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The term of art in our industry for this type of option pricing formula is a series solution. As Farahvartish indicates in the comments, a series solution is not considered to be an "analytical solution" due to the reliance on a converging infinite sum for actual numeric output.(*) Series solutions have been employed at least since the 1990s, when they ...


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You can write $$\mathbb{E}\left[ \max(a X_T + b X_S -K,0)\right] = \mathbb{E}\left[ \max(a X_S Y_{S,T} + b X_S -K,0)\right],$$ with $Y_{S,T} = X_T/X_S.$ For a given value of $X_S$ we can write $$\mathbb{E}\left[ \max(a X_S Y_{S,T} + b X_S -K,0)\right] = X_S \mathbb{E}\left[ \max(a Y_{S,T} + b -K/X_s,0)\right],$$ since $Y_{S,T}$ is log-normal this can ...


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Let $$I= \mathbb{E}_t^\mathbb{Q}\left[\text{exp}(-2\sigma W_{T-t}) \cdot\mathbb{1}_{S_T\ge K}\right] = \frac{1}{\sqrt{2\pi}} \int_{\hat{d}_2}^{\infty} e^{-2\sigma x} e^{-x^2/2} dx.$$ So $$ I = \frac{1}{\sqrt{2\pi}} \int_{\hat{d}_2}^{\infty} e^{-(x-2\sigma)^2/2} dx \, e^{2\sigma^2}. $$ Change variables $y = x-2\sigma$ and you are done.


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I don't believe you will necessarily find a cite-able source as, I believe, this comes from a practical rather than theoretical motivation. As you know option prices are a function of: future prices, discount rates and implied volatility, volatility surface skew and other supple/demand factors. So when you are trading these instruments, you need to ...


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When a pay-off is piecewise linear plus jumps, it the same as the portfolio of calls and digital calls. Its price must agree with that of the portfolio by no arbitrage. Every time there is a jump we add in a digital call and every time there is a change in gradient we add in calls equal to the gradient change. Here we have a call struck at $K$. Just below ...


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Using the usual arbitrage arguments, we can write option prices as discounted expectations of future values under risk-neutral probabilities. That is $$ V(S,0) = B(0,T) E\left[ V(S,T) \right] $$ Start by re-writing your particular payoff as the following sum $$ C_K+aC_{2K}+KD_K $$ where $C_x$ is a call struck at $x$ and $D_x$ is a digital option struck at ...


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Ftse100 would not have a smooth dividend yield, as your formula has, it would be discrete, being much higher on certain days of year than others. In pricing options on ftse, u need to take into account implied dividends (dividends that are implied by put call parity)


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To answer your question consider the following example using actual prices for SPY ETF on 7/31/15: "hopey.netfonds.no" By looking at the last 19 trades that occurred at the very last second, you will see a notable price movement on prices. If you go to Google/Yahoo Finance the Closing Price for the ETF is 210.50 (largest trade at the close?) but the very ...


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Essentially the question is importance sampling $$ \int f(S_T) \psi_{r}(S_T) dS_T = \int f(S_T) \psi_{\alpha}(S_T) \frac{\psi_r}{\psi_\alpha}(S_T) dS_T $$ Here $\psi_{\mu}$ denotes the log-normal density with drift $\mu.$ So when you simulate with drift $\alpha$ each sample used is $$ f(S_T) \frac{\psi_r}{\psi_\alpha}(S_T) $$ instead of $f(S_T).$ You ...


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Sorry to disagree but if interest rates is 0, the binary is still not worth $1 now. Suppose spot $S(0) = 100$, assume $x = 110$ and upon touch (whenever it happens as the option has no maturity) you receive one dollar. Suppose I buy 1 stock. If the barrier hits, i sell the stock and receive 110 USD. What if I buy N stocks at t=0? upon hit of barrier i ...



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