Then D t is constant in t between consecutive payment dates and jumps up by a j

Then d t is constant in t between consecutive payment

This preview shows page 182 - 185 out of 185 pages.

Then D ( t ) is constant in t between consecutive payment dates, and jumps up by a j S ( t j - ) at time t j . By (5.27) the sum S ( t ) + D ( t ) is continuous in t . Therefore S ( t ) jumps down by a j S ( t j - ) at time t j . Also dS ( t ) = S ( t ) α ( t ) dt + S ( t ) σ ( t ) dW ( t ) = S ( t ) R ( t ) dt + S ( t ) σ ( t ) d f W ( t ) for t ( t j , t j +1 ) between consecutive payment dates. Thus S ( t ) = S ( t j ) e R t t j σ ( u ) d f W ( u )+ R t t j ( R ( u ) - 1 2 σ ( u ) 2 ) du if t ( t j , t j +1 ) , S ( t j +1 ) = (1 - a j +1 ) S ( t j ) e R t j +1 t j σ ( u ) d f W ( u )+ R t j +1 t j ( R ( u ) - 1 2 σ ( u ) 2 ) du .
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By recursion it follows for all t [0 , T ] that S ( t ) = S 0 Y t j t (1 - a j ) · e R t 0 σ ( u ) d f W ( u )+ R t 0 ( R ( u ) - 1 2 σ ( u ) 2 ) du (5.32) For constant coefficients R ( t ) = r , σ ( t ) = σ , and a j , S ( t ) = S 0 Y t j t (1 - a j ) · e σ f W ( t )+ ( r - 1 2 σ 2 ) t . The risk-neutral pricing formula (5.30) then yields for the price V ( t ) of the European call with strike K and maturity T V ( t ) = ¯ S ( t d + ( t , ¯ S ( t ) ) - e - r ( T - t ) K Φ d - ( t , ¯ S ( t ) ) where ¯ S ( t ) = S ( t ) Y t < t j T (1 - a j ).
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Remark. We can compute the value X ( t ) of a portfolio strategy which starts with one share of stock and instantaneously reinvests all dividends in the stock. Let Δ( t ) be the number of shares in the portfolio at time t . This is an increasing process with Δ(0) = 1 and d Δ( t ) = Δ( t - ) dD ( t ) S ( t ) . Case 1) D ( t ) = R t 0 A ( u ) S ( u ) du . Then we find Δ( t ) = e R t 0 A ( u ) du . Case 2) D ( t ) = t j t a j S ( t j - ). Then Δ( t ) is constant between consecutive payment dates and at payment date t j we have Δ( t j ) - Δ( t j - ) = Δ( t j - ) a j S ( t j - ) S ( t j ) = Δ( t j - ) a j 1 - a j . Thus Δ( t j ) = Δ( t j - ) 1 1 - a j = Δ( t j - 1 ) 1 1 - a j and so Δ( t ) = Q t j t 1 1 - a j . Now since X ( t ) = Δ( t ) S ( t ), using either (5.31) or (5.32) we obtain in each case that X ( t ) = S 0 exp R t 0 σ ( u ) d f W ( u ) + R t 0 ( R ( u ) - 1 2 σ ( u ) 2 ) du .
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In conclusion, if the dividend paying stock model is given by (5.27), then the associated value process X ( t ) from continuous dividend reinvestment follows the same stochastic process as the non-dividend paying stock S ( t ) in (5.14). In particular, the discounted value process X ( t ) B ( t ) = S 0 exp R t 0 σ ( u ) d f W ( u ) - R t 0 1 2 σ ( u ) 2 du is again a martingale under the risk-neutral measure e P .
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  • Fall '09
  • J.WISSEL
  • Probability theory

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