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European Contracts It's a really important question and as @noob2 commented, the FTAP is normally applied to European-style derivatives, even if they are (strongly) path-dependent, including barrier options and Asian options. The idea is always the same, $V_t=B_t\mathbb{E}^\mathbb{Q}\left[\frac{\xi_T}{B_T}\Big|\mathcal{F}_t\right]$, that is the derivative's ...


4

You could but there are difficulties associated with this approach. The main one is that $\tau$ is stochastic, ie it is different for different paths of $S$, so the standard Black-Scholes formula does not apply. For example some $\tau$s you need to check are of the form $\tau =\inf\{t : S(t) <B\}$ in which case you need to value a barrier option with ...


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He circled the volume of the option in question. It was roughly 50,000 and one option contract is usually for 100 shares. Price was 0.13. $50k*100*0.13 = 650k$


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You can use the standard black-scholes formula to price an european option. The only parameter you do not know to use the formula is the volatility. If you have the price of an american option then you can use the Cox-Ross-Rubinstein (CRR) model to backout the implied volatility. Then just use black scholes. The CRR model: In the framework of the CRR model, ...


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Not sure what version you have, but if you check the QuantLib documentation (link), that class was deprecated in version 1.17: Use FdBlackScholesVanillaEngine instead. Deprecated in version 1.17. To get it working you just have to replace the this line: engine = ql.FDDividendAmericanEngine(process, time_steps, grid_points) with: engine = ql....


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Lets assume the price of the underlying equals the strike at some point prior to expiry. Then the probability of the price being still greater or equal the strike at expiry is 0.5. So the probability of the European option paying out is exactly half of the probability for the American option.


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No. For example, consider a call option struck on a stock with current price \$150, strike price \$100, 1 year to maturity, 0% risk free rate, 5% dividend yield and 40% implied volatility. The intrinsic value is \$50 but the price of the call is \$47.52.


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$$S(t_n)-D_n-Ke^{-r(T-t_n)}\geqslant S(t_n)-K$$ means that if LHS (lower bound of option price) $\geqslant$ RHS (what you get if you exercise early), it cannot be optimal to exercise. This is an assumption, not a claim that it is true or must hold under any circumstances. The rest is just reformulation to have $D$ on one side. Hence, iff $$D_n \leqslant RHS$$...


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Implied Volatility represents uncertainty. While the election is over, there is no certainly in future laws that will help or hurt any company. There are certainly other factors other then the election which can increase uncertainty for the company. Same for NetFlix. The election is one of only many factors which contribute to uncertainty. Again, no future ...


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In both cases, you should argue along the lines of options arbitrage. If you think an asset (or a portfolio) is relatively cheap (as in: arbitrageable) then you simply buy low, sell high. In your first case, it seems that the call is too cheap, as in: $$ C^E<P^E+S-Ke^{-rT} $$ So let's buy the call, sell a put, short a unit of stock, and borrow some money: ...


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Usually, you would use the volatility from a fitted volatility curve or surface. Those are based on implied volatilities. You can use historical volatility, but then your valuation is likely to be off because the volatility curve/surface is not constant and at historical vol. You should use a yield curve to present value nodes. This is unlikely to make a big ...


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Following the notation in Hull, let $H$ be the barrier level. I list the prices of European-style down-and-out barrier options with continuously observed barrier. If $H\leq K$, then $$c_{di}=S_0e^{-qT}(H/S_0)^{2\lambda}N(y)-Ke^{-rT}(H/S_0)^{2\lambda-2}N(y-\sigma\sqrt{T})$$ and $$c_{do}=c-c_{di}.$$ If $H>K$, then $$c_{do}=S_0N(x_1)e^{-qT}-Ke^{-rT}N(x_1-\...


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Your understanding is correct. It makes no sense to exercise a call (as well as a put) when there is time premium remaining because you are throwing away that time premium by doing so. Sell the call and buy the stock if you want to own it in order to capture the dividend. Buying the stock to capture the dividend makes no sense to me if it's a non sheltered ...


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To add to @DM63's answer, as a secondary characterisitc, vol may also matter in deciding the european approximation impact. As vol goes to 0, you want to exercise as soon as possible, because the underlying future rate becomes a constant (a martingale with no vol). As you'll receive the same payoff at at date, better to get it earlier (if rates are.positive)....


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For this, let us consider two portfolios: A:One American Call (c) and Kexp(-r(T-t)) in money mkt and B: The stock S. Now, let us say we decide to exercise the Call early, Value of portfolio A (by exercising at t1): (S(t1) - K) + Kexp(-r(T-t1)) < S = B Now if we consider the portfolio A at expiry (time T): Value of portfolio A (by exercising at T): max(S-...


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