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2

In Hull's textbook, the stock price dynamics is lognormal: $S_T = S_0 \exp(\mu T - \frac{1}{2}\sigma^2T + \sigma W_T)$, where $W_t$ is a standard brownian motion. And so the mean of this is the mean of a lognormal random variable with the log mean as $\ln S_0 + \mu T - \frac{1}{2}\sigma^2T$ and the log standard deviation as $\sigma \sqrt{T}$, and so the ...


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I came across this thread while searching for a similar topic. In Nualart's book (Intoduction to Malliavin Calculus), it is asked to show that $\int_0^t B_s ds$ is gaussien and it is asked to compute its mean and variance. This exerice should rely only on basic brownian motion properties, in particular, no Itô calculus should be used (Itô calculus is ...


4

Note that \begin{align*} \int_0^t e^{B_s}ds &= t\int_0^1 e^{B_{tu}}du\\ &=t\int_0^1 e^{\sqrt{t}\frac{1}{\sqrt{t}}B_{tu}}du\\ &=t\int_0^1 e^{\sqrt{t}W_u}du, \end{align*} where $\{W_u=\frac{1}{\sqrt{t}}B_{tu}, \, u\ge 0\}$ is a standard Brownian motion. Then \begin{align*} \frac{1}{\sqrt{t}} \ln \int_0^t e^{B_s}ds &= \frac{\ln t}{\sqrt{t}} + \...


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It's a special case of the AM-GM inequality, assuming that market returns follow a lognormal distribution. Consider the simple example of a stock that has a 50% probability of rising and falling 10% every period. Its arithmetic average is obviously 0: (50% * +10%) + (50% * -10%) = 0 Its geometric average is (1+10%)^0.5*(1-10%)^0.5 -1 = -0.5% Or a more ...


2

If $S$ is the solution to geometric brownian motion SDE: \begin{equation} dS=\mu S dt + \sigma S dW(t) \end{equation} then \begin{equation} S=S_0e^{(\mu - \sigma^2/2)t + \sigma W(t)} \end{equation} Then if you take expectation \begin{equation} \mathbb{E}[S(t)]=S_0e^{(\mu - \sigma^2/2)t}\mathbb{E}[e^{\sigma W(t)}] \end{equation} Now since $W$ is a wiener ...


4

An integral with respect to a stochastic process is the theme of stochastic calculus for which you ought to get an introductory textbook as it is the key to financial models. A Brownian motion $(W_t)$ is the easiest integrand and typically the first example one encounters. Then, $\int_t^T 1\mathrm{d}W_s=W_T-W_t=W_{T-t}=\sqrt{T-t} Z$ where $Z\sim N(0,1)$. ...


1

In its simplest form, an option is a combination of two binary options. The buyer of a call option is long of an "asset-or-nothing" binary call. I.E. if Spot>Strike, it is worth Spot; else 0. To fund that, he is selling a "cash-or-nothing" call: worth Strike if Spot>Strike, else 0. The positive value of the option obviously derives from the fact that as ...


2

The Black Scholes (1973) model assumes that $\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma S_t\mathrm{d}W_t$. Thus, $$S_t=S_0\exp\left(\left(r-\frac{1}{2}\sigma^2\right)t+\sigma W_t\right).$$ Please note the factor $-\frac{1}{2}\sigma^2t$ in the exponential. If you incorporate dividends, replace $r$ by $r-q$. You do not need an extra term $\sqrt{t}$ in front of the ...


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