Geometric Brownian motion probability questionLet S(t), t ≥ 0 be a geometric Brownian motion with drift parameter μ = 0.3 and volatility parameter σ = 0.3. Find ( a ) P ( S ( 1 ) &gt; S ( 0 ) ) ( b ) P ( S ( 2 ) ) &gt; S ( 0 ) )

Question

Geometric Brownian motion probability questionLet S(t),   t  ≥  0 be a geometric Brownian motion with drift parameter   μ  =  0.3 and volatility parameter   σ  =  0.3. Find  (  a  )  P  (  S  (  1  )  &amp;gt;  S  (  0  )  )  (  b  )  P  (  S  (  2  )  )  &amp;gt;  S  (  0  )  )

hanfydded1c · Accepted Answer

Step 1I think you can solve (b) using the same way as you did in (a).                    P        (        S        (        2        )        &amp;gt;        S        (        0        )        )                            =        P        (        S        (        2        )                  /                S        (        0        )        &amp;gt;        1        )                                          =        P        (        log        ⁡        (        S        (        2        )                  /                S        (        1        )        )        &amp;gt;        0        )                                          =        P        (        Y        &amp;gt;        0        )            where Y is also Gaussian, which can be derived from the property of the brown motion.Step 2Be careful about the expectation and variance of Y, which are different with the expectation and variance of Z.

raapjeqp · Accepted Answer

Step 1Let   (  B  (  t  )      )          t      ≥      0       be the standard brownian motion.The individual random variable B(t) has thus mean 0 and variance       σ          B      (      t      )        2    =  t.Then we have   S  (  t  )  =  S  (  0  )  exp  ⁡      (          (      μ      −              1        2                    σ        2            )        t    +    σ    B    (    t    )    )       ..Step 2The probabilities to be computed are then (for a more general time t):                              P                (                 S        (        t        )        &amp;gt;        S        (        0        )                 )                            =                  P                          (                                               S              (              t              )                                      S              (              0              )                                &amp;gt;          1                     )                                                  =                  P                          (                     exp          ⁡                      (                          (              μ              −                              1                2                                            σ                2                            )                        t            +            σ            B            (            t            )            )                    &amp;gt;          1                     )                                                  =                  P                          (                                 (            μ            −                          1              2                                      σ              2                        )                    t          +          σ          B          (          t          )          &amp;gt;          0                     )                                                  =                  P                          (                     B          (          t          )          &amp;gt;          −                      1            σ                                (            μ            −                          1              2                                      σ              2                        )                    t                     )                                                  =                  P                          (                                               B              (              t              )                                      t                                &amp;lt;          +                      1            σ                                (            μ            −                          1              2                                      σ              2                        )                                t                               )                                                  =        Φ                  (                                 1            σ                                (            μ            −                          1              2                                      σ              2                        )                                t                               )                         .            Here,   Φ is the repartition of an   N  (  0  ,      1    2    ) random variable.

Let S(t), t ge 0 be a geometric Brownian motion with drift parameter mu=0.3 and volatility parameter sigma=0.3. Find (a) P(S(1)>S(0)) (b) P(S(2))>S(0))

Answered question

Answer & Explanation

New Questions in High school geometry