup s<=t X s X t inf s<=t X s 0.05 X t ρ(0.3) Time t

6 The cases a = b The cases a > b The cases a < b Probability distribution On {T D (a) T U (a)}, X TD (a) [ a,0). It suffices to determine P 0 (T D (a) dt,t U (a) > t,x t du), a u < 0. Connection with the hitting probability P 0 (T D (a) dt,t U (a) > t,x t du) = P 0 (τ u dt,supx s du + a) s t = a P 0(τ u dt,supx s < u + a)du s t = a P 0(τ u dt,τ u+a > t)du. We have a closed-form formula for P 0 (T D (a) T U (a) T) under drifted Brownian motion dynamics. (RW is similar)

7 The cases a = b The cases a > b The cases a < b Laplace transform under general diffusion dynamics: a = b Consider a linear diffusion X on I = (l,r) with continuous generator coefficients and natural (or entrance) boundaries. The goal: Laplace transform E x { e λt D (a) 1I {TD (a)<t U (a)}}. For a + x u < x, λ > 0 with X 0 = x, Lx X (λ,u;a,a)du { } E x e λt D (a) 1I {TD (a)<t U (a),x TD (a) du} = a E { } x e λτu 1I {sups τu X s<u+a} du = a E x { } e λτu 1I {τu <τ u+a} du. The last conditioned Laplace transform of first hitting time is known through solutions of an ODE.

8 The cases a = b The cases a > b The cases a < b The conditioned Laplace transform of first hitting time Consider the SDE governing the linear diffusion X dx t = µ(x t )dt + σ(x t )dw t, X 0 = x. For l < L x H < r and λ > 0, (Lehoczky 77 ) { } E x e λτl g λ (x)h λ (H) g λ (H)h λ (x) 1I {τl <τ H } = g λ (L)h λ (H) g λ (H)h λ (L), where g λ and h λ are any two independent solutions of the ODE 1 2 σ 2 (x) 2 f f + µ(x) x2 x = λf. For constant parameter case (X is a drifted Brownian motion), g λ and h λ are exponential functions.

9 The cases a = b The cases a > b The cases a < b Path decomposition If a > b, the strong Markov property of linear diffusion facilitate the use of Laplace transform and path decomposition. For any path in {T D (a) < T U (b)} 1. {X t ;0 t T D (b)} Range process. 2. {X t+td (b) X TD (b);0 t T D (a) T D (b)} Hitting time with drawup constraint. T D (a) = T D (b) + τ XTD (b)+b a θ TD (b).

10 The cases a = b The cases a > b The cases a < b Illustration of a Brownian sample path with a = 0.4,b = Brownian motion X t =0.1t+0.2W t, t [0,1]; T D (0.4) 1<T U (0.2) sup s<=t X s X t inf s<=t X s 0.05 X t P T (0.2) D Time t

11 The cases a = b The cases a > b The cases a < b Illustration of a Brownian sample path with a = 0.4,b = Brownian motion X =0.1t+0.2W, t [0,1]; T t t (0.4) 1<T D U (0.2) 1 sup s<=t X s X t inf s<=t X s 0.05 X t P Q T (0.2) D T D (0.4) Time t

12 The cases a = b The cases a > b The cases a < b Laplace transform a > b Recall that on {T D (a) < T U (b)} with a > b, T D (a) = T D (b) + τ XTD (b)+b a θ TD (b) Conditioning on {X TD (b) = u}, E x { e λτ u+b a θ TD (b) 1I {τu+b a θ TD (b)<t U (b) θ TD (b)} X TD (b) = u } = E u { e λτ u+b a 1I {τu+b a <T U (b)}}. For a + x < u < x, λ > 0 with X 0 = x, Lx X (λ,u;a,b)du { } E x e λt D (a) 1I {TD (a)<t U (b),x TD (b) du} = Lx X { } (λ,u;b,b) E u e λτ u+b a 1I {τu+b a <T U (b)} du. }{{} The strong Markov property

