6
votes
Accepted
Attributing change in option prices to greek components
Suppose that the fair value of your option is a function $f$ of 3 inputs: the price of the underlying, the implied volaility, and time. You want to understand why the function value changed from time $...
- 11.3k
6
votes
Accepted
Bartlett's delta gives wrong signs for calls and puts
Bartlett's delta as computed in your code is a simple finite difference (FD), also called bump and reprice, of the Black values. I do not think there is anything wrong here, besides the fact that you ...
- 6,869
6
votes
Theta changes over time
The classic textbook theta decay shows that it accelerates until expiry. It is frequently shown with regards to the option value as shown below.
This only holds for ATM options though, because an ITM ...
- 6,869
6
votes
Theta changes over time
This old question Why we consider second derivative w.rt price but only first derivative w.r.t time and volatility suggests that it may just be called "acceleration".
If I were pricing some ...
- 11.3k
6
votes
Accepted
Since $S = e^{(\mu-\frac{\sigma^2}{2})t+\sigma W_t}$, why treat it as a constant when calculating the greek Theta (dC/dt) for a European call option?
Let me heed @Bob's suggestion and turn my comment into a full answer:
Like other disciplines, finance uses lots of shortcuts to achieve brevity and convenience. That can be awfully confusing for ...
- 15k
5
votes
Accepted
Gamma for a basket option in Python - Finite Differences vs. AAD Autograd library using Heaviside Approximation
1) have I applied Heaviside correctly?
I'd say yes, as the result matches with the final difference calculation. Although, I'm puzzled why you use so overcomplicated smoothing for Heaviside function. ...
- 393
4
votes
Accepted
The derivation of vega/gamma relationship
Just want to add the observation that the pricing PDE solution can be formally written as
$$
C(\tau) = e^{\tau \mathcal H} C(0) \quad (*)
$$
where $\tau$ is time to maturity and $\mathcal H$ is a ...
- 850
4
votes
Accepted
Can european call option on stock have positive theta? (assume positive interest rate)
@nbbo2 and @Quantuple already answered the question in their comments but if in doubt, I always think computer coding is very helpful because you can simply try it out and run a lot of calculations in ...
- 6,869
4
votes
Attempt of an analytical proof that a call price decreases as its strike increases
One interesting property among the variables in the Black-Scholes formula is
$$ S_0 \varphi(d_1) = K e^{-rT} \varphi(d_2), $$
where $\varphi(x) = \Phi'(x)$ is the normal distribution PDF.
This is ...
- 81
3
votes
Accepted
Option pricing Greeks in Python - incorrect Gamma with MC option pricing (Black) using AAD autograd / JAX libraries - but works with closed form?
I think the issues is because of the payoff function. You should replace the maximum() with HeavisideApprox(). Read this paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1626547
Here's the ...
2
votes
Vanna vs volga and vega
I am not sure I agree with @dm63 - this is way too long for a comment, not necessarily a definitive answer though to be honest. I think the fungibility argument is mainly applied / applicable for ...
- 6,869
2
votes
The derivation of vega/gamma relationship
Even though it is true that the volatility is constant in this setting, the relationship is valid for all terminal condition or pay-off function -- beyond the typical $(\pm(S-K))_+$ -- so long as the ...
- 2,641
2
votes
Black Scholes Theta Finite difference
@Sanjay's answer is correct but there is an important consideration from a practical perspective.
Closed form theta in BS is the change per unit time (the change after one year). In other words, ...
- 6,869
1
vote
Is Nassim Taleb wrong about his DdeltaDvol dynamics in his Dynamic Hedging book?
You’ve probably figured this out by now. But you’re confusing OTM calls with below the money calls. An OTM call is above the money, not below.
- 11
1
vote
Accepted
Get strikes from delta works with put but no with call function
If you have dividends, then $\Delta_C - \Delta_P = e^{-qt}$. As you note, you have $\Delta_C = e^{-qt}N(d_1)$ and $\Delta_P = e^{-qt}(N(d_1)-1)$.
For both $\Delta_C$ and $\Delta_P$ you then get $N(d_1)...
- 1,398
1
vote
delta-gamma-vega VaR approximation: how to calculate the delta volatility?
If you use historical VaR, i.e. reprice the portfolio under many historical market move scenarios, then you need not assume anything about the distributions of the market factors. But you need ...
- 11.3k
1
vote
Derivation of Call Theta from Black Scholes Model
In the original Black-Scholes-Merton model, with the interest rate $r$ and the dividend yield $q$ constant, you have
$$
c = S e^{-q \left(T - t\right)} \Phi \left(d_1\right) - Ke^{-r \left(T - t\right)...
- 2,520
1
vote
Derivation of Call Theta from Black Scholes Model
You have a great website that show the derivation, step by step.
It involves both the chain rule and product rule.
https://quantpie.co.uk/bsm_formula/bs_theta.php
If there is a step you don't ...
- 66
1
vote
Dollar gamma formula and its derivation
The correct formula is:
$$ \Gamma_{DV$} = { 1 \over 2 } \Gamma (S * 1 \%)^2 $$
Gamma dollars is the change in the delta dollars for a 1% change in underlying around price S. Depending on what you're ...
- 111
1
vote
Derive vega for Black-Scholes call from this formula?
The answer by @Gordon is pretty complete, but let me add one more point. Let $n(x) = N'(x)$ be the PDF of standard normal distribution.
In the derivation, note that
$$ e^{d_+^2/2 - d_-^2/2} = \frac{n(...
- 1,145
Only top scored, non community-wiki answers of a minimum length are eligible