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$ \sigma S $ is in units of dollars per square root of a unit of time. $ \sigma $ is usually quoted as an annual or daily percentage. $ dX ^2 $ is in units of time, as $ E[(dX)^2] = dt $. Here is an online tutorial which you may find helpful. EDIT by kotozna: $\sigma$ has dimensions 1/(square root of time) and $dX$ has dimensions square root of time. ...


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The Black Scholes formula still holds without any hedging (as long as the underlying follows a GBM) by the first fundamental theorem of asset pricing. The original paper by Black and Scholes actually made an equilibrium argument as well as a no-arbitrage argument to show that the formula is valid.


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I use the Black-Scholes formula here http://www.seleno.us/options.php often when I buy and sell options. Sometimes the Black-Scholes price equals the price of the option on the exchange. A lot of times when companies grant stocks options they use the Black-Scholes to report the expense of the options or determine how much the stock price might be diluted by ...


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Well the terminal FX rate is lognormally distributed and lognormals are skewed. So this is not surprising.


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Here is to continue the above answer of Emcor to make it more explicit. Note that the fact given in the question should instead be \begin{align*} P(\inf \big\{t \in [0, T], B_t +ct = a \big\} \geq T) = 1- \Phi\Big(\frac{a-cT}{\sqrt{T}}\Big) + e^{2ac}\Phi\Big(\frac{-a-cT}{\sqrt{T}}\Big). \end{align*} Then, for $0<t_0\leq T$, \begin{align*} P(\tau \leq ...


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If you assume the payoff is paid at time T, you just have to compute P(tau < T). In this case, you have everything you need to do it. If the payoff is paid at time tau, you need to compute the density of the stopping time.


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$S_t$ is already under $Q$ (riskfree drift), so you not need to change the measure here. Note that $c:=\left(\frac{r}{\sigma}-\frac{1}{2}\sigma\right)$ and $E\left(1_A\right)=P(A)$. So one computes the European option price as the discounted payoff expectation: $$C=e^{-rT}E\left(1_{\tau\leq T}\right)=e^{-rT}P(\tau\leq T).$$ The option price equals the ...


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the length of rate which closest corresponds to the maturity of the option. This will be true opportunity cost of having capital tied up in option positions with regard to the risk free rate.


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You need to replace Z by Brownian motion at time t. Also, the expectation should be conditional expectation with respect to the sigma-algebra at time t. See http://kalx.net/fms/fms.html for a more complete explanation.


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I do this question to death in Concepts and ... If (discounted price of) everything is a martingale then every trading strategy is a martingale. Therefore any self-financing portfolio of initial value zero and has expectation zero. Therefore there are no arbitrages (since these have positive expectation and initial value zero). So there is no arbitrage in ...


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put call parity implies no arbitrage , along with a expected value of stock price of $pe^{rt}$ . working out the integrals yields this outcome


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The fact that one claim has an arbitrage-free price, does not imply that the entire market (for all claims) is arbitrage-free. E.g. $C_T=0$ is always arbitrage-free.


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The stock weight $\pi_t\geq0$ is nonnegative as $C_T=\max(S_T-K,0)$ is increasing in $S_T$ (always long). $\pi_t\leq 1$ cannot be greater than one, because one can receive at most 1 stock from the call at maturity, so you dont pay more than 1 stock price for it. Otherwise, one could buy the (cheaper) call and sell the replicating portfolio for riskfree ...


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Part 1: Show that there exists a trading strategy which replicates a European Call. Proof: I am actually going to prove a stronger statement: that there exists an admissible trading strategy which replicates any payoff in this market. By the First Fundamental Theorem of Asset Pricing, there is no arbitrage if there exists a change of measure such that, ...


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Actually BS model is still applicable in the market where the upwards/downwards move is much more probable than move in the opposite direction. The Black-Scholes price process model has the form: $\frac{dS}{S} = \mu dt + \sigma dW$ And with significantly non-zero $\mu$ (called drift) it will capture just what you are talking about. Quite surprisingly, the ...


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Well if you think that this model represents reality more accurately than the Black-Scholes assumptions. A lot of people do indeed think so. But I wouldn't say you're "tweaking" Black-Scholes... you're just assuming another model altogether and you will use risk-neutral pricing to compute the fair value of the option at time $t$, just like BS. Frankly, I'm ...


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A portfolio $V_t(\alpha_t,\beta_t)$ (for stock $S_t$ and zerobond $B_t$) is self-financing iff: $$V_t=\alpha_tS_t+\beta_t B_t$$ It further implies $$dV_t=\alpha_tdS_t+\beta_tdB_t$$ To replicate a derivative $C(S_t,t)$ by a self-financing portfolio of stock and bond, set: $$dV_t=dC_t$$ The dynamics of $dC$ can be specified using Ito's Lemma on ...



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