Constructing Markov models for barrier options

Transcription

1 Constructing Markov models for barrier options Gerard Brunick joint work with Steven Shreve Department of Mathematics University of Texas at Austin Nov. 14 th, rd Western Conference on Mathematical Finance UCSB - Santa Barbara, CA

2 Outline Introduction General Mimicking Results Idea of Proof Application to Barrier Options Conclusion

3 Introduction This is really a talk about Markovian projection or constructing Markov mimicking processes. Main point: It often possible to construction Markov processes which mimick properties of more general non-markovian processes. This can be useful for a number of reasons. 1. Difficult and expensive to compute with non-markovian models or models of large dimension 2. To determine the correct nonparametric form for a given application 3. As a tool to understand the general model (calibration) application (which models allow perfect calibration )

4 Introduction Local volatility is a mimicking result. Consider a linear pricing model where the risk-neutral dynamics of the stock price are given by for some process σ. ds t = σ t S t dw t, There is often is local volatility model where the risk neutral dynamics of the stock price are given by: dŝt = σ(t, Ŝt) Ŝt dw t with the same European option prices.

5 Local Volatility Why are local volatility models attractive? simple dynamics low dimensional Markov process general enough to allow for perfect calibration to wide range of option prices Markovian projection - one can use the local volatility model to characterize the set models consistent with a given set of prices

6 The local volatility function σ. Dupire (1994) as well as Derman & Kani (1994) σ 2 (t, x) = T C(t, x) 1 2 x2 2 C(t, x) K 2 Gyöngy (1986), Derman & Kani (1998) as well as Britten-Jones & Neuberger (2000). If σ 2 (t, x) = E [ σt 2 St = x ], then dŝt = σ(t, Ŝt) Ŝt dw t has the some one-dimensional marginal distributions as ds t = σ t S t dw t.

7 Local Volatility The relationship between European option prices and the 1-dimensional risk-neutral marginals of the underlying asset has been understood since at least Breeden and Litzenberger (1978). If C(T, K) denotes the price of a European call option with maturity T and strike K and p(t, x) = P[S t dx], then 2 C(T, K) = 2 K2 = K 2 (x K) + p(t, x) dx δ(x K) p(t, x) dx = p(t, K)

8 Krylov (1984) and Gyöngy (1986) Theorem Let W be an R r -valued Brownian motion, and let X solve where dx t = µ s ds + σ s dw s, 1. µ is a bounded, R d -valued, adapted process, and 2. σ is a bounded, R d r -valued, adapted process such that σσ T is uniformly positive definite (i.e., there exists λ. > 0 with x T σ t σ T t x λ x for all t R + and x R d ).

9 Krylov (1984) and Gyöngy (1986) Theorem If the conditions on the last slide are met by dx t = µ s ds + σ s dw s, then there exists a weak solution to the SDE: d X t = µ(t, X t ) dt + σ(t, X t ) dŵt where 1. µ(t, X t ) = E[ µ t X t ] for Lebesgue-a.e. t, 2. σ σ T (t, X t ) = E[σ t σt T X t ] for Lebesgue-a.e. t, and 3. Xt has the same distribution as X t for each fixed t.

10 General Mimicking Results 1. Given a (non-markov) Ito process it is possible to find a mimicking process which preserves the distributions of a number of running statistics about the process. 2. If futher technical conditions are met, the mimicking Itô process drives a Markov process whose dimension is equal to the number of running statistics. 3. To understand the kinds of running statistics that can be preserved, we need to introduce the notion of an updating function.

11 Some Notation We let C 0 (R + ; R d ) denotes the paths in C(R + ; R d ) that start at zero, and we let denote the map such that : C(R +, R d ) R + C 0 (R +, R d ) u (x, t) = x(t + u) x(t) So (x, t) is the path in C 0 (R +, R d ) that corresponds to the changes x after the time t.

12 Updating Functions Definition Let E be a Polish space, and let Φ : E C 0 (R + ; R d ) C(R + ; E) be a function. We say that Φ is an updating function if 1. x(s) = y(s) for all s [0, t] implies that Φ s (e, x) = Φ s (e, y) for all s [0, t], and 2. Φ t+u (e, x) = Φ u ( Φt (e, x), (x, t) ) t, u R +. If Φ is also continuous as map from E C 0 (R + ; R d ) to C(R + ; E), then we say that Φ is a continuous updating function.

13 Example: Process Itself A trivial updating function: take E = R d, and Φ(e, x) = e + x, e R d, x C0 d, ( so X t = Φ t X0, (X, t) ). The updating property reads X t+u = X t + u (X, t) So Φ t+u is function of Φ t and (X, t).

