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13

This is an interesting and not so easy question. Here's my 2 cents: First, you should distinguish between mathematical models for the dynamics of an underlying asset (Black-Scholes, Merton, Heston etc.) and numerical methods designed to calculate financial instruments' prices under given modelling assumptions (lattices, Fourier inversion techniques etc.). ...


10

If your working modelling assumptions are such that the dynamics of the log price process $\ln(S_t)$ is space homogeneous, you have that the price of a European vanilla option is itself a space-homogeneous function of degree one. You can then appeal to Euler theorem to get the relationship you need. More specifically, define the price at time $t$ of the ...


7

Options on interest rates futures in the listed markets are always traded 1-yield (100-yield) just like the futures which are traded 1-yield. So negative rates aren't an issue and its always black volatility. In the OTC market, both normal and black volatility are quoted, but the common practice is to use black volatility is what is way more frequently used....


7

the problem is that the pay-off has discontinuous first derivative. Try a contract with pay-off that is twice differentiable and it will probably work. The problem is that all the value comes from the tiny number of paths within $\Delta S$ of the strike, and these paths have huge value. This is a well-known problem. As the bump size goes to zero, the ...


7

[Mathematically] Risk-neutral pricing means that \begin{align} C_0(K,T) &= \mathbb {E}_0\left[\frac{1}{B_T} (S_T - K)^+\right] \\ &= \mathbb {E}_0\left[\left(\frac {S_T}{B_T} - \frac {K}{B_T}\right)^+\right] \end{align} Now simply notice that the dynamics of $$\tilde{S}_t := \frac {S_t}{B_t},\ \forall t \geq 0$$ is independent of $r$ (see the ...


7

I feel like your notations are not accurate enough to write what you would like to write. Let $\Sigma(S;K,T)$ denote the implied volatility of a European vanilla of strike $K$ and maturity $T$ now that the underlying spot price is worth $S$. Sticky strike translates to $$ \Sigma(S+\delta S;K,T) = \Sigma(S;K,T) \iff \color{blue}{\frac{\partial \Sigma}{\...


6

Options have an asymmetric payoff profile: The payoffs are zero for almost all cases and positive else (as we well know). If the option is OTM, most of its payoffs are zero. A rise in volatility will hence increase the likelihood for instead positive payoffs from a change in the underlying price (i.e. delta increases). If the option is already ITM, most(...


6

The time to expiry is required, but it's included in the inputs: the two discounts $e^{-rT}$ and $e^{-qT}$ and the standard deviation $\sigma\sqrt{T}$. You might argue it could be documented more clearly, and I might agree with you.


5

This is a little more complicated than the answer provided above since this is FX and the convention for determining the strike matters. https://www.researchgate.net/publication/275905055_A_Guide_to_FX_Options_Quoting_Conventions Most pairs take premium in the foreign (i.e. left hand side) currency. This means that you are paying for an option in the ...


5

Basically, the author is saying that the delta of an option, $dC/dS = \frac{\partial C}{\partial S} + \frac{\partial C}{\partial v}\frac{\partial v}{\partial S}$, where the $\frac{\partial C}{\partial S}$ is the delta assuming constant volatility, the $\frac{\partial C}{\partial v}$ is the vega of the option, and the $\frac{\partial v}{\partial S}$ ...


4

I've started thinking about this, too. My gedanken conclusion turned out to be too simple once I found what I was after: http://www.investment-and-finance.net/derivatives/o/option-beta.html, which I've confirmed in Black & Scholes (1973) p10 (eq 15). In short: $$ \beta_{\text{option}} = \frac{S\cdot\Delta}{O}{\beta_S} $$ where $S$ is the underlying ...


4

It is false. Here is an example. Let $$ dS_t = rS_t dt + f(S_0) S_t dW_t, $$ $$ dB_t = r dt. $$ The price is then the Black-Scholes price with volatility $f(S_0).$ The delta is the BS delta plus $$ f'(S_0) \times \operatorname{BS Vega}. $$ Picking $f$ appropriately, we can make the Delta as big as we like. Note that the example is highly artificial in ...


4

Since the volatility is not changing, we can assume that the only change is the underlying asset price $S$. Then \begin{align*} C(S+\Delta) &\approx C(S) + Delta \times\Delta +\frac{1}{2} Gamma \times \Delta^2 \\ &=11.50 + 0.58 \times 0.5 + \frac{1}{2}\times 2 \times (0.5)^2\\ &=12.04. \end{align*}


4

The strangle vol defined in your formula \begin{align*} Strangle(∆) = 0.5[Call Vol(∆) + Put Vol(∆)] - ATM Vol \end{align*} is the smile butterfly volatility. Then you have the volatility quote. Your confusion is caused by the misuse of notations. Note that, other treatments are also available. See for example, FX Volatility Smile Construction by UWe Wystup ...