13 The cases a = b The cases a > b The cases a < b The strong Markov property and discrete approximation Brownian motion X t =0.1t+0.2W t, t [T D (0.2),T D (0.4)] 0.1 Conditioning on {X TD (b) = u}, partition the interval [u a + b,u] into n subintervals with equal length = (a b)/n. Use conditioned hitting times to approximate. Pass to the limit. The continuity of the sample path and bounded convergence theorem justifies this u X t u Δ+b u 2Δ+b u 3Δ+b P u 4Δ+b 0.15 u Δ 0.2 u 2Δ 0.25 u 3Δ Q 0.3 T D (0.2) u 4Δ T D (0.4) Time t

14 The cases a = b The cases a > b The cases a < b Path decomposition Relationship between Laplace transforms { } E x e λt D (a) 1I {TD (a)<t U (b)} { = E x e λt D (a) } { } E x e λt D (a) 1I {TD (a)>t U (b)} To get the very last Laplace transform, observe that on {T D (a) > T U (b)}, T D (a) = T U (b) + T D (a) θ TU (b). Using strong Markov property of linear diffusion E x { e λt D (a) θ TU (b) X TU (b)} = EXTU (b){ e λt D (a) }. We can use reflection to find E x { e λt U (b) 1I {TD (a)>t U (b),x TU (b) du}}.

15 The cases a = b The cases a > b The cases a < b Illustration of a Brownian sample path with a = 0.15,b = sup s<=t X s Brownian motion X t =0.1t+0.2W t, t [0,1]; T D (0.3) 1<T U (0.15) X t inf s<=t X s P 0.15 X t T U (0.3) Time t

16 The cases a = b The cases a > b The cases a < b Illustration of a Brownian sample path with a = 0.15,b = Brownian motion X t =0.1t+0.2W t, t [0,1]; T D (0.3) 1<T U (0.15) 1 sup X s<=t s 0.25 X t inf s<=t X s P X t Q T U (0.3) T D (0.15) Time t

17 Detection of two-sided alternatives We sequentially observe a process {ξ t } with the following dynamics: dw t t < τ dx t = α(x t )dt + σ(x t )dw t or α(x t )dt + σ(x t )dw t T t τ probability of misidentification P 0,+ x (T D (a) < T U (b) T) = = 0 0 Px 0,+ (T D (a) < T U (b) t) λe λt dt e λt Px 0,+ (T D (a) dt,t U (b) > t)dt = L X0,+ x (λ;a,b), (1)

18 Detection of two-sided alternatives(cont) aggregate probability of misidentification P τ,+ y (T D (a) θ(τ) < T U (b) θ(τ) T)f Xτ (y x)dy (2) = Ly X0,+ (λ,a,b)f Xτ (y x)dy,

19 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Digital call insurance Financial assets are risky S&P500, 2007 Now Historical High SP500 Historical Low Jan Dec Dec Nov 2009 A digital call that pays 1I {TD (K) T U (K) T} can be perceived as an insurance against adverse movement in the market.

20 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Pricing and replication The previously defined digital call only pays out one dollar (compensation) if the price process X draws down by K dollars before it draws up by the equal amount. Under no transaction cost and no arbitrage, the price of an option with payment at time T is just the expectation of the discounted cashflow in the future. Let B t (T) be the price of a bond maturing at T, consider its equivalent martingale measure Q T t. The arbitrage-free price of the previously defined digital call at time t is DCt D<U (K,T) = B t (T)Q T t (T D (K) T U (K) T). In simple models (e.g., constant parameters market model), the previous work computes the price at time 0. The contribution of the work: develop replication strategy to hedge the risk of the above digital call.

