Geometric Brownian Motion Probability of hitting uper boundaryLet ( S t ) be a geometric Brownian Motion, i.e. S t = S 0 e ( μ − σ 2 2 ) t + σ W t . Let α &gt; 0 and τ = inf { t &gt; 0 | S t ≥ α }. Compute P ( τ ≤ t ).

Question

Geometric Brownian Motion Probability of hitting uper boundaryLet   (      S    t    ) be a geometric Brownian Motion, i.e.       S    t    =      S    0        e                  (        μ        −                              σ            2                    2                )            t      +      σ              W        t            . Let   α  &amp;gt;  0 and   τ  =  inf  {  t  &amp;gt;  0      |        S    t    ≥  α  }. Compute   P  (  τ  ≤  t  ).

Lisa Acevedo · Accepted Answer

Step 1  P  (  τ  ≤  t  )  =  P  (      S    s    ≥  α  ,  s  ∈  [  0  ,  t  ]  )  =  P  (  log  ⁡  (      S    0    )  +      (    μ    −                  σ        2            2        )    s  +  σ      W    s    ≥  log  ⁡  (  α  )  ,  s  ∈  [  0  ,  t  ]  )  =  P      (          W      s        ≥                  log        ⁡                  (                      α                          S              0                                )                −                  (          μ          −                                    σ              2                        2                    )                s            σ        ,    s    ∈    [    0    ,    t    ]    )  Now set       α                          ′              =            log      ⁡              (                  α                      S            0                          )              σ   and   β  =  −            (      μ      −                        σ          2                2            )        σ  .Step 2Then you have   P      (          W      s        ≥          α                                  ′                      +    β    s    ,    s    ∈    [    0    ,    t    ]    )  .Define       X    t    =      W    t    −  β  t and       X    t    ∗    =      sup          0      ≤      s      ≤      t            X    s   and use the formula   P  (      X    t    ∗    ≥      α                          ′              )  =  1  −  Φ      (                            α                                                  ′                                      +        β        t                    t              )    +      e          −      2              α                                          ′                              β        Φ      (                  β        t        −                  α                                                  ′                                                  t              )

Let (S_t) be a geometric Brownian Motion, i.e. S_t=S_0e(mu-sigma^2/2)t+sigma^W t. Let alpha>0 and tau=inf {t>0|S_t ge alpha\}. Compute P(tau le t).

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