4

Just to strengthen the intuition in the perfect answer above: With r going very high (and hence F), all prices on cash instruments are expected to gain fast with time (to compensate for the carry) and the call-strike is expected to be deep[er] in the money; hence with a delta close[r] to one.


4

You are looking for the Greek commonly referred to as Charm. This is a quick visualization with a good chart I found on Google: https://www.optiontradingtips.com/greeks/charm.html


4

@Kiwiakos gave you the intuition. Here is the corresponding analysis that you asked for. The European plain vanilla call delta is given by \begin{equation} \frac{\partial C_0}{\partial S_0} = \mathcal{N} \left( d_+ \right), \end{equation} where \begin{equation} d_+ = \frac{1}{\sigma \sqrt{T}} \left( \ln \left( \frac{S_0}{K} \right) + \left( r - \frac{1}{2}...


4

Strictly speaking, you cannot aggregate (i.e. sum) deltas. However, equity traders often provide their net exposure in currency units, which is a useful number. The same reasoning is possible with equity options: You can compute the 'delta equivalent position', i.e. delta times number of contracts (times multiplier) for each stock. Taking the delta ...


4

First note that delta is the derivative w.r.t. to the spot and not the strike. The latter is often called "dual delta". Also, you don't need any knowledge of Black-Scholes as this is a model-independent result. The result follows from the general expression of the call price \begin{equation} C_0 = e^{-r T} \mathbb{E}_\mathbb{Q} \left[ \left( S_T - K \right)...


4

Gamma and vega have the same general shape , peaking at ATM and tapering to the tails. But gamma concentrate as the option gets closer to expiry (when vega is small). For options a long way from maturity, vega increases and gamma is small. Consequently for short dated options, if the price is close to strike, the option will have to be rehedged often (...


4

This is an approximation (to first order) based on the idea that the option gives you access to the underlying, but with leverage. Let the duration of the underlying be $D_B$. The expression $\lambda=\Delta_c\frac{B}{C}$ is called the elasticity of the option (link), defined as "the percentage change in option value per percentage change in the underlying ...


3

Interest rate options (swaptions, caps, floors, spread options, mid-curves, etc) that are traded over-the-counter (OTC), as well as those listed on the Liffe/CME exchanges, have been quoted using Normal volatility (basis points, annualised) for quite some time for several reasons, not least of which is the lack of a real zero-bound in yields that you ...


3

Delta is a linear approximation of the change in price due to a small move of the relevant interest rate. Typically a parallel move of the whole interest curve is assumed here. This applies to all kind of fixed income instruments, in particular IRS. Interest rates can be given as coupon rates (these are the so called par rates, based on prices observable in ...


3

If you're asking what the FX Outright for 1M EUR/PLN is, given that table, then yes the answer is just outright = spot + fwd points, which is 3.4550 + 0.0079 = 3.4629 (you had the wrong column for your 1M value). Usually fwd points are quoted directly (i.e. not as an outright), using a divisor set by market convention. I expect EUR/PLN divisor to be 10,000, ...


3

For Black-Scholes, $\Delta_C=\partial_{S} C=N(d_1)$, $d_1= \frac{\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)}{\sigma\sqrt{T - t}}$ You may fit the volatility $\sigma$ to this term by $$\Delta_C({\hat{\sigma}})=0.25$$Note that $\Delta_P=1-\Delta_C$ by Put-Call-Parity.


3

Given that by delta means that if the price goes up by 0.01% i.e. one basis point, you gain 15 and vice versa if the price goes down by one basis point. You know that the daily standard deviation is 2.2%, than again you know that $ 220*15 = 3300$ is the standard deviation of your portfolio. So, since we are using a normal distribution you can look at a table ...


3

Note that, at time $t$, \begin{align*} d_1(t) = \frac{\ln \frac{S_t}{K} + (r+\frac{\sigma^2}{2}) (T-t)}{\sigma \sqrt{T-t}}, \end{align*} which is a function of $S_t$, and then, it is a random quantity. Consequently, the delta $N(d_1(t))$ is also a random quantity. Note that, at time $t$, $N(d_1(t))$ is known. However, $\{N(d_1(t)) \mid t \ge 0\}$ is a ...


3

Since we are dealing with the change in the value of the call option in relation to the change in the value of the stock price, we are looking at the simple slope of the function. We do not need to know how much the stock price will change in order to estimate the slope. In the absence of gamma, the stock price could change 2% or 1% and this wouldn't ...


3

IMHO the 'definition' you mention is not a mathematical definition per se, but rather an approximation used by some practitioners. Mathematically, it is $N(d_2)$ in the BS formula which figures the conditional probability that the terminal asset price $S_T$ will finish above the strike level $X$ given the information we possess today (represented by $S_t$), ...


3

Here is a page from The Options Guide with an understandable picture. They explain, As the time remaining to expiration grows shorter, the time value of the option evaporates and correspondingly, the delta of in-the-money options increases while the delta of out-of-the-money options decreases. This happens because the shorter expiration that is deep in-...


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