How to compute the dynamic of stock using Geometric Brownian Motion?
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I have been given the following question:
Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$
For the first part of the question, I have got this answer:
$$dS_t = mu S_tdt + sigma S_t dWt$$
Is it correct?
And for the second part, I know that the price $f(t,S_t)$ follows the process
$$df = (frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2} sigma ^2S_tfrac{partial^2f}{partial S_t^2})dt +sigma S_t dWt$$
I am having trouble finding the answer using this process and given the information.
Any help is appreciated.
black-scholes stochastic-processes stochastic-calculus
$endgroup$
add a comment |
$begingroup$
I have been given the following question:
Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$
For the first part of the question, I have got this answer:
$$dS_t = mu S_tdt + sigma S_t dWt$$
Is it correct?
And for the second part, I know that the price $f(t,S_t)$ follows the process
$$df = (frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2} sigma ^2S_tfrac{partial^2f}{partial S_t^2})dt +sigma S_t dWt$$
I am having trouble finding the answer using this process and given the information.
Any help is appreciated.
black-scholes stochastic-processes stochastic-calculus
$endgroup$
add a comment |
$begingroup$
I have been given the following question:
Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$
For the first part of the question, I have got this answer:
$$dS_t = mu S_tdt + sigma S_t dWt$$
Is it correct?
And for the second part, I know that the price $f(t,S_t)$ follows the process
$$df = (frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2} sigma ^2S_tfrac{partial^2f}{partial S_t^2})dt +sigma S_t dWt$$
I am having trouble finding the answer using this process and given the information.
Any help is appreciated.
black-scholes stochastic-processes stochastic-calculus
$endgroup$
I have been given the following question:
Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$
For the first part of the question, I have got this answer:
$$dS_t = mu S_tdt + sigma S_t dWt$$
Is it correct?
And for the second part, I know that the price $f(t,S_t)$ follows the process
$$df = (frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2} sigma ^2S_tfrac{partial^2f}{partial S_t^2})dt +sigma S_t dWt$$
I am having trouble finding the answer using this process and given the information.
Any help is appreciated.
black-scholes stochastic-processes stochastic-calculus
black-scholes stochastic-processes stochastic-calculus
edited 6 hours ago
Emma
27312
27312
asked 6 hours ago
Rito LoweRito Lowe
164
164
add a comment |
add a comment |
1 Answer
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$begingroup$
The above equation should correctly read as follows:
$df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$
Using:
(a) $frac{partial f}{partial t}=S_t^2f$
(b) $frac{partial f}{partial S_t}=2S_ttf$
(c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$
The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:
$frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The above equation should correctly read as follows:
$df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$
Using:
(a) $frac{partial f}{partial t}=S_t^2f$
(b) $frac{partial f}{partial S_t}=2S_ttf$
(c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$
The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:
$frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$
$endgroup$
add a comment |
$begingroup$
The above equation should correctly read as follows:
$df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$
Using:
(a) $frac{partial f}{partial t}=S_t^2f$
(b) $frac{partial f}{partial S_t}=2S_ttf$
(c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$
The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:
$frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$
$endgroup$
add a comment |
$begingroup$
The above equation should correctly read as follows:
$df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$
Using:
(a) $frac{partial f}{partial t}=S_t^2f$
(b) $frac{partial f}{partial S_t}=2S_ttf$
(c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$
The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:
$frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$
$endgroup$
The above equation should correctly read as follows:
$df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$
Using:
(a) $frac{partial f}{partial t}=S_t^2f$
(b) $frac{partial f}{partial S_t}=2S_ttf$
(c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$
The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:
$frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$
edited 5 hours ago
Emma
27312
27312
answered 5 hours ago
ZRHZRH
593112
593112
add a comment |
add a comment |
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