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5

I think the easiest way to derive the solution to the GBM is via Ito's Lemma. The GBM: $dS_t = \mu S_t dt + \sigma S_t dW_t$ is a short hand for: $$S_t = S_0 + \int_{h=0}^{h=t}\left(\mu S_h\right)dh + \int_{h=0}^{h=t}\left(\sigma S_h\right)dW_h$$ Ito process is defined as: $$X_t = S_0 + \int_{h=0}^{h=t}\left(a(X_h,h)\right)dh + \int_{h=0}^{h=t}\left(b(... 5 You're right but a GBM doesn't assume that percentage returns are normally distributed. It's about log-returns. If the log-return r_t=\ln\left(\frac{S_{t+dt}}{S_t}\right) is normally distributed (GBM assumption), then r_t can indeed be any arbitrarily large (positive or negative) number with positive probability. This also implies that stock prices are ... 3 The return R_i as expressed in$$R_{i+1,i}=\frac{S_{i+1}-S_i}{S_i}=\mu \Delta t + \sigma \Delta W(t_{i+1},t_i)$$is not possible. To see this, let's get the returns over two small time steps of \Delta t each. Then$$R_{i+2,i+1}=\frac{S_{i+2}-S_{i+1}}{S_{i+1}}= \mu \Delta t + \sigma \Delta W(t_{i+2},t_{i+1})$$but$$R_{i+2,i}=\frac{S_{i+2}-S_{i}}{S_{i}}= ...

3

I assume you want to real-world probability, because the risk-neutral probability is not a probability in the 'likelihood' sense. Under the real-world measure, we model the stock under the B-S model as: $$X(t)=X(0)+\int^{t}_{0}\mu X(h)dh+\int^{t}_{0}\sigma X(h)dW(h)$$ If the market demands a 30% annual return, I will take that as the real-world rate $\mu$....

2

Call option: $$\mathbb{P}\left(S_t\geq K\right)=\mathbb{P}\left(S_0e^{(rt-0.5\sigma^2t+\sigma W_t)}\geq K\right)=\\=\mathbb{P}\left(W_t\geq \frac{ln\left(\frac{K}{S_0}\right)-rt+0.5\sigma^2t}{\sigma}\right)=\\=\mathbb{P}\left(Z\geq \frac{ln\left(\frac{K}{S_0}\right)-rt+0.5\sigma^2t}{\sigma\sqrt{t}}\right)=\mathbb{P}(Z\leq d2)$$ So we have shown the well-...

1

Another sketch of proof: If you move to the equivalent PDE (using Feynman-Kac), you can assume that S is positive, find the solution by log-transfomation. Then as the solution is unique given initial conditions, and it is the solution of the original PDE, S must be positive.

1

Asset prices follow a random walk, so assuming probabilities and forecasting stock prices are not that accurate. Hence, investors try to project volatility rather than asset prices (i.e. implied vol) using GARCH, EWMA, or other vol forecasting models. The optimal portfolio is to invest long term in a globally diversified portfolio with a focus on ...

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