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Asset Dynamics:Geometric Brownian motion (GBM)

$$dA_t=r A_t \, dt+σ A_t \, dZ_t$$

$r$ is the risk free interest rate, $σ$ is the diffusion coefficient. $Z$ is a standard Brownian motion. Maturity is $T$. The initial asset value is $A_0$. We set a constant threshold $D$ ($0<D< A_0$). We have another threshold $D+S$ (with $S>0$)

My question is, how to get the following joint probability?

$$P\left(A_T≤D+S \;, \;\min_{0≤t≤T}A_t ≤D \right)$$

I need to calculate this result in my thesis. So I really need to know how to do it. Any comment or answer is very very appreciated!

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1 Answer 1

$A_t=A_0 e^{(r-s^2/2)t+sZ_t}$.

Then $$P\left(A_T≤D+S \;, \;\min_{0≤t≤T}A_t ≤D \right) =$$ $$=P\left(\theta T+Z_T≤\frac{1}{s}\log(\frac{D+S}{A_0}) \;, \;\min_{0≤t≤T} \theta t+Z_t≤\frac{1}{s}\log(\frac{D}{A_0})\right);$$

where $\theta=\frac{1}{s}(r-s^2/2)$.

Now, using Girsanov, define a new measure $Q$ under which $\{B_t=\theta t+Z_t\}_{t \in[0,T]}$ is a standard BM. This is because the joint density $\min_{t \leq T}B_t,B_T$ it is known (can be compute with the reflection principle).

Then, calling W the events you want, you get $$E^P[1_W]=E^Q\left[\frac{dP}{dQ}|_T1_W\right]$$

where $\frac{dP}{dQ}|_t=e^{\theta B_t -\theta^2 t/2}$.

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Oh, thanks very much for that! Now I know how to do it! –  Li Zekai May 8 '12 at 11:22
    
Yes. I have verified. The answer provided by Kolmo is totally correct! –  user28138 Jun 6 '12 at 21:02
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