# Questions tagged [black-scholes]

Black-Scholes is a mathematical model used for pricing options.

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### Can derivatives of non-lognormal securities be priced using risk-neutral evaluations assuming risk-free drift?

The Black-Scholes model essentially says that, if we assume some things (lognormal, constant variance, etc.) then the following the fair price of a $$C = N(d_1) S_t - N(d_2) K e^{-rt}$$ However, ...
1 vote
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### Change in Option Price given Change in Implied Volatliity, Moneyness, and Maturity

I have an implied volatility surface parametrized into moneyless-maturity coordinates. At each period of time, I only have access to an option's moneyness (K/S), maturity, and change in implied ...
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### How to perform volatility arbitrage between two instruments with different prices but the same realized volatility

Suppose We have two assets $S_1$ and $S_2$. They have different price, but share the same realized vol. They have corresponding options $O_1$ and $O_2$. When the ATM IV of $O_1$ and $O_2$ differ too ...
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### Comparing implied volatility in 2 different correlated assets

The general idea here is that I am trying to compare the volatility surface of two different financial assets whose prices and returns time series exhibit a strong relationship/correlation : The ...
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### Monte Carlo simulations with extremely high volatility

I am using monte Carlo simulations to price a forex option. This is a standard model and works very well with less than 1 % error from black scholes price for 10000 simulations. But, as I increase ...
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### BSM replication with expiry delta

I’ve been thinking about this problem and I’m missing something. Assuming a BSM world, I sell an OTM option at strike K. I then proceed to delta hedge it at the strike K each time K is touched. Why ...
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### FX risk reversal approximation

i see this risk reversal approximation in Uwe Wystup's https://www.mathfinance.com/wp-content/uploads/2020/09/wystup_vannavolga_eqf.pdf in which the approximation of a risk reversal is given by: vega ...
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### Geometric Brownian Motion as the limit of a Binomial Tree?

Consider the price of a stock whose drift and volatility parameters are $\mu, \sigma$ respectively, over the time interval $[0, t]$. Suppose we use an $n$-stage binomial tree to model the price ...
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### Negative Dupire Variance

I want to compute Dupire Local volatility using the identity that links Dupire local variance to BS implied total variance. I calibrated an SVI on options data to get the implied total variance ...
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### Preferred Option pricing model [closed]

I am at Uni studying mathematical finance and wanted to know which is most preferred /widely used model by Finance Industry Practitioners from the list below. Fourier Transform for option pricing ...
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### stdev of delta hedged portfolio for call option !=0. Why?

I wrote a Monte-Carlo simulation of delta hedging for a european call. R and Sigma are fixed. I start simulation with zero money and short call option. At each step I borrow money to buy 'delta' of ...
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### Is the Black Scholes PDE actually immediate from Ito's lemma?

Ito's lemma replaces $dS^2$ by $vol^2*dt$, however it is repeatedly mentioned that the lemma manifests in the integral form and the differential form below is merely a short hand for the integral form:...
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### Can a Call and a Put with same strike price and expiration date and underlying asset have different implied volatility?

Furthermore, assume that the current price of the underlying asset equals the strike price of the options. If volatility measures variance without a direction, it doesn't make sense to me that the ...
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### Barrier Puts Pricing (down-and-in put)

I am trying to price the down-and-in put option with European Style (when barrier level < strike price) by using Black Scholes Option Pricing model. but after checking the formula several times, I ...
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### How to access the Black Sholes Formula through the Distributive Law?

