Valuation of derivative assets Lecture 6

Valuation of derivative assets Lecture 6 Magnus Wiktorsson September 14, 2017 Magnus Wiktorsson L6 September 14, 2017 1 / 13

Feynman-Kac representation This is the link between a class of Partial Differential Equations (PDE:s) and stochastic differential equations. Say we want to solve the PDE (boundary value problem): { tf(t, x) + µ(t, x) x f(t, x) + σ2 (t,x) 2 2 f(t, x) = 0 x 2 f(t, x) = Φ(x) (1) Suppose X satisfies the SDE: dx u = µ(u, X u ) du + σ(u, X u ) dw u, 0 u T. Then f(t, x) = E[Φ(X T ) X t = x], is a solution to the PDE (1). Magnus Wiktorsson L6 September 14, 2017 2 / 13

Connection between Feynman-Kac and finance Say we want to solve the PDE: { tf(t, x) + r(t)x x f(t, x) + σ2 (t,x) 2 2 f(t, x) x 2 f(t, x) = r(t)f(t, x), = Φ(x). (2) Suppose Φ is a pay-off function for some derivative where the underlying satisfies X satisfies the SDE (under Q) dx u = r(u)x u du + σ(u, X u ) dw u, 0 t u T. Then f(t, x) = E[e T t r(u) du Φ(X T ) X t = x], is a solution to the PDE (2). We will soon see where this PDE comes from. Magnus Wiktorsson L6 September 14, 2017 3 / 13

The Market Consider a market consisting of N + 1 assets (S 0, S 1,..., S N ) = S. S 0 is usually the bank account. We assume that S is a solution to the SDE: ds(t) = diag(s(t))µ(t, S(t)) dt + diag(s(t))σ(t, S(t)) dw (t) S(0) = s. σ : (N + 1) d Matrix µ : (N + 1) 1 Vector W : d-dim Brownian Motion. Magnus Wiktorsson L6 September 14, 2017 4 / 13

Portfolio (B: Def 6.2 p. 87) Definition Let {S(t)} t 0 be an N + 1-dimensional price process. 1 A portfolio {h(t)} t 0 is an N + 1-dim adapted process. 2 The corresponding value process {V h (t)} t 0 is given by V h (t) = N h i (t)s i (t) i=0 3 A portfolio is self-financing if V h (t + ) V h (t) = dv h (t) = N h i (t)(s i (t + ) S i (t)) discrete time i=0 N h i (t) ds i (t) continuous time i=0 Magnus Wiktorsson L6 September 14, 2017 5 / 13

Relative Portfolio Definition For a given portfolio h the relative portfolio u is given by u i (t) = h i(t)s i (t) V h, (t) i.e. the fraction of the value coming from asset i. We have n i=0 u i(t) = 1 but note that we allow u i 0 and u i 1. It is self-fin if dv h (t) = N h i (t) ds i (t) = i=0 = V h (t) N i=0 N i=0 u i (t) ds i(t) S i (t) h i (t)s i (t) ds i (t) V h (t) S i (t) V h (t) Magnus Wiktorsson L6 September 14, 2017 6 / 13

Contingent Claim (B: Def 7.4 p. 94) Definition Let {Ft S } t 0 be the filtration generated be the asset process S. A contingent claim with maturity T is any FT S -measurable r.v. X. X is a simple claim if X = Φ(S(T )), where Φ is called a contract function. Magnus Wiktorsson L6 September 14, 2017 7 / 13

Arbitrage (B: Def 7.5 p. 96) Definition An arbitrage opportunity is a self-financing portfolio h with value process V h such that i) V h (0) = 0, ii) P(V h (t) 0) = 1, iii) P(V h (t) > 0) > 0, for some t > 0. If there does not exist any arbitrage opportunities on a market, the market is called free of arbitrage. Magnus Wiktorsson L6 September 14, 2017 8 / 13

Locally risk-free assets Definition A self-financing portfolio h is locally risk-free if dv h (t) = k(t)v h (t) dt, where k is an adapted process. Theorem (Proposition 7.6 (B: p. 97)) If h is locally risk-free then k(t) should equal the short rate r(t) for almost all t in order to avoid arbitrage opportunities. Magnus Wiktorsson L6 September 14, 2017 9 / 13

Black-Scholes equation (B: Thm 7.7 p. 101) Assume that we have a market consisting of a risky asset S and a bank-account B, where the corresponding P-dynamics are given as ds(t) = S(t)µ(t, S(t)) dt + S(t)σ(t, S(t)) dw (t), S(0) = s 0 db(t) = r(t)b(t) dt, B(0) = 1. Assume that F (t, s) is the value at time t of a simple claim with maturity T of the form F (T, s) = Φ(s). Then the value F is a solution to the boundary value problem { tf (t, x) + r(t)x x F (t, x) + x2 σ 2 (t,x) 2 2 F (t, x) = r(t)f (t, x) x 2 F (T, x) = Φ(x) Magnus Wiktorsson L6 September 14, 2017 10 / 13

Black-Scholes equation 2, Why this PDE? Let (h S, h B, 1) be the portfolio in the stock S, the bank account B and the derivative Π (with value F (t, S(t))). We want to choose the portfolio such that it is locally riskfree. The self-fin condition gives dv h (t) = h S (t) ds(t) + h B (t) db(t) dπ(t) To simplify the notation on the blackboard we use F := F (t, S(t)), F t := F t(t, S(t)) F S := F S (t, S(t)), F SS := F SS (t, S(t)) µ := µ(t, S(t)), σ := σ(t, S(t)), S := S(t), r := r(t). Magnus Wiktorsson L6 September 14, 2017 11 / 13

Feynman-Kac representation (B: Prop 5.6 p 74) Assume that F is a solution to the boundary value problem F (t, x) F (t, x) = r(t)x 1 t x 2 x2 σ(t, x) 2 2 F (t, x) x 2 + r(t)f (t, x) F (T, x) = Φ(x) [ ( ) ] T 2 and assume that E 0 X(u)2 σ(u, X(u)) 2 F (u,x) x du <. x=x(u) Then F has the representation (RNVF) F (t, x) = E [e ] T t r(u)du Φ(X(T )) X t = x, where dx(u) = r(u)x(u) du + X(u)σ(u, X(u)) dw (u), 0 t u T, X(t) = x. Magnus Wiktorsson L6 September 14, 2017 12 / 13

Replicating portfolio Let X be a simple claim with contract function Φ with a value function F that satisfies the Black-Scholes equation on the previous slide. A portfolio h = (h S, h B ) with value process where V h (t) = h S (t)s(t) + h B (t)b(t), h S t = F (t, s), s s=s(t) h B t = F (t, S(t)) hs (t)s(t), B(t) is a replicating portfolio for the claim X. Magnus Wiktorsson L6 September 14, 2017 13 / 13