We see that the investment in the nonrisky asset is

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, we see that the investment in the nonrisky asset is given by X t θ t , and the variation of the portfolio value under the self-financing condition is given by dX θ t = θ t dS t S t + ( X t θ t ) rdt, t 0 . We say that the portfolio θ is admissible if in addition the corresponding port- folio value X θ is bounded from below. The admissibility condition means that the investor is limited by a credit line below which he is considered bankrupt. We denote by A the collection of all admissible portfolios. Using again Itˆ o’s formula, we see that the discounted portfolio value process ˜ X t := X t e rt satisfies the dynamics: d ˜ X θ t = e rt θ t d ˜ S t ˜ S t where ˜ S t := S t e rt , (6.8)
74 CHAPTER 6. ITO DIFFERENTIAL CALCULUS and then d ˜ S t = e rt ( rS t dt + dS t ) = ˜ S t ( ( μ r ) dt + σ dW t ) . (6.9) We recall the no-arbitrage principle which says that if a portfolio strategy on the financial market produces an a.s. nonegative final portfolio value, starting from a zero intial capital, then the portfolio value is zero a.s. A contingent claim is an F T measurable random variable which describes the random payo ff of the contract at time T . The following result is specific to the case where the contingent claim is g ( S T ) for some deterministic function g : R + −→ R . Such contingent claims are called Vanilla options . In preparation of the main result of this section, we start by Proposition 6.13. Suppose that the function g : R + −→ R has polynomial growth, i.e. | g ( s ) | α (1 + s β ) for some α , β 0 . Then, the linear partial di ff erential equation on [0 , T ) × (0 , ) : L v := v t + rs v s + 1 2 σ 2 s 2 2 v s 2 rv = 0 and v ( T, . ) = g, (6.10) has a unique solution v C 0 ([0 , T ] × (0 , )) C 1 , 2 ([0 , T ) , (0 , )) in the class of polynomially growing functions, and given by v ( t, s ) = E e r ( T t ) g ˆ S T | S t = s , ( t, s ) [0 , T ] × (0 , ) where ˆ S T := e ( r μ )( T t ) S T . Proof. We denote V ( t, s ) := E e r ( T t ) g ˆ S T | S t = s . 1- We first observe that V C 0 ([0 , T ] × (0 , )) C 1 , 2 ([0 , T ) , (0 , )). To see this, we simply write V ( t, s ) = e r ( T t ) R g ( e x ) 1 2 πσ 2 ( T t ) e 1 2 x ln ( s )( r 1 2 σ 2 )( T t ) σ T t « 2 dx, and see that the claimed regularity holds true by the dominated convergence theorem. 2- Immediate calculation reveals that V inherits the polynomial growth of g . Let ˆ S t := e ( r μ ) t S t , t 0 , so that d ˆ S t ˆ S t = rdt + σ dW t , an consider the stopping time τ := inf u > t : | ln ( ˆ S u /s ) | > 1 . Then, it fol- lows from the law of iterated expectations that V ( t, s ) = E t,s e r ( τ h t ) V τ h, ˆ S τ h
6.4. Black-Scholes by verification 75 for every h > t , where we denoted by E t,s the expectation conditional on { ˆ S t = s } . It then follows from Itˆ o’s formula that 0 = E t,s τ h t e ru L V ( u, ˆ S u ) du + τ h t e ru V s ( u, ˆ S u ) σ ˆ S u dW u = E t,s τ h t e ru L V ( u, ˆ S u ) du .

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