A digression The problem Ideas and questions Results Towards the end Modelling option prices Martin Schweizer Department of Mathematics ETH Z¨ urich Workshop and Mid-Term Conference on A dvanced Ma thematical Me thods for F inance Vienna, September 17-22, 2007 18.09.2007 based on joint work with Johannes Wissel (ETH Z¨ urich) Martin Schweizer Modelling option prices
A digression The problem Ideas and questions Results Towards the end Call for Papers for a Special Issue of Finance and Stochastics “Computational Methods in Finance” http://www.math.ethz.ch/ ∼ finasto Martin Schweizer Modelling option prices
A digression The problem The problem Ideas and questions Martingale models ? Results Market models Towards the end The problem Martin Schweizer Modelling option prices
A digression The problem The problem Ideas and questions Martingale models ? Results Market models Towards the end The problem Basic goal: Construct a joint dynamic model for a stock S a bond ( ≡ 1 in discounted units) several call options C ( K , T ) on S in such a way that the model is arbitrage-free, gives explicit joint dynamics of stock and calls, is (perhaps) practically usable. Where is the problem ? Martin Schweizer Modelling option prices
A digression The problem The problem Ideas and questions Martingale models ? Results Market models Towards the end Martingale models ? Write down model only for S , directly under a pricing measure Q . Define C t ( K , T ) := E Q [( S T − K ) + |F t ]. This martingale model is obviously arbitrage-free. But . . . . . . typically no explicit expressions for C t ( K , T ) . . . . . . hence no explicit joint dynamics for S and C . . . . . . and good calibration can be difficult. This is no solution to our problem ! So what now ??? Martin Schweizer Modelling option prices
A digression The problem The problem Ideas and questions Martingale models ? Results Market models Towards the end Market models Basic idea: specify dynamics (SDEs) for all tradable assets . Joint model for S and all C ( K , T ) with K ∈ K and T ∈ T . Advantages: we know joint dynamics of S and C by construction. calibration is automatic since market prices of options at time 0 are input as initial conditions. But: must observe arbitrage restrictions: No static arbitrage − → restrictions on state space of processes . No dynamic arbitrage − → restrictions on SDE coefficients ( drift restrictions ` a la HJM). How to handle these constraints ? Martin Schweizer Modelling option prices
A digression The simplest example The problem Multiple maturities Ideas and questions Problems with multiple maturities Results Multiple strikes Towards the end Ideas and questions Martin Schweizer Modelling option prices
A digression The simplest example The problem Multiple maturities Ideas and questions Problems with multiple maturities Results Multiple strikes Towards the end The simplest example Consider one stock S and one call C ( K , T ). Restrictions are ( S t − K ) + ≤ C t ≤ S t and C T = ( S T − K ) + . static: dynamic: S and C both martingales under some Q ≈ P . How to write down explicit SDE for ( S , C ) satisfying this ??? Way out ( − → Lyons 1997, Babbar 2001): reparametrize: Instead of C t , use implied volatility ˆ σ t via � σ 2 � C t = c BS S t , K , ( T − t )ˆ . t σ 2 more precisely: work with V t := ( T − t )ˆ t . static arbitrage constraint is equivalent to 0 ≤ V t < ∞ and V T = 0; so state space is nice. dynamic arbitrage constraint reduces to drift restriction for V in SDE model for ( S , V ). SDE is still tricky (nonlinear), but feasible; explicit examples. Martin Schweizer Modelling option prices
A digression The simplest example The problem Multiple maturities Ideas and questions Problems with multiple maturities Results Multiple strikes Towards the end Multiple maturities Now consider one stock S and many calls C ( K , T ) with one fixed strike K and maturities T ∈ T . Use new parametrization: forward implied volatilities defined by ∂ ∂ � σ 2 � X t ( T ) := ( T − t )ˆ t ( K , T ) = ∂ T V t ( T ) . ∂ T static arbitrage constraints are equivalent to (i) V t ( T ) ≥ 0 and V T ( T ) = 0 as before. (ii) T �→ V t ( T ) is increasing, i.e., 0 ≤ X t ( T ) < ∞ . So: state space is nice. dynamic arbitrage constraints reduce to drift restrictions for � � all X ( T ) in SDE model for S , X ( T ) T ∈T . − → Sch¨ onbucher 1999, Brace et al. 2001, Ledoit et al. 2002 Martin Schweizer Modelling option prices
