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The comments already give links to many top answers and articles outlining the answer. Here's the summary: The Black-Scholes formula for European-style call options is $$C = Se^{-qT}\Phi(d_1)-Ke^{-rT}\Phi(d_2).$$ The option delta (sensitivity to changes in the stock price) is $$\Delta=\frac{\partial C}{\partial S} =e^{-qT}\Phi(d_1).$$ Firstly, the delta of ...

3

Let's assume that the relevant pillar $t$ of your curve is currently (exclusively) calibrated using the reference instrument $f_0$ at market quote $q_0$. The instrument could be a swap, a forward rate agreement, tenor basis swap... In what follows, I simplify somewhat in using scalar expressions; in practice you may see gradients / vector valued functions ...

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I'll rewrite the first payoff in the more common form $(S_T/S_{t^*} - k)_+$, where $t^*$ is the forward start date, $T$ the expiry date, and today is $t$. So $t < t^* < T$. I'll assume a pure stochastic volatility model (quite important to specify the model). Then the forward start option price today is \begin{align} E_t \left[ \left( \frac{S_T}{S_{t^*}...

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If you want to split hairs, as I like to do, there are 2 ways to express Delta. "Pure Delta" is a fraction, i.e. a number between 0 and 1 (for a Call). In terms of units, it is a pure number. "Position Delta" is equal to Delta times the size of your Call position. If you have Calls on 100 shares and Delta is 0.5 then Position Delta is 100 ...

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You'll find the same issue with all of the greeks. I would say that the standard rule for delta is the following: If you're talking about a single option/strategy, then people will talk about delta percentage*. If you're talking about a single option/strategy, and you don't plan to hedge the delta when it tades (i.e. you want to trade it live), then you'll ...

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