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4

An actual option with an independent existence cannot have a negative price. But we are talking here about 'embedded options' that are part of another security (in this case a USTR bond) and cannot be separated from their parent. Their price is not quoted in the marketplace but is found by a calculation. The problem is that this calculation comes up with a ...


2

Agree with @Brian B. With BS, you cannot have the issue in (1). Tree, grid, Monte Carlo could all result in errors though. (2) is a likely reason. I just tried in Julia for ATM, 0 div and rates plus 0.2 vol and 1 year tenor. Shifts smaller than ~ 0.00008 result in an error for Gamma. Delta seems to be less sensitive for this, and it is fine for at least 1e-7 ...


0

Volatility = Vega = Supply/Demand (Marginal Cost of FOMO) Just hypothetical stuff here. Control the price through controlling the volume in the most important near-mark strikes. You set the price to what you want, which pushes out or pulls in the price of the strikes that are further out. You just control the volatility by changing what you're willing to buy ...


0

These observations do not hold through time or moneyness in general. A few remarks below: EMINI and SPY are not the same underlying! The CME has a document highlighting some key differences. Trading hours: SPY: Primary exchange for SPY options is in San Fran and local trading hours are 6:30 - 16:15 according to Bloomberg. ES - Emini options on CME trade ...


3

You are correct, in that you need not (explicitly) specify real world dynamics to calculate option prices. Indeed in many rates derivatives models, you simply assume a unique risk neutral measure exists (completeness), specify the dynamics under the risk neutral measure (risk neutral probabilities) and price your options. At the same time, it is important to ...


2

Delta is not the probability of finishing in the money as suggested in another answer, N(d2) is. The foot note mentions this. See Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model by Lars Tyge Nielsen for a detailed explanation. If time and vol is low, $d1 \approx d2$ and delta will be closer to the risk-adjusted ...


2

Everything (warning: I have not checked 3rd order greeks) that is not delta is in terms of ccy2 in the standard Garman Kohlhagen model. Gamma is not in CCY1 by default either (some vendors like Bloomberg display it like that to be consistent with Delta). Why Let's start by not looking at FX but equity to help build intuition. The actual price of an option is ...


3

The best way is to start with definitions (instantaneous and their finite difference versions) of Greeks. For a currency pair $(FOR,DOM)$ with FX rate $S$, the number of $[DOM]$ (domestic, numeraire, right-side) units needed to buy one $[FOR]$ (foreign, asset, left-side) unit, let $V(S)$ be an option's price in $[DOM]$ units. Note that the unit of $S$ is: $$ ...


1

Copy pasting parts of an answer I did here as it illustrates the limits of call and put option premia. $N(d2)$ is the probability that a call option with an exercise price of $K$ is exercised in a risk-neutral world. Therefore, $(1− N(d2)$ or $N(-d2)$ is the probability that a put with the same exercise price will be exercised. Let's plot this as a function ...


2

Not sure if you meant only short puts with "when option is ITM increase in volatility will decrease the delta, whereas for OTM option increase in volatility will increase the delta". Either way, you cannot generalize it like that as you figured out yourself. Adding a few remarks to @Kermittfrog's excellent answer. If you plot delta and the ...


2

I guess generally what ATM means depends a lot on asset classes. FX vols are quoted as ATM DNS (delta neutral straddles). This in itself can be Spot, Forward, Spot premium adjusted, forward premium adjusted with the following formulas retrieved from the working paper FX volatility smile construction : However, based on your wording I assume you think 50D ...


3

Tricky right? The reason is that when dealing with stocks or options or any sort of spot market asset, there is an original cash settlement. For instance: You have an account with $4000 and you buy AMZN for 3500. 3500 leaves your account and 1 share of AMZN comes in. Your account is now 500 + 1 share AMZN. AMZN price goes to 4000. Your account is now worth ...


-1

When referring to a long straddle, atm means the 50 delta strike.


0

For straddles ATM usually implies 0 delta. In general, ATM is determined by the market conventions in question.


5

In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, continuously compounded risk-free rate $r$, continuously compounded dividend yield $y$, time-to-maturity (in year fractions) $\tau$ and implied volatility $\sigma$, our $\Delta$ is defined as $$ \Delta\equiv \frac{\partial V}{\partial S_t}...


0

Since you write under BS model, it is a tautology in this model. Assume you use Black, which you can without loss of generality, as one can easily be transformed into the other (FX is easier to show with covered interest rate parity in my opinion). Looking at $$d1 = ( log(F/K) + 0.5*σ^2*t ) / (σ*sqrt(t))$$ $$ d2 = d1 - σ*sqrt(t)$$ it is not immediately ...


1

The V that you see is only at expiry (like any hockey stick) and completely independent of vol or tenor. All that matter is notional. Assuming put backspread, you sell a put with higher strike, and buy it back with lower strike(same maturity). The more you buy the steeper. Vol will only impact the position of V. The more expensive the long positions are, the ...


1

Not that it will add much to the above, but I always kind of took this as given and did not think of it too much. Your question made me look into this a bit more. I assume you mainly talk about equity or indices here. So I did some searches and read some papers. Here is a summary of what I found (in an arguably short time period) but it largely confirms the ...


3

It really depends on the market you are interested in. Currently, almost every market has some peculiar shapes in the volatility smile driven by different dynamics or upcoming events. One famous example is the equity index market. If you have access to some current market data, just check the volatility curve for large indices like S&P 500 or EuroStoxx ...


1

How do you define higher payoff? Could you show what you compute? Do you look at what the options cost at the moment? If you want the same payoff (graphically and expiry), you can do the two things: buy call (say ATM) and sell call (OTM) buy put (ATM) & sell Put (ITM with same strike as OTM call) Now, it is intuitive that (although same strike and ...


3

Short answer: Gamma is computed differently; price is quoted by the exchange in a way QuantLib does not compute Long answer: BBG's DES page computes Gamma as 1% chg in underlying - there is also no flexibility with the DES page and frequently this is not the best calc they offer. However you can load any listed option in their OTC pricers to get listed ...


3

Some categories of bond issuers, particularly some financial institutions, have regulatory requirements to jump through some mildly annoying liquidity hoops when they have outstanding bonds with less than 1 year left to maturity. Such issuers often find it more convenient to issue bonds that have a call date 1 year (or sometimes more) before maturity. You ...


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