CS476/676 Mar 6, Today s Topics. American Option: early exercise curve. PDE overview. Discretizations. Finite difference approximations

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1 CS476/676 Mar 6, Today s Topics American Option: early exercise curve PDE overview Discretizations Finite difference approximations

2 CS476/676 Mar 6, American Option American Option: PDE Complementarity problem V t σ2 S 2 2 V S 2 +(r q)s V S V(S,t) payoff(s) 0 rv 0, (V(S,t) payoff(s))( V t σ2 S 2 2 V S 2 +(r q)s V S rv) = 0 0 < S < +, 0 t < T and V(S,T) = payoff(s).

3 CS476/676 Mar 6, The early exercise boundary {S t} is implicitly determined. Assume that V(S, t) solves the complementarity problem. For a put, S t = max{s : V(S,t) = payoff(s)}. For a call (with dividend yield), S t = min{s : V(S,t) = payoff(s)}.

4 CS476/676 Mar 6, Pricing European Options: PDE Approach Assume ds t = µs t dt+σ(s t,t)s t dz t (assume no dividend for simplicity) Any European option value V(S, t) satisfies V t σ2 S 2 2 V S +(r q)s V rv = 0, 2 S 0 < S < +, 0 t < T A PDE has an infinite number of solutions. It needs proper combinations of initial/final conditions to uniquely determine a solution. Numerically, boundary conditions are also required.

5 CS476/676 Mar 6, Assume that σ,r,q are constants. Consider transformations: S = Ke x, t = T 2τ σ 2, ˆδ = 2r 2(r q) σ2, δ = σ 2, ( V(S,t) = V Ke x,t 2τ ) def = v(x,τ) σ 2 v(x,τ) = Ke 1 2 (δ 1)x (1 4 (δ 1)2 +ˆδ)τ u(x,τ)

6 CS476/676 Mar 6, It can be shown that the BS PDE is equivalent to which is called a heat equation. u τ = u xx It models the diffusion of heat on an (uniform) infinite bar from an initial temperature distribution. The solution to a heat equation is infinitely smooth, even if initially the distribution is discontinuous. European option value function V(S, t) is infinitely smooth

7 CS476/676 Mar 6, A partial differential equation with two independent variables can be written as Au xx +Bu xy +Cu yy +F(u x,u y,u,x,y) = 0 Note. Subsequently subscripts denote partial derivatives with respect to the independent variables. Parabolic: B 2 4AC = 0 (BS Eqns) Hyperbolic: B 2 4AC > 0 Elliptic: B 2 4AC < 0 Different types of PDE have different properties.

8 CS476/676 Mar 6, The heat equations (thus BS equation) has an explicit solution. But for a LVF model and American options, numerical methods based on discretizations are required to compute their values.

9 CS476/676 Mar 6, Numerical methods for option pricing: Lattice methods Actually simple explicit finite difference methods Not very efficient, very complex for certain cases Monte-Carlo Method of choice for > 3 underlying stochastic factors Generally difficult to handle American early exercise (ideas: Longstaff-Schwartz) 3 stochastic factors, not very efficient

10 CS476/676 Mar 6, Numerical PDE Method of choice for < 3 stochastic factors Complex features easy to handle (i.e. complex American type contracts) Get value, hedging parameters for whole range of asset prices, not just a single point Useful for simulating hedging strategies Can handle more complex nonlinear PDE problems arising in finance Can build up general library so that many different contracts can be quickly priced

11 CS476/676 Mar 6, In folloing discussion, we consider the transformation τ = T t (time to expiry) and BS PDE has the form: V τ = σ2 2 S2 V SS +rsv S rv V = value of option T = expiry time of option τ = T t σ = volatility S = asset price r = risk free interest rate Note: V(S,τ) now represents the option value when time to expiry is τ and underlying price is S.

12 CS476/676 Mar 6, Numerical Methods for PDE: an overview Initial condition: V(S,0) = payoff(s). Use a finite computational domain [0,S max ] [0,T] to approximate [0,+ ] [0,T] Apply some boundary conditions at S = 0 and S = S max ; the effect of the boundary conditions is small if S 0 is sufficiently away from the boundary

13 CS476/676 Mar 6, Discretize the continuous region by a finite set of grids {(S i,τ n ) : n = 0, N,i = 0,,M} Discretize PDE at each grid point to obtain an algebraic equation relates values at n with values at n+1 As N,M +, computed values approximate the solution to the PDE

14 CS476/676 Mar 6, Discretization PDE Methods Finite difference method Replace derivatives by finite differences, Taylor series error analysis Standard FD analysis requires smooth payoffs Alternative FE analysis allows discontinuous payoffs FD,FE give the same discrete equations