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Look here for a detailed derivation of the formula for $\Delta$ (be aware that this particular website uses $r_d$ to denote the risk-free rate and $r_f$ to denote the dividend yield). You can always ask for more specific help regarding a particular step in the derivation. It is easy to see that $\mathbb{Q}[\{S_T\geq K\}]= \Phi(d_2)$. Just replace $S_T=S_0\...


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In the Black Scholes (1973) model, the stock price is assumed to follow a geometric Brownian motion $\mathrm{d}S_t=S_t\mu \mathrm{d}t + S_t \sigma \mathrm{d}W_t$. If you solve the SDE, $(S_t)$ is log-normally distributed for every $t$. Alternative, you can model the returns by a normal distribution and then take the exponential function to obtain the stock ...


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In one sentence, time value has to do with the probability of crossing the strike before expiration (whether from below or above). Doesn’t matter whether the crossing results in the option being in the money or not.


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The linear/non-linear classification is concerned about the dependent variables, and its derivatives. To verify whether the equation is linear, you should be checking that the equation is linear in each of these variables, and the coefficients of these are functions of the independent variables (t and x in your example). In your example, the dependent ...


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I don't understand the question but I can try. I think the problem is to find the price of a contingent claim that has payoff $(S_T^3 - K)^+$. The well-known pricing formula is: \begin{equation} \pi(t)=\mathbb{E}^\mathbb{Q}[e^{-r(T-t)}(S_T^3 - K)^+|\mathcal{F}_t] \end{equation} Now put $Y=S^3$, by using Ito's Lemma \begin{equation} dY(t)=dS^3(t)=3S^2(t)dS(t) ...


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If $\sigma=0$, the stock price is deterministic and grows at rate $r$. In one year, it is thus worth $100\cdot e^{0.05}\approx 105.13$. The strike is $K=100$. Your payoff is thus $5.13$. Discounting at rate $r$, you get as today’s fair option price $5.13\cdot e^{-0.05}\approx4.88$. Note that there is no randomness and the stock price is perfectly predictable....


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We assume that the stock price process $\{S_t,\,t>0\}$ satisfies, under the real-world probability measure $P$, an SDE of the form \begin{align*} dS_t=S_t\big((\mu-q)dt+\sigma dW_t\big), \end{align*} where $\{W_t, \, t >0\}$ is a standard Brownian motion. Here, we need to consider the total return asset $e^{qt}S_t$, that is, the asset with the dividend ...


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The only difference in the derivation when you have a dividend-yield paying stock lies in the value of the Riskless Portfolio $\Pi_t$. The financial meaning here is the key: to delta-hedge your option you buy a quantity $\Delta$ of the stock $S$, and only the stock is paying you the dividend, so you have to add this contribution in time to your hedge. The ...


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A fundamental premise of the BS model is that equity prices move according to a Weiner process. Moreover, when there are a series of many small random movements in the share price the track that it is tracing can be assumed to be geometric Brownian motion. This process is then symbolically defined as (see Ito's lemma) $$dS = \mu S dt + \sigma dz$$ Where ...


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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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Intuitively, in a (log)-space homogenous diffusion model $$ S_t \propto S_0, \forall t \geq 0 $$ such that implied volatilities will only depend on the moneyness level and not on the absolute spot level, which is precisely the definition of sticky delta. Mathematically, consider a (log)-space homogeneous diffusion model (be it stochastic or not) $$ \frac{...


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You can find the answers to most of your questions in the Taylor's series and/or approximation theory articles, but I will add a bit more detail below (in order): A simplistic example would be $y=a+bx$ vs $z=bx$, so greeks being equal does not necessarily mean that the prices will be equal. But you can use hedging/replicating argument, though it needs more ...


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"Whose price function $V$ fluctuates according to the actual market price of that derivative"—this is not true. The reason being that we are 'modeling' the derivative price (where a model is a simplified version of reality). So $V$ tells us what the derivative price would be under our model—and since this model doesn't use the actual derivative price as an ...


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Check out Approximation of CRR as Black Scholes PDE. I show the derivation in my post there


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I think everything is related to the concept of Risk Neutral measure $\mathbb{Q}$. In deriving Black- Scholes equation you use the dynamics \begin{equation} dS(t)=\mu S(t)dt + \sigma S dW(t) \end{equation} where $W$ is a brownian motion under the $\mathbb{P}$ measure, and you get the following PDE for the price $f$ of a certain derivative: \begin{equation} ...


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In discrete time the (annualized) logarithmic return is defined as $\frac{\Delta \ln(S)}{\Delta t}=\frac{\ln(S_{t+\Delta t})-\ln(S_t)}{\Delta t}$ In continuous time this becomes $\frac{d \ln(S)}{d t}=\frac{1}{S}\frac{d S}{d t}$ Note that $\frac{d S}{dt}$ is the instantaneous rate of change in price, dividing it by $S$ turns it into a percentage change in ...


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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 ...


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This is easy to answer with the meta theorem given in the same chapter. Here you have two sources of randomness (W and N), and one risky asset. Q1: Arbitrage generally happens when you have more assets than the number of random sources, but here it is the other way around, so the answer is yes. Q2: You have one risky asset so you can delta hedge one ...


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