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You can derive these formulae by tweaking the black scholes derivation. If you are using PDE method, you will use different boundary conditions. If you are using integration over the risk neutral probability , you will use a different payoff function but the same risk neutral density. Alternatively , you can observe that these payoffs are combinations ...

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Formally, a long call payoff can be split as follows: $$(S_T-K)^+ = (S_T-K)\cdot 1_{\{S_T>K\}}$$ $$= S_T\cdot 1_{\{S_T>K\}} - K\cdot 1_{\{S_T>K\}},$$ that is, long an asset-or-nothing digital call payoff and short a cash-or-nothing digital call payoff. Here, $1_A$ is $1$ if event $A$ takes place, and it is $0$ otherwise.

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You can use such an approximation but there are known analytical prices. You have a special case in which the stock price is normally distributed. See Bachelier Model. Set $\mu=r-q$ (if you have dividends, or simply $\mu=r$ if there are no dividends). So if you change from the real worl probability measure $\mathbb{P}$ to the risk-neutral measure $\mathbb{Q}... 3 As Daneel mentioned in his comment, you can't simply split your expectation of product into a product of two expecations as the two quantities are far from being independent... Now, to answer your question w.r.t. how you could compute the expectation of the joint event of being in the money while having hit the barrier, you were right in using the reflexion ... 2 I have laid out below one way of solving this kind of problem. You have your timeline right and I have reproduced it with the correct amounts. The way to discount your 30Ks is the same as discounting 1,500K if you do it this way. Basically, you need to compute a discount factor. To calculate this discount factor, you need to de-annualize your interest rate ... 2 To add a bit to Will Gu's answer: Compute$\mathbb{E} \left[ \left. S_T \right| S_T > K \right]$using the fact that$S_T$is lognormally distributed with mean$ln(S_0) + (r - \sigma^2/2)T$and variance$\sigma^2 T$. Then find the pdf of the lognormal distribution on, e.g., Wikipedia, and compute the expectation integral. You may find the following ... 2 The value of an cash-or-nothing option is just the discounted expected payoff of the option. So the value of such a call should be$e^{-r (T - t)} N \mathbb{P} \left\{ S_T > K \right\}$, where$\mathbb{P} \left\{ S_T > K \right\} = \mathcal{N} \left( d_2 \right)$, and$N$is the cash agreed to be paid. The asset-or-nothing is a bit more complicated ... 1 Black–Scholes usually assumes your time and volatility are annualised. Accordingly, when you calculate the volatility term you would usually annualise it to 252 or 260 (or however many trading days a year are applicable to your situation). Accordingly, the time remaining term of the Binary Option must also be expressed as a fraction of a year (again, 252, or ... 1 The payoff of an European call option is$(S_T-K)^+$. At maturity, if the spot price is greater than (or equal to) the strike price, then holding an asset-or-nothing call option has payoff$S_T$, writing a cash-or-noting call option$K\$, which together give the payoff of the European call in this scenario. If the spot price is less than the strike, then all ...

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