Do *all* non-dividend paying assets have the risk-free instantaneous return rate under the risk-neutral...
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For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by
$$dB_t^{Bbb Q} = lambda dt + dB_t$$
where $lambda$ is the unique market price of risk, or the so-called Sharpe ratio.
Under the risk-neutral measure, any non-dividend paying stock price process $S_t$ thus follows
$$frac{dS_t}{S_t} = rdt + sigma_SdB_t^{Bbb Q}.$$
However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $C_t$ is also $r$, i.e.
$$frac{dC_t}{C_t} = rdt + sigma_C d B_t^{Bbb Q}$$
where $sigma_C$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $C_t$ is $rC_tdt$) to derive the BS PDE.
Question: is it true that under the risk neutral measure, any non-dividend paying asset price $X_t$ must have its instantaneous rate of return equal to $r$? If so, what would be a rigorous explanation for this?
Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $Y_t=e^{-rt}X_t$ must be a martingale or equivalently an Ito integral without drift. Hence
$$frac{dY_t}{Y_t}=sigma_Y dB_t^{Bbb Q}.$$
where $sigma_Y$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes,
$$frac{dY_t}{Y_t}=-rdt+frac{dX_t}{X_t}$$
Therefore it follows
$$frac{dX_t}{X_t}=rdt+sigma_Y dB_t^{Bbb Q}.$$
option-pricing black-scholes arbitrage risk-neutral-measure
asked Feb 7 at 10:24
VimVim
383111
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add a comment |
$begingroup$
For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by
$$dB_t^{Bbb Q} = lambda dt + dB_t$$
where $lambda$ is the unique market price of risk, or the so-called Sharpe ratio.
Under the risk-neutral measure, any non-dividend paying stock price process $S_t$ thus follows
$$frac{dS_t}{S_t} = rdt + sigma_SdB_t^{Bbb Q}.$$
However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $C_t$ is also $r$, i.e.
$$frac{dC_t}{C_t} = rdt + sigma_C d B_t^{Bbb Q}$$
where $sigma_C$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $C_t$ is $rC_tdt$) to derive the BS PDE.
Question: is it true that under the risk neutral measure, any non-dividend paying asset price $X_t$ must have its instantaneous rate of return equal to $r$? If so, what would be a rigorous explanation for this?
Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $Y_t=e^{-rt}X_t$ must be a martingale or equivalently an Ito integral without drift. Hence
$$frac{dY_t}{Y_t}=sigma_Y dB_t^{Bbb Q}.$$
where $sigma_Y$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes,
$$frac{dY_t}{Y_t}=-rdt+frac{dX_t}{X_t}$$
Therefore it follows
$$frac{dX_t}{X_t}=rdt+sigma_Y dB_t^{Bbb Q}.$$
option-pricing black-scholes arbitrage risk-neutral-measure
asked Feb 7 at 10:24
VimVim
383111
$endgroup$
add a comment |
$begingroup$
For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by
$$dB_t^{Bbb Q} = lambda dt + dB_t$$
where $lambda$ is the unique market price of risk, or the so-called Sharpe ratio.
Under the risk-neutral measure, any non-dividend paying stock price process $S_t$ thus follows
$$frac{dS_t}{S_t} = rdt + sigma_SdB_t^{Bbb Q}.$$
However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $C_t$ is also $r$, i.e.
$$frac{dC_t}{C_t} = rdt + sigma_C d B_t^{Bbb Q}$$
where $sigma_C$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $C_t$ is $rC_tdt$) to derive the BS PDE.
Question: is it true that under the risk neutral measure, any non-dividend paying asset price $X_t$ must have its instantaneous rate of return equal to $r$? If so, what would be a rigorous explanation for this?
Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $Y_t=e^{-rt}X_t$ must be a martingale or equivalently an Ito integral without drift. Hence
$$frac{dY_t}{Y_t}=sigma_Y dB_t^{Bbb Q}.$$
where $sigma_Y$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes,
$$frac{dY_t}{Y_t}=-rdt+frac{dX_t}{X_t}$$
Therefore it follows
$$frac{dX_t}{X_t}=rdt+sigma_Y dB_t^{Bbb Q}.$$
option-pricing black-scholes arbitrage risk-neutral-measure
asked Feb 7 at 10:24
VimVim
383111
$endgroup$
For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by
$$dB_t^{Bbb Q} = lambda dt + dB_t$$
where $lambda$ is the unique market price of risk, or the so-called Sharpe ratio.
Under the risk-neutral measure, any non-dividend paying stock price process $S_t$ thus follows
$$frac{dS_t}{S_t} = rdt + sigma_SdB_t^{Bbb Q}.$$
However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $C_t$ is also $r$, i.e.
$$frac{dC_t}{C_t} = rdt + sigma_C d B_t^{Bbb Q}$$
where $sigma_C$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $C_t$ is $rC_tdt$) to derive the BS PDE.
Question: is it true that under the risk neutral measure, any non-dividend paying asset price $X_t$ must have its instantaneous rate of return equal to $r$? If so, what would be a rigorous explanation for this?
Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $Y_t=e^{-rt}X_t$ must be a martingale or equivalently an Ito integral without drift. Hence
$$frac{dY_t}{Y_t}=sigma_Y dB_t^{Bbb Q}.$$
where $sigma_Y$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes,
$$frac{dY_t}{Y_t}=-rdt+frac{dX_t}{X_t}$$
Therefore it follows
$$frac{dX_t}{X_t}=rdt+sigma_Y dB_t^{Bbb Q}.$$
option-pricing black-scholes arbitrage risk-neutral-measure
option-pricing black-scholes arbitrage risk-neutral-measure
asked Feb 7 at 10:24
VimVim
383111
asked Feb 7 at 10:24
VimVim
383111
edited Feb 7 at 11:46
Vim
asked Feb 7 at 10:24
VimVim
383111
asked Feb 7 at 10:24
VimVim
383111
asked Feb 7 at 10:24
VimVim
383111
383111
add a comment |
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Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.
answered Feb 7 at 11:16
Antoine ConzeAntoine Conze
3,770149
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1
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Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
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– Vim
Feb 7 at 11:35
add a comment |
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1 Answer
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Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.
answered Feb 7 at 11:16
Antoine ConzeAntoine Conze
3,770149
$endgroup$
1
$begingroup$
Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
$endgroup$
– Vim
Feb 7 at 11:35
add a comment |
$begingroup$
Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.
answered Feb 7 at 11:16
Antoine ConzeAntoine Conze
3,770149
$endgroup$
1
$begingroup$
Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
$endgroup$
– Vim
Feb 7 at 11:35
add a comment |
$begingroup$
Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.
answered Feb 7 at 11:16
Antoine ConzeAntoine Conze
3,770149
$endgroup$
Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.
answered Feb 7 at 11:16
Antoine ConzeAntoine Conze
3,770149
answered Feb 7 at 11:16
Antoine ConzeAntoine Conze
3,770149
answered Feb 7 at 11:16
Antoine ConzeAntoine Conze
3,770149
answered Feb 7 at 11:16
Antoine ConzeAntoine Conze
3,770149
3,770149
1
$begingroup$
Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
$endgroup$
– Vim
Feb 7 at 11:35
add a comment |
1
$begingroup$
Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
$endgroup$
– Vim
Feb 7 at 11:35
1
1
$begingroup$
Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
$endgroup$
– Vim
Feb 7 at 11:35
$begingroup$
Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
$endgroup$
– Vim
Feb 7 at 11:35
add a comment |
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