GBM with adjusted normal distribution
GBM with adjusted normal distribution
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Staf · External communityPost link
External question — Quantitative Finance Stack Exchange
Author: Staf
Original post: https://quant.stackexchange.com/questions/85269
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
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To model a structured product, I thought of using a geometric Brownian motion model, where I choose a certain mean and variance for the normal distribution to make sure that a certain percentage of paths (as a result from Monte Carlo) cross a threshold value where different conditions apply. However my question is does this violate the risk-free and arbitrage free assumption? Meaning I can no longer discount using the risk-free rate?
Please let me know if i need to provide any more information.
Kind regards
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QuantCalc.net · External communityPost link
External answer — Quantitative Finance Stack Exchange
Author: QuantCalc.net
Original post: https://quant.stackexchange.com/a/85277
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In the geometric Brownian motion model setting, choosing any specific volatility or variance does
not
violate no-arbitrage; the only requirement is that under the risk-neutral measure the drift equals the risk-free rate (minus dividends). The arbitrage-free dynamics must satisfy
$$
dS_t = S_t (r - q)\, dt + S_t \sigma\, dW_t^{\mathbb{Q}},
$$
which ensures that the discounted price process
$$
e^{-(r-q)t} S_t
$$
is a martingale. You are free to choose any real-world drift
$\mu_{\mathbb{P}}$
for forecasting and any volatility function
$\sigma(t,S_t)$
, but
the risk-neutral drift must stay at $r-q$
to avoid arbitrage.
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Quoted from Forex.com.bd-Editorial External answer — Quantitative Finance Stack Exchange Author: QuantCalc.net Source score (net votes, not local likes): 0 Original post: https://quant.stackexchange.com/a/85277 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. In the geometric Brownian motion model setting, choosing any specific volatility or variance does not violate no-arbitrage; the only requirement is that under the risk-neutral measure the drift equals the risk-free rate (minus dividends). The arbitrage-free dynamics must satisfy $$ dS_t = S_t (r - q)\, dt + S_t \sigma\, dW_t^{\mathbb{Q}}, $$ which ensures that the discounted price process $$ e^{-(r-q)t} S_t $$ is a martingale. You are free to choose any real-world drift $\mu_{\mathbb{P}}$ for forecasting and any volatility function $\sigma(t,S_t)$ , but the risk-neutral drift must stay at $r-q$ to avoid arbitrage.
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