A digression The simplest example The problem Multiple maturities Ideas and questions Problems with multiple maturities Results Multiple strikes Towards the end Problems with multiple maturities I Structure of model: start with dS t = S t µ t dt + S t σ t dW t , dX t ( T ) = α t ( T ) dt + v t ( T ) dW t . Dynamic arbitrage constraints: ( S t ) and all � T � � C t ( T ) = c BS S t , K , X t ( s ) ds t must be (local) martingales under some Q ≈ P . Drift restrictions: µ t = − σ t b t , X t ( t ) , S t , v t ( . ) � � σ t = f , X t ( T ) , S t , v t ( . ) , X t ( . ) � � α t ( T ) = g . Martin Schweizer Modelling option prices
A digression The simplest example The problem Multiple maturities Ideas and questions Problems with multiple maturities Results Multiple strikes Towards the end Problems with multiple maturities II Structure of model with drift restrictions: X t ( t ) , S t , v t ( . ) � � dS t = S t f ( dW t − b t dt ) , X t ( T ) , S t , v t ( . ) , X t ( . ) � � dX t ( T ) = g dt + v t ( T ) dW t . Recall: specifying a joint model means that we want to choose the volatility structure v t ( T ) in some way. f and g are nonlinear ; so even if v is Lipschitz and of linear growth, dt -coefficients and σ are not ! Existence problem for (infinite, nonlinear) SDE system ! No results in the literature (except classical HJM, with severe conditions: bounded and Lipschitz). Martin Schweizer Modelling option prices
A digression The simplest example The problem Multiple maturities Ideas and questions Problems with multiple maturities Results Multiple strikes Towards the end Multiple strikes: even more problems Next consider one stock S and many calls C ( K , T ) with one fixed maturity T and strikes K ∈ K . Arbitrage constraints: dynamic: as usual some drift restrictions. static: K �→ C t ( K , T ) is convex and satisfies ∂ − 1 ≤ ∂ K C t ( K , T ) ≤ 0 . state space for C ( K , T ) very complicated. using (classical or forward) implied volatilities does not help either. Before even thinking about SDEs: How to choose parametrization ?? Martin Schweizer Modelling option prices
A digression The problem Infinite SDE systems Ideas and questions Multiple maturities Results Multiple strikes Towards the end Results Martin Schweizer Modelling option prices
A digression The problem Infinite SDE systems Ideas and questions Multiple maturities Results Multiple strikes Towards the end Infinite SDE systems Key mathematical tool: consider SDE system θ, X . ( . ) θ, X . ( . ) � � � � dX t ( θ ) = A t dt + B t dW t � � with θ ∈ Θ usually [0 , T ∗ ] or [0 , ∞ ) and 0 ≤ t ≤ T 0 . − → J. Wissel 2006: Existence and uniqueness result for strong solution under only local Lipschitz-type conditions on A , B . Includes sufficient conditions on growth for non-explosion. Key idea: work on product space Θ × Ω. Important: global Lipschitz condition is too strong for the required applications. Martin Schweizer Modelling option prices
A digression The problem Infinite SDE systems Ideas and questions Multiple maturities Results Multiple strikes Towards the end Multiple maturities SDE system with drift restrictions is X t ( t ) , S t , v t ( . ) � � dS t = S t f ( dW t − b t dt ) , X t ( T ) , S t , v t ( . ) , X t ( . ) � � dX t ( T ) = g dt + v t ( T ) dW t . Theorem: Sufficient conditions on v ( volatility structure of forward implied volatilities) for existence and uniqueness of solution. Not direct from general SDE results, because only dW -coefficient v can be chosen here. in addition, must have X ≥ 0. Classes of explicit examples for such models, for first time in literature. − → S/Wissel 2006 Martin Schweizer Modelling option prices
A digression The problem Infinite SDE systems Ideas and questions Multiple maturities Results Multiple strikes Towards the end Explicitly: � T − 1 �� Z t ( T ) − 1 1 � � 2 − α t ( T ) = R t ( T ) v t ( T ) · v t ( s ) ds 2 4 t � X t ( T ) � T 2 � � + 1 �� � 2 − 1 1 � � R t ( T ) v t ( s ) ds � � 2 2 Z t ( T ) Z t ( T ) � � t � R t ( T ) − 1 � σ t v 1 + t ( T ) 2 � T − R t ( T ) X t ( T ) v 1 t ( s ) ds − b t · v t ( T ) Z t ( T ) σ t t with � T R t ( T ) := Y t ( t ) − log K Y t ( t ) := log S t , , Z t ( T ) := X t ( s ) ds . Z t ( T ) t Martin Schweizer Modelling option prices
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