The choice of portfolio in the proof of the Black-Scholes formula
$begingroup$
Consider a stock whose price $S$ satisfies $$dS_t=mu S_tdt+sigma S_tdW_t$$ for constants $mu,sigma$ and where $W$ is a $mathbb{P}$-Brownian motion. Further assume that the stock pays out dividends continuously at a rate of $d$ proportional to the current stock price.
Let $p_t$ denote the price at time $t$ of a European-style derivative which has a payoff of $f(S_T)$ at time $T$. In order to determine a formula for $p_t$ we essentially carry out the following steps:
- Use Girsanov's theorem to determine the risk-neutral probability measure $mathbb{Q}$ such that $widetilde{W}_t=left(frac{mu+d-r}{sigma}right)t+W_t$ is a $mathbb{Q}$-Brownian motion.
- Define $P_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$. Show that both $hat{S}_t=e^{-(r-d)t}S_t$ and $hat{P_t}=e^{-rt}P_t$ are $mathbb{Q}$-martingales.
- Use the Martingale Representation Theorem to conclude the existence of a predictable process $A$ such that $hat{P}_t=hat{P}_0+int_0^tA_sdhat{S}_s$ under $mathbb{Q}$.
- Construct the portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ which consists of $hat{P}_t-A_that{S}_t$ units of cash and $A_te^{dt}$ units of the stock at time $t$. The value of this portfolio is $P_t$.
- Since $P_T=p_T$ we conclude from the Law of One Price that $P_t=p_t$ for all $0leq tleq T$. In other words, $p_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$.
After going through the above steps I am wondering why the portfolio needs to be $(hat{P}_t-A_that{S}_t,A_te^{dt})$. It seems like we could simply choose $(hat{P}_t,0)$ as our portfolio and this would still have a value of $P_t$ at time $t$.
black-scholes stochastic-processes risk-neutral-measure
asked Jan 10 at 4:46
user375366user375366
1233
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$begingroup$
Consider a stock whose price $S$ satisfies $$dS_t=mu S_tdt+sigma S_tdW_t$$ for constants $mu,sigma$ and where $W$ is a $mathbb{P}$-Brownian motion. Further assume that the stock pays out dividends continuously at a rate of $d$ proportional to the current stock price.
Let $p_t$ denote the price at time $t$ of a European-style derivative which has a payoff of $f(S_T)$ at time $T$. In order to determine a formula for $p_t$ we essentially carry out the following steps:
- Use Girsanov's theorem to determine the risk-neutral probability measure $mathbb{Q}$ such that $widetilde{W}_t=left(frac{mu+d-r}{sigma}right)t+W_t$ is a $mathbb{Q}$-Brownian motion.
- Define $P_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$. Show that both $hat{S}_t=e^{-(r-d)t}S_t$ and $hat{P_t}=e^{-rt}P_t$ are $mathbb{Q}$-martingales.
- Use the Martingale Representation Theorem to conclude the existence of a predictable process $A$ such that $hat{P}_t=hat{P}_0+int_0^tA_sdhat{S}_s$ under $mathbb{Q}$.
- Construct the portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ which consists of $hat{P}_t-A_that{S}_t$ units of cash and $A_te^{dt}$ units of the stock at time $t$. The value of this portfolio is $P_t$.
- Since $P_T=p_T$ we conclude from the Law of One Price that $P_t=p_t$ for all $0leq tleq T$. In other words, $p_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$.
After going through the above steps I am wondering why the portfolio needs to be $(hat{P}_t-A_that{S}_t,A_te^{dt})$. It seems like we could simply choose $(hat{P}_t,0)$ as our portfolio and this would still have a value of $P_t$ at time $t$.
black-scholes stochastic-processes risk-neutral-measure
asked Jan 10 at 4:46
user375366user375366
1233
$endgroup$
add a comment |
$begingroup$
Consider a stock whose price $S$ satisfies $$dS_t=mu S_tdt+sigma S_tdW_t$$ for constants $mu,sigma$ and where $W$ is a $mathbb{P}$-Brownian motion. Further assume that the stock pays out dividends continuously at a rate of $d$ proportional to the current stock price.
Let $p_t$ denote the price at time $t$ of a European-style derivative which has a payoff of $f(S_T)$ at time $T$. In order to determine a formula for $p_t$ we essentially carry out the following steps:
- Use Girsanov's theorem to determine the risk-neutral probability measure $mathbb{Q}$ such that $widetilde{W}_t=left(frac{mu+d-r}{sigma}right)t+W_t$ is a $mathbb{Q}$-Brownian motion.