14 Example: Process and Running Max Let E = {(x, m) R 2 : x m}. x Process position m Maximum-to-date Given x, m E and changes y C 0 (R + ; R d ), we update the current location and current maximum-to-date by: Φ t (x, m; y) = ( ( )) x + y(t), m max x + y(s). 0 s t

15 Example: Process and Running Max If we take M t = max s t X t, then we have Φ t ( X0, X 0 ; (X, 0) ) = (X t, M t ) The second property in the definition of updating function amounts to ( (X t+u, M t+u ) = X t + u (X, t), ( M t max Xt + s (X, t) )) s u So Φ t+u is function of Φ t and (X, t).

16 Example: Entire History Take E = { (x, s) C(R + ; R d ) R + ; x is constant on [s, ) }. Given an initial path segment (x, s) E and changes y C 0 (R + ; R d ), let (x, s) y denote the path obtained by appending y to x after time s: ( (x, s) y ) (t) = {x(t) if t s, and x(s) + y(t s) if t > s. Then Φ t (x, s; y) = ( (x, s) y t, s + t ) is an updating function, where y t is the path y stopped at time t.

17 Example: Entire History With E = { (x, s) C(R + ; R d ) R + ; x is constant on [s, ) }, and Φ t (x, s; y) = ( (x, s) y t, s + t ), we have Φ t (X 0, 0; (X, 0)) = (X t, t), so Φ tracks the whole path history. The updating property amounts to (X t+u, t + u) = ( (X t, t) u (X, t), t + u ), so again Φ t+u is a function of Φ t and (X, t).

18 General Mimicking Result (B. and Shreve) Let Y be a R d -valued process with Y t t 0 µ s ds + t 0 σ s dw s, where W be an R r -valued B.M. and µ and σ be an adapted processes with [ t ] E µ s + σ s σs T ds < t R +, (1) 0 Let E be a Polish space, and let Z be a continuous, E-valued process with Z = Φ(Z 0, Y ) for some continuous updating function Φ. (Z tracks the running statistics of Y that we care about.)

19 General Mimicking Result (B. and Shreve) Then there exists a weak solution to the stochastic system where t t Ŷ t = µ(s, Ẑs) dt + σ(s, Ẑs) dŵs, and 0 0 Ẑ t = Φ(Ẑ0, Ŷ ), 1. µ(t, z) = E[µ t Z t = z] a.e. t, 2. σ σ T (t, z) = E[σ t σ T t Z t = z], a.e. t, and 3. Ẑ t has the same law as Z t for each t.

20 Corollary: Process Itself Suppose X solves dx t = µ t dt + σ t dw t and the integrability condition (1) is satisfied. Then there exists a weak solution to d X t = µ(t, X t )dt + σ(t, X t )dw t where 1. µ(t, x) = E[µ t X t = x] a.e. t, 2. σ σ T (t, x) = E[σ t σt T X t = x], a.e. t, and 3. Xt has the same law as X t for each t.

21 Corollary: Process and Running Max Suppose X solves dx t = µ t dt + σ t dw t, M t = sup s t X s, and the integrability condition (1) is satisfied. Then there exists a weak solution to d X t = µ(t, X t, M t )dt + σ(t, X t, M t )dŵt, M t = max s t X t, where 1. µ(t, x, m) = E[µ t X t, M t = x, m] a.e. t, 2. σ σ T (t, x, m) = E[σ t σt T X t, M t = x, m], a.e. t, and 3. ( X t, M t ) has the same law as (X t, M t ) for each t.

22 Main Idea of Proof Let S be an Itô process S that solves ds t = σ t S t dw t. We construct processes S 1, S 2, and S 3 on some space with L (S 1 ) = L (S 2 ) = L (S 3 ) = L (S). We then piece these processes together to form a process S with L ( S t ) = L (S t ) for all t.

23 Main Idea of Proof Suppose S solves ds t = σ t S t dw t.

24 Main Idea of Proof Let L (S 1 ) = L (S).

25 Main Idea of Proof Forget everything about S 1 except S 1 t 1.

26 Main Idea of Proof Let L (S 2 S 1 t 1 ) = L (S S t1 =S 1 t 1 ).

27 Main Idea of Proof Let L (S 2 St 1 1 ) = L (S S t1 =St 1 1 ). Taking any measurable A C(R + ; R), notice that P[S 2 A] = P[S 2 A S 1 t 1 = x] P[S 1 t 1 dx] R = P[S A S t1 = x] P[S t1 dx] R = P[S A]. In particular, S 2 is distributed according to L (S).

