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2

Hints: You know the vega of a digital call option formula: $V=-\frac{e^{-r(T-t)}}{\sigma} d_1 n\left(d_2\right)$ Where n is the standard normal density, which is positive. Sigma and exponential are also positive, so the sign of V is down to the sign of $d_1$. Which is negative when: $d_1 <0$ $\ln \frac{S}{X}+\left(r+0.5\sigma^2\right)(T-t)<0$ $S&...


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Let $C\left(S,t\right) $ represent the price of the call option when the underlying price is S at time t. Now if S changes by h instantaneously, the call price becomes $C\left(S+h, t\right) $. So the change in the call option price is: $C\left(S+h, t\right) - C\left(S,t\right) $ Which you can approximate via first order Talyor series: $C\left(S+h, t\right)...


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If $\sigma=0$, the stock price is deterministic and grows at rate $r$. In one year, it is thus worth $100\cdot e^{0.05}\approx 105.13$. The strike is $K=100$. Your payoff is thus $5.13$. Discounting at rate $r$, you get as today’s fair option price $5.13\cdot e^{-0.05}\approx4.88$. Note that there is no randomness and the stock price is perfectly predictable....


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In the B&S world, interest rates are constant and thus deterministic. In particular, they are not correlated to the stock price whatsoever. Thus, firstly interest rates don’t change in the first place in the B&S world. You can generalise the model and allow for time dependent (but still deterministic) interest rates. The interest rates are then ...


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When you’re short an American option, the buyer of that option may wish to exercise that option early - at any time point. You have no control about that. So there is no terminal payoff, the payoff can occur at any time. For an European-style option it is clear that it may only be exercised on maturity date. If you have a portfolio of several options with ...


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Can I suggest a much more pedestrian explanation? HML is a "risk premium"... so it doesn't always pay off; and there are regimes in which it can pay off negatively by a significant degree for a significant time because the conditions are all wrong, right ;-) Consider the cliche that large-caps tend to be multi-national exporters more than smaller-caps who ...


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Preliminary This answer provides evidence (and confirms your supposition!) that the HML factor in Germany is no longer rewarded with higher returns in the cross-section of German stocks. From July 1992 - December 2018, the average HML-premium is 0.49% (2.26) per month, but insignificantly -0.28% (-1.01) from July 2011 - December 2018. Momentum (WML) and ...


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Futures are in "zero net supply", or "for every long there is a short", which means that at any time there are investors who are long a certain number of contracts and other investors who are short an (exactly matching!) number of contracts. This number is called the Open Interest. It starts at zero when the exchange introduces a new contract (like Sep 2019 ...


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To calculate the average YTM of the index, you need to use the duration-adjusted market value weight ($mv_{wd}$) for any bond $i$: $$mv_{wd;i} = \frac{mv_i \cdot d_i}{\sum_{i=1}^n mv_i \cdot d_i}$$ , where $d_i$ denotes the duration of a bond $i$ and $mv_i$ the market value. The average index YTM, $Ind_{YTM}$, is than calculated by: $$Ind_{YTM} = \sum_{i ...


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