Constructing Markov models for barrier options Gerard Brunick joint - PowerPoint PPT Presentation

Constructing Markov models for barrier options Gerard Brunick joint work with Steven Shreve Department of Mathematics University of Texas at Austin Nov. 14 th , 2009 3 rd Western Conference on Mathematical Finance UCSB - Santa Barbara, CA

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

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”)

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

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

The local volatility function � σ . ◮ Dupire (1994) as well as Derman & Kani (1994) ∂ ∂T C ( t, x ) σ 2 ( t, x ) = � 2 x 2 ∂ 2 1 ∂K 2 C ( t, x ) ◮ Gy¨ ongy (1986), Derman & Kani (1998) as well as Britten-Jones & Neuberger (2000). If � � � � S t = x σ 2 ( t, x ) = E σ 2 � , t then d � σ ( t, � S t ) � S t = � S t d W t has the some one-dimensional marginal distributions as d S t = σ t S t d W t .

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 ∈ d x ] , then � ∂ 2 ∂ 2 ( x − K ) + p ( T, x ) d x ∂K 2 C ( T, K ) = ∂K 2 � = δ ( x − K ) p ( T, x ) d x = p ( T, K )

Krylov (1984) and Gy¨ ongy (1986) Theorem Let W be an R r -valued Brownian motion, and let X solve d X t = µ s d s + σ s d W s , where 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 t x ≥ λ � x � for all t ∈ R + and x ∈ R d ). x T σ t σ T

Krylov (1984) and Gy¨ ongy (1986) Theorem If the conditions on the last slide are met by d X t = µ s d s + σ s d W s , then there exists a weak solution to the SDE: d � µ ( t, � σ ( t, � X t ) d � X t = � X t ) d t + � W t where 1. � µ ( t, X t ) = E [ µ t | X t ] for Lebesgue-a.e. t , σ T ( t, X t ) = E [ σ t σ T 2. � σ � t | X t ] for Lebesgue-a.e. t , and 3. � X t has the same distribution as X t for each fixed t .

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 Itˆ o 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.

Some Notation We let C 0 ( R + ; R d ) denotes the paths in C ( R + ; R d ) that start at zero, and we let ∆ : C ( R + , R d ) × R + → C 0 ( R + , R d ) denote the map such that ∆ 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 .

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 Φ t ( 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 .

Example: Process Itself A trivial updating function: take E = R d , and e ∈ R d , x ∈ C d Φ( e, x ) = e + x, 0 , � � so X t = Φ t X 0 , ∆( 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 ) .

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

Example: Process and Running Max If we take M t = max s ≤ t X t , then we have � � Φ t X 0 , 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 X t + ∆ s ( X, t ) s ≤ u So Φ t + u is function of Φ t and ∆( X, t ) .

Example: Entire History Take � � ( x, s ) ∈ C ( R + ; R d ) × R + ; x is constant on [ s, ∞ ) E = . 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 ( t ) if t ≤ s , and ( x, s ) ⊕ y ( t ) = x ( s ) + y ( t − s ) if t > s . � � ( x, s ) ⊕ y t , s + t Then Φ t ( x, s ; y ) = is an updating function, where y t is the path y stopped at time t .

Example: Entire History With � � ( x, s ) ∈ C ( R + ; R d ) × R + ; x is constant on [ s, ∞ ) E = , and � � ( x, s ) ⊕ y t , s + t Φ t ( x, s ; y ) = , 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 ) .

General Mimicking Result (B. and Shreve) Let Y be a R d -valued process with � t � t Y t � µ s d s + σ s d W s , 0 0 where W be an R r -valued B.M. and µ and σ be an adapted processes with � � t � � µ s � + � σ s σ T s � d s < ∞ ∀ t ∈ R + , (1) E 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.)

General Mimicking Result (B. and Shreve) Then there exists a weak solution to the stochastic system � t � t � µ ( s, � σ ( s, � Z s ) d � Y t = Z s ) d t + W s , and � � 0 0 Z t = Φ( � � Z 0 , � Y ) , where 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. � Z t has the same law as Z t for each t .

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

Corollary: Process and Running Max Suppose X solves d X t = µ t d t + σ t d W t , M t = sup s ≤ t X s , and the integrability condition (1) is satisfied. Then there exists a weak solution to d � µ ( t, � X t , � σ ( t, � X t , � M t )d � X t = � M t )d t + � W t , � � M t = max X t , s ≤ 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 σ T 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 .

Main Idea of Proof Let S be an Itˆ o process S that solves d S t = σ t S t d W 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 .

Main Idea of Proof Suppose S solves d S t = σ t S t d W t .

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

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

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

Main Idea of Proof Let L ( S 2 | S 1 t 1 ) = L ( S | S t 1 = S 1 t 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 ∈ d x ] R � = P [ S ∈ A | S t 1 = x ] P [ S t 1 ∈ d x ] R = P [ S ∈ A ] . In particular, S 2 is distributed according to L ( S ) .

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

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