Recently I read a comment on how to interpret the Black Sholes Formula and more specifically how to wrap your head around the d1/d2. Although there were many good comments, this one stood out when one ...
1 vote
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### Martingale property of the CEV model

I am a bit confused about the martingale property of the CEV model. Given $dS(t)=σS(t)^βdW(t)$, is $S$ a martingale for values of $β<1$?
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### Obtain B-S-M from a binomial tree as n goes to infinty using Lebesgue integral

My question is simple, consider a European call with payoff max(S_T-K, 0), Let's suppose that the underlying stock follows a binomial tree with up and down factors I know as we take n goes to infinity ...
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### How to estimate Dealers’ Gamma Positioning

I am new here so please forgive my basic question. There are many websites and experts out there that estimate dealer gamma positions, but I don't know what they are doing. I think I understand the ...
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### Real options: discount rate for the value of the underlying security

This is an example inspired by Chapter 3, sub-chapter "Combining decision trees with real options(DTRO)", sub-sub-chapter "Case 4 Part Two", of Boer, F.P., 2004. Technology ...
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### Black scholes, issues inferring T(time to expiry) andS (underlying price)) from wrds SPX dataset

I'm working on a project with the SPX option data from wrds. This data doesn't provide the underlying price at the time of the observation, or the time to expiry at the time of the observation. ...
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### How do I calculate the implied dividend yield and/or the forward rate for an equity ETF?

I am interested in building an implied volatility surface for a given ETF given a set of option prices for several combinations of (call/put,strike,expiry). I am interested in different ways to arrive ...
1 vote
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### How should I go about computing the 30-day model free implied volatility (MFIV) daily?

As the title suggests, how can I calculate the MFIV daily (for a market index)? My MFIV follows the procedure described in DeMiguel et al. (2013) Improving Portfolio Selection Using Option-Implied ...
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### Does it make sense to use Black Scholes greeks to attribute P/L given the Black Scholes assumptions don't hold?

I've seen some takes from experts in the industry (Benn Eifert for example) who say that we should treat Black Scholes as a translation mechanism for putting price into a more workable form (IV). They ...
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### Positive Theta for an At The Money option (with real data)

Ive been doing some work on looking at historical options prices on a stock index using real data, and I came across an odd example that I cant really get my head around. I am aware that for extreme ...
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### Black-Scholes model portfolio position

Question: I am a physicist currently learning about the Black-Scholes model in a statistical mechanics course. I have been teaching myself financial terminology and was reading the "Derivation of ...
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### About Hedging of One-touch Options

The pricing of American Digital Call (one-touch Calls) has the following formulas, taken from P13, the textbook \begin{aligned} C_{\mathrm{d}}^{\mathrm{Am}}(S, t ; E) & =\left(\frac{S}{E}\right)^{\...
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### Seeking Advice on Normalizing Implied Volatility Change for Options Modeling

I'm working with a substantial dataset spanning five years of weekly options data, with records down to the second. My goal is to develop a model that can accurately predict the probability mass ...
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### In Black-1976, why is the differential equation missing a term relative to B-S?

In the notation of the original Black-Scholes paper, let $w(x, t)$ be the price of an option with underlying priced at $x$, and let $w_1$ denote the derivative of $w$ w.r.t. to $x$ and $w_2$ denote ...
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### Black Scholes derivation and Ito's Product

In this derivation of the Black Scholes equation can someone please explain the last step where the author uses Ito's product rule? I do not understand where the "rC" term comes from.
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### Risk free rate for Black and Scholes model: Incorporating inflation?

I am new to quantitative finance and I am trying to create a model for option pricing. Naturally the Black and Scholes equation is front and center for this sort of thing, but that raises the question ...
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### Connection between the $\sigma$ parameters of the spot price and the forward price

It is well known, that under the Black-Scholes framework: $$F\left(t,T\right)=\exp\left(r\left(T-t\right)\right)S\left(t\right),$$ where $S\left(t\right)$ is the spot price of an asset at time $t$, \$F\...
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### perpetual American-style call option

Greek “phi” for a derivative f is defined as its sensitivity to the changes in dividend yield q : $$\phi = \frac{\partial f}{\partial q}$$ HOW CAN I FIND PHI WITHOUT THE CORRELATION?
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### The metric to evaluate the efficiency of ANN based option pricing over mathematical option pricing models

The stock exchanges provide the data of option prices using theoretical formulations such as Black-Scholes formula. The dataset necessary for training an artificial neural network (ANN) to address ...
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