28 Main Idea of Proof Let L (S 2 S 1 t 1 ) = L (S S t1 =S 1 t 1 ).

29 Main Idea of Proof Forget everything about S 2 except S 2 t 2.

30 Main Idea of Proof Let L (S 3 S 1 t 2 ) = L (S S t2 =S 1 t 2 ).

31 Main Idea of Proof Set Ŝ S1 1 [0,t1 ) + S 2 1 [t1,t 2 ) + S 3 1 [t2, ).

32 Main Idea of Proof This still works when we track additional information.

33 Main Idea of Proof Let L (S 1 ) = L (S).

34 Main Idea of Proof Forget everything about S 1 except S 1 t 1 and M 1 t 1.

35 Main Idea of Proof Let L (S 2 S 1 t 1, M 1 t 1 ) = L (S S t1 =S 1 t 1, M t1 =M 1 t 1 ).

36 Main Idea of Proof Set S S 1 1 [0,t1 ) + S 2 1 [t1, ).

37 General Mimicking Result (B. and Shreve) Then there exists a weak solution to the stochastic system where t t Ŷ t = µ(s, Ẑs) dt + σ(s, Ẑs) dŵs, and 0 0 Ẑ t = Φ(Ẑ0, Ŷ ), 1. µ(t, z) = E[µ t Z t = z] a.e. t, 2. σ σ T (t, z) = E[σ t σ T t Z t = z], a.e. t, and 3. Ẑ t has the same law as Z t for each t.

38 Example: Barrier Options Definition Given an exercise time, T, an upper barrier, U, and strike, K, the holder of an up-and-out call option has the right to exercise a call option at time T with strike K if the stock price has remained below the barrier U. If the stock price crosses the barrier, the option becomes worthless. Calibration Problem Given a collection {B(T, U, K)} T,U,K of prices for up-and-out call options, we would like to construct a linear pricing model which is consistent with these prices.

39 Example: Barrier Options Previous results suggest that we may want to look for a (risk-neutral) model of the form: with σ choosen so that ds t = σ(t, S t, M t )S t dw t M t = max s t S t, E [ 1 {MT U} (S T K) +] = B(T, L, K).

40 Dupire Formula Formally, we may recover σ from the prices of corridor options with a Dupire-type formula. So B(T, K, U) = E Q[ 1 {MT U}(S T K) +] B(T, K, U) = E Q[ δ U (M T )(S T K) +] U 2 B(T, K, U) = E Q[ 1 T U 2 σ2 (T, K, U)K 2 δ U (M T )δ K (S T ) +] 3 B(T, K, U) K 2 = E Q[ δ U (M T )δ K (S t ) ] U σ 2 (T, K, U) = 2 2 B(T, K, U)/ T U 3 B(T, K, U)/ K 2 U

41 Markov Property Theorem Let E be a Polish space and let Φ be a continuous updating function Φ. Consider the stochastic differential equation: Ŷ t = t t 0 µ(s, Ẑs) dt + Ẑ t = Φ(Ẑt 0, Ŷ ). t t 0 σ(s, Ẑs) dŵs, and If weak uniqueness holds for each initial condition Z t0 = z 0 E, then the process Z is strong Markov.

42 Markov Property Corollary Suppose σ is Lipshitz continuous, then weak uniqueness holds for the stochastic differential equation ds t = σ(t, S t, M t ) dw t M t = max s t S t, and the process Z = (S, M) is strong Markov.

43 Conclusions It is often possible to construct reduced form models which preserve the prices of path-dependent options. Weak uniqueness results allow one to conclude that the reduced form models are Markov.

44 Open Question? Let σ be continuous with 1/C σ C for some constant C. Is this sufficient to ensure weak uniqueness for the stochastic differential equation: dx t = σ(t, X t, M t ) dw t M t = max s t X t?

45 References I M. Britten-Jones and A. Neuberger. Option prices, implied price processes, and stochastic volatility. The Journal of Finance, 55(2): , E. Derman and I. Kani. Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility. International Journal of Theoretical and Applied Finance, 1(1):61 110, B. Dupire. Pricing with a smile. Risk, 7(1):18 20, 1994.

46 References II I. Gyöngy. Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probability Theory and Related Fields, 71(4): , N. V. Krylov. Once more about the connection between elliptic operators and Itôs stochastic equations. Statistics and Control of Stochastic Processes, Steklov Seminar, pages , 1984.