21 Insuring against drawing down before drawing up Robust replication Semi-robust hedges The Laplace Transform Approach Laplace transform (FFT) pricing formula ] E QT S 0 [e λt (S0 +K) D(K) 1(T D (K) T U (K)) = f (H K,H,λ)dH, S 0 where f (L,H,λ) = [ ] H EQT S 0 e λτs L 1(τL S τs H ) for L < H. Back to time domain What about the replication at t > 0? Q T S 0 {T D (K) T U (K) T} (S0 +K) = H QT S 0 {τ H K < τ H T}dH. S 0

22 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Model-free Decomposition Let X t, M t and m t be spot price, the historical high and the historical low at time t [0,T] of the underlying, respectively. On any path in the event {T D (K) T U (K) T}, at t < T D (K) T U (K), If the spot does not reach a new high by T D (K), M TD (K) = M t. Otherwise, M TD (K) (M t,m t + K). Replicate payoff based on the historical high when there is a crash: M TD (K) 1I {TD (K) T U (K) T} =1I {TD (K)=τ MTD (K) K T,M TD (K) [M t,m t +K)} =1I {τm t K T,M τm t K =M t} (mt +K) + 1I M t + {τh K T}δ(M τh K H)dH. Find instruments with desired payoffs.

23 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Hedging instruments An one-touch knockout is a double barrier digital option with a (low) in-barrier L and a (high) out-barrier H, the price of this options at time t before its maturity date T is OTKO t (L,H,T) =B t (T)Q T t (τ L τ H T) =B t (T)Q T t (τ L T,M τl < H). The payoff indicator of an one-touch knockout can be modified OTKO t (L,H +,T) = B t (T)Q T t (τ L T,M τl H). A touch-upper-first down-and-in claim is a spread of one-touch knockouts. It has a low barrier L and a high barrier H. TUFDI t (L,H,T) = lim ε 0 + OTKO t (L,H + ε,t) OTKO t (L,H,T) ε =B t (T)Et QT [1I {τl T}δ(M τl H)], which pays one dollar at expiry if and only if the spot touches the upper barrier H and then hits L from above before T.

24 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Semi-static replication of one-touch knockouts Although the previous replication is fairly robust (no model assumption), the instruments used are rather exotic. Under skip-freedom and symmetry assumption, any one-touch knockout can be replicated by single barrier one-touch options: OT t (B,T) = B t (T)Q T t (τ B T). We assume that the barriers of an one-touch knockout are skip-free, and when X exit the corridor (L,H), the risk-neutral probability of hitting X t before T is the same as the risk-neutral probability of hitting X t + before T, for any 0. This is satisfied by dx t = α t dw t with dα t dw t = 0. Then,...

25 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Replication of One-touch Knockouts under Arithmetic Symmetry We can show that at t [0,τ L τ H T] { } OTKO t (L,H,T) = OT t (H (2n + 1),T) OT t (H + (2n + 1),T), n=0 where = H L. A sketched proof. If the spot hits L first, t L H S T L H S 0

26 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Replication of One-touch Knockouts under Arithmetic Symmetry We can show that at t [0,τ L τ H T] { } OTKO t (L,H,T) = OT t (H (2n + 1),T) OT t (H + (2n + 1),T), n=0 where = H L. A sketched proof. If the spot hits H first, t L H H + S T L H S 0

27 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Replication of One-touch Knockouts under Arithmetic Symmetry We can show that at t [0,τ L τ H T] { } OTKO t (L,H,T) = OT t (H (2n + 1),T) OT t (H + (2n + 1),T), n=0 where = H L. A sketched proof. If the spot hits L first, t L 2 L H H + S T L H S 0

28 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Replication of One-touch Knockouts under Arithmetic Symmetry We can show that at t [0,τ L τ H T] { } OTKO t (L,H,T) = OT t (H (2n + 1),T) OT t (H + (2n + 1),T), n=0 where = H L. A sketched proof. If the spot hits H first, t L 2 L H H + H + 3 S T L H S 0

