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N(d2) < N(d1). See Nielsen 1992. All else equal, the formula for a put (and call) according to Black Scholes will always increase in t.


3

The greeks are non-linear and only give you the instantaneous rate of change. The larger the change in underlying is, the less accurate the change based on the greeks will be. A doubling of the underlying will certainly not be predictable by the greeks alone. Delta and Gamma can be used to estimate the new price using a second-order taylor series ...


4

In addition to the good answer given below: think of options with almost no maturity left (i.e. 1-day before expiry): for these options, the time value of the option is almost zero, so the option value as a function of the underlying has almost fully converged to the "step-function" of the pay-off at maturity (i.e. the classical "hockey-stick&...


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Your greeks are derivatives, not absolute price differences, when your underlying changes by 1%. Also, the put price and delta are not linear in the underlying (in fact in your case they are highly non linear) and you cannot expect for a large change (which the 1% change is) that the delta increases by the linear amount that you expected. You need to look at ...


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It is fairly standard to hedge a sold option as follows: at any time $t$ buy $\alpha(t)=\frac{\partial}{\partial S}c(t,S(t))$ amounts of stock $S(t)\,,$ and invest $\beta(t)=\frac{c(t,S(t))-\alpha(t)S(t)}{B(t)}$ into the money market account $B(t)=e^{rt}$ By definition, the hedge portfolio $X(t)=\alpha(t)S(t)+\beta(t)B(t)$ exactly matches the option value ...


4

Yes, conceptually, you can replicate options with stocks and bonds, but why would you when options are readily available? Put another way, you can replicate a Coke with the right amount of sugar, water, flavorings, etc., but why would you do that when you can just buy a bottle. How easy are those ingredients to get individually (and in the minute quantities ...


10

"But at the same time, it has left me confused. If I could achieve such a portfolio using stocks/bonds then why are call options studied separately?" Good question! The short answer: because it turns out options cannot be synthesized using stocks and bonds only, except in the highly idealized case of Black-Scholes which assumes constant volatility, ...


1

An old question, but I'd like to add to the answer by Kiwiakos that it is not only correlation that results in skew (defined as the slope of the ATM implied volatility), but actually the product of correlation and the "vol of vol", which explains why/how skew exists and its magnitude when volatility is stochastic. (Of course there could be other ...


3

The value of an option is based on its intrinsic value plus its time value. Intrinsic value is simply based on, for example for a plain option, the strike price of the option and the underlying instrument’s spot price. Intrinsic value remains unchanged as the maturity is approached as long as the underlying instrument’s price remains unchanged. Time value, ...


0

The premium on a put option may be estimated very easily using the following equation that I have discovered, which is fairly accurate compared to BS when strike price is on the x-axis and premium is on the y-axis: $2\sigma e^{-\frac{\left|k-S\right|}{4\sigma}}\ +\left|\frac{k-S}{2}\right|+\frac{k-S}{2}$ For a call option, reflect the equation in the y axis ...


2

Your end goal of obtaining "fair" theoretical option prices will unfortunately require a lot of effort if you want to get this done properly on your own. Here are a few reasons why: All Nasdaq marketplace stock options are American-style IVOL exhibits a skew You need to have reliable interest rates Dividend assumptions will influence your pricing ...


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They are usually "mid" of bid/ask. For illiquid options (OTM, very long dated) I wouldn't necessarily trust either the bid or offer. Traders are known to paint the market on both sides.


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First, let's go back to basics to answer why theta can be both positive and negative, and why it's referred to as time decay? At it's core, an option's value is composed of two components: intrinsic value, and time value. As time passes, the proportion of the 'time value' gradually decreases until the option is worth exactly its intrinsic value at its ...


4

It’s just the effect of interest. If you are long a deep ITM European put, it is worth the PV of K minus the stock price. But one day later the PV of K has grown a bit. That’s it. It’s the opposite for calls because you have to pay the K, so bringing the date closer costs you money. This is all assuming interest rates are positive.


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You never know for sure what will be the next market move. You assuming that underlying will continue to decline and in this case additional hedge will eat your collected premium. But what if underlying will reverse? In this case you delta risk will eat you premiun even faster. You should always try to find right balance between frequency of hedging (which ...


4

If you think of New York and London as cutoff, that doesn't exist as a market quote (white instead of amber in OVDV as it is interpolated). BGN and BGNL stands for New York and London daily close according to the times shown on XDF. Never use CMPN, that is composite and just a hard coded list of contributors without any quality check (timeliness, spikes etc)....


3

You are confusing listed options with OTC (over the counter options). Conventional option chains only exist for listed options. Equity and commodity are liquid listed option markets. FX and IRS are predominantly OTC markets and mostly volatility quoted. Focusing on FX, there are listed option markets as well. The CME offers both, vol quoted and premium (...


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The fx interbank option works by dealers quoting standard runs of implied volatilities on specific tenors and product types (fly/strangle, risk reversal and at the money) which are currency pair dependent. The strikes that make up these structures are negotiated as part of the deal. Some market data providers publish these implied volatilities (like ...


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