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The Black-Scholes price of this option is approximately $14.8$. When I run a Monte Carlo simulation with $10000$ paths and "exact" time stepping, I get results very close to this value. You are simulating the terminal asset price with the first-order Euler approximation over multiple time steps: $$S(t+\Delta t)= S(t) + rS(t)\Delta t + \sigma ...


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In general, if one can create a portfolio with the same payoff as the derivative, their prices must be equal. This is also called "Law of One Price". Here an excerpt from my script: Here EMM = Equivalent Martingale Measure (Q), NA = No-Arbitrage.


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Is the one in red supposed to be the proof of the Pricing Principle 1? Or merely an intuitive explanation? It is not a proof. The explanation/reasoning in this paragraph lets the author state the pricing principle. It has hints on how to prove Prop 2.9 (for instance, see the line ...no difference between holding the claim and the portfolio...). If ...


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You can also include variance reduction techniques in you monte carlo simulations, such as control variates or antithetic variates. Both aim at reducing the variability of your simulated option price and are very popular for monte carlo simulations. http://en.wikipedia.org/wiki/Antithetic_variates http://en.wikipedia.org/wiki/Control_variates Both are ...


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It is not possible to derive the joint distribution from the expectation under the given information here. The fact that you have the expectation for all $K$ says nothing about the joint distribution $f(x,y)$ because $K$ just shifts the mean of $Y$ but gives no information on the joint probability for $(x,y)$. You may particularly note if $f(x,y)$ have >1 ...


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Assume the price follows a lognormal process. We can convert it into a problem of finding the probability of a standard Brownian motion particle starting from $0$ and hitting $x$ before time $t$, or its first passage time $\tau_x$ being less than $t$. This can be derived through the reflection principle. The paths crossing $x$ are exactly paired up by the ...



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