29 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Semi-Robust Replication of Digital Call on Maximum Drawdown (Carr) The maximum drawdown MD T = sup s [0,T] DD s, is commonly used as a measure of the risk of holding the underlying asset over a period [0,T]. A risk adverse investor or a portfolio manager can get protection against a loss from the market if he or she holds a claim which pays 1(MD T K), for some strike K > 0. The maximum drawdown and the maximum drawup over a period [0,T] are related to two stopping times: for K > 0 T D (K) = inf{t 0,DD t K}, T U (K) = inf{t 0,DU t K}. Let us denote by MU T = sup s [0,T] DU s, then {MD T K} = {T D (K) T}, {MU T K} = {T U (K) T}. Introduce another digital call for K > 0: Digital call on maximum drawdown DCt MD (K,T) = B t (T)Q T t {T D (K) T}.

30 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Replication of Digital Call on Maximum Drawdown (Carr) Under the above CAHS assumption, a digital call on maximum drawdown can be replicated with double-one-touches (DOT): DCt MD (K,T) := B t (T)1 {TD (K) t} + 1 {TD (K)>t}DOT t (M t K,M t + K,T). A double-one-touch is a double barrier digital option with a high barrier H and a low barrier L, the price of this option at time t before its maturity date T is DOT t (L,H,T) = B t (T)Q T t {τ S L τs H T}. In the Bachelier model, using Lévy isomorphism, we have ( law sup W s W t = Wt sup s [0,t] t [0,T] ) law sup W s W t = sup W t, s [0,t] t [0,T] where W is a standard Brownian motion starting at 0. A double-one-touch can be replicated with two one-touch knockouts: DOT t (L,H,T) = OTKO t (L,H,T) + OTKO t (H,L,T).

31 Insuring against drawing down before drawing up Robust replication Semi-robust hedges Remark The payoff of a digital call on the drawdown of K preceding the drawup of equal size can be semi-statically replicated with one-touches under arithmetic symmetry assumption. The replication can also be done with vanilla options (payoff only depends on the value of the stock at maturity). This is the reflection principle: If H > M t OT t (H,T) = B t (T)Q T t (τ H T) = 2B t (T)Q T t (S T H). We also developed replicating strategies under geometric symmetry. In particular, under the Black-Scholes model ds t = rs t dt + σs t dw t and its independent time-changes S βt (β t is a continuous increasing process and dβ t dw t = 0), the replicating strategies work well.

32 Hongzhong Zhang* Peter Carr Libor Pospisil Jan Vecer

33 The joint distribution of drawdown and drawup: Zhang, H., Hadjiliadis, O.: Drawdowns and rallies in a finite time horizon, accepted by Methodology and Computing in Applied Probability, special issue (2010). Pospisil, L., Vecer, J., Hadjiliadis, O.: Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups, Stochastic processes and their applications 119(8), Zhang, H., Hadjiliadis, O.: Formulas for the Laplace transform of stopping times based on drawdowns and drawups, submitted to Annals of Applied Probability on , revised on Maximum drawdown protection: Carr, P., Zhang, H., Hadjiliadis, O.: Insuring against maximum drawdown and drawing down before drawing up, to be submitted to Finance and Stochastics.

34 Summary We study drawdown and drawup processes in this work. The probability that a drawdown of size a precedes a drawup of size b is fully characterized for biased simple random walk, drifted Brownian motion and more general linear diffusion with continuous generator coefficients. Digital insurance can be considered in terms of drawdowns and drawups. Pricing can be done analytically for classical models. Robust and semi-robust replicating strategies of the digital insurance are developed. We considered both the arithmetic symmetry and the more involved geometric symmetry in the paper. These strategies are robust to independent continuous time-changes. The drawup of log-likelihood ratio process has optimal property when used as a means of detecting abrupt changes. We proved the asymptotic optimality of N-CUSUM stopping rule in the multi-source observation setting. In the paper we considered both the Brownian motion system and the discrete-time observation system.