- Define $P_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$. Show that both $hat{S}_t=e^{-(r-d)t}S_t$ and $hat{P_t}=e^{-rt}P_t$ are $mathbb{Q}$-martingales.
- Use the Martingale Representation Theorem to conclude the existence of a predictable process $A$ such that $hat{P}_t=hat{P}_0+int_0^tA_sdhat{S}_s$ under $mathbb{Q}$.
- Construct the portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ which consists of $hat{P}_t-A_that{S}_t$ units of cash and $A_te^{dt}$ units of the stock at time $t$. The value of this portfolio is $P_t$.
- Since $P_T=p_T$ we conclude from the Law of One Price that $P_t=p_t$ for all $0leq tleq T$. In other words, $p_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$.
After going through the above steps I am wondering why the portfolio needs to be $(hat{P}_t-A_that{S}_t,A_te^{dt})$. It seems like we could simply choose $(hat{P}_t,0)$ as our portfolio and this would still have a value of $P_t$ at time $t$.
black-scholes stochastic-processes risk-neutral-measure
asked Jan 10 at 4:46
user375366user375366
1233
$endgroup$
Consider a stock whose price $S$ satisfies $$dS_t=mu S_tdt+sigma S_tdW_t$$ for constants $mu,sigma$ and where $W$ is a $mathbb{P}$-Brownian motion. Further assume that the stock pays out dividends continuously at a rate of $d$ proportional to the current stock price.
Let $p_t$ denote the price at time $t$ of a European-style derivative which has a payoff of $f(S_T)$ at time $T$. In order to determine a formula for $p_t$ we essentially carry out the following steps:
- Use Girsanov's theorem to determine the risk-neutral probability measure $mathbb{Q}$ such that $widetilde{W}_t=left(frac{mu+d-r}{sigma}right)t+W_t$ is a $mathbb{Q}$-Brownian motion.
- Define $P_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$. Show that both $hat{S}_t=e^{-(r-d)t}S_t$ and $hat{P_t}=e^{-rt}P_t$ are $mathbb{Q}$-martingales.
- Use the Martingale Representation Theorem to conclude the existence of a predictable process $A$ such that $hat{P}_t=hat{P}_0+int_0^tA_sdhat{S}_s$ under $mathbb{Q}$.
- Construct the portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ which consists of $hat{P}_t-A_that{S}_t$ units of cash and $A_te^{dt}$ units of the stock at time $t$. The value of this portfolio is $P_t$.
- Since $P_T=p_T$ we conclude from the Law of One Price that $P_t=p_t$ for all $0leq tleq T$. In other words, $p_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$.
After going through the above steps I am wondering why the portfolio needs to be $(hat{P}_t-A_that{S}_t,A_te^{dt})$. It seems like we could simply choose $(hat{P}_t,0)$ as our portfolio and this would still have a value of $P_t$ at time $t$.
black-scholes stochastic-processes risk-neutral-measure
black-scholes stochastic-processes risk-neutral-measure
asked Jan 10 at 4:46
user375366user375366
1233
asked Jan 10 at 4:46
user375366user375366
1233
asked Jan 10 at 4:46
user375366user375366
1233
asked Jan 10 at 4:46
user375366user375366
1233
asked Jan 10 at 4:46
user375366user375366
1233
1233
add a comment |
add a comment |
1 Answer
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The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.
answered Jan 10 at 5:49
Bob Jansen♦Bob Jansen
3,44752145
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1 Answer
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$begingroup$
The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.
answered Jan 10 at 5:49
Bob Jansen♦Bob Jansen
3,44752145
$endgroup$
add a comment |
$begingroup$
The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.
answered Jan 10 at 5:49
Bob Jansen♦Bob Jansen
3,44752145
$endgroup$
add a comment |
$begingroup$
The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.
answered Jan 10 at 5:49
Bob Jansen♦Bob Jansen
3,44752145
$endgroup$
The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.
answered Jan 10 at 5:49
Bob Jansen♦Bob Jansen
3,44752145
answered Jan 10 at 5:49
Bob Jansen♦Bob Jansen
3,44752145
answered Jan 10 at 5:49
Bob Jansen♦Bob Jansen
3,44752145
answered Jan 10 at 5:49
Bob Jansen♦Bob Jansen
3,44752145
3,44752145
add a comment |
add a comment |
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