Questions tagged [black-scholes]
Black-Scholes is a mathematical model used for pricing options.
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Why do the Greeks not converge to the strike as the volatility tends to zero? [closed]
So, I was playing around with the Greeks in Python with some made up data for a European call option assuming the Black-Scholes model. I plotted the graphs to see what happens to the Greeks when ...
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Closed form / analytical solution for bespoke (but vanilla) Option
Question:
I want to derive closed form expression (similar to the Black Scholes formula for a call price) for the payoff below. I would like to do it from first principles starting with Expectations ...
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how to use ratio spread?
If I sell more options, then my gamma risk will be more difficult to control, but if I sell too few options, then when I judge the wrong direction, I will leave the market with a loss. I try to ...
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How can I price this option? [closed]
In the Black-Scholes model, I want to price the so called Butterfly option, where the payoff $P(x)$ is the following function: $P(x)=0$ if $0\leq x\leq 40$, $P(x)=x-40$ for $40\leq x\leq 60$, $P(x)=-x+...
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0DTE volatility and greeks
When european stock options have very little time until expiration (less than 2-3 hours), they can exhibit extreme sensitivity to changes in the underlying asset's price. This behavior leads to ...
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If we use BSM to generate a vol surface, and the vol surface isn't flat, doesn't that contradict BSM assumptions itself? [duplicate]
If we use BSM to generate a vol surface, and the vol surface isn't flat, doesn't that contradict BSM assumptions itself?
Or does the contradiction not matter because we'll still get the same rough ...
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Straddle Approximation - Directly from Integral
The ATMF straddle approximation formula, given by
$V_\text{Str}(S, T) \approx \sqrt{\frac{2}{\pi}} S_0 \sigma \sqrt{T}$
where $S_0$ is the current underlying spot price, $T$ is the time remaining ...
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Black Scholes/American Put/Martingale Condition
Consider a Black Scholes model with $r \geq 0$. Show that the price of an American Put Option with maturity $T > 0$ is bounded by $\frac{K}{1 + \alpha} {(\frac{\alpha K}{1 + \alpha})}^{\alpha}{S_{0}...
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Analyzing the Impact of S&P Volatility Shift on ATM Straddle Sale: Calculating Loss/Gain[black scholes]
Black scholes:The 1-month implied volatility of S& ;P is 16. The slope of the skewness curve is -1 point per 1%; For example, the 99% exercise trades at a premium of 1 vol point. regarding the ...
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When to use total derivative and when not to?
as I was trying to teach myself financial mathematics, I came across this topic on transforming black scholes pde to a heat equation. I had the exat same question as this post Black Scholes to Heat ...
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How to calculate option premium stop loss if underlying reaches a certain value near the strike price given the current implied volatility
I have sold a put option. The market is likely to open negative on Monday, the expiry of option is on Thursday. I have a certain stop loss level in my mind to exit this position if the index reaches ...
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Pricing an option with a certain payoff
Suppose an option with a payoff function
$$ \max((1+k)S_1,kS_2) $$ where $S_1, S_2$ are stock prices and $k>0$ is a constant value.
To value such an option, one would decompose this payoff function ...
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Theta using black scholes when time to maturity approaches 0
When time to maturity tends to 0, like on expiry day, denominator $\sqrt t$ in becomes 0 and the first term in the formula becomes large enough to make theta of the contract more than its premium. How ...
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How to calculate profit loss curve of a put option [closed]
I am using the black scholes method to calculate the premium for selling put option using the py_vollib package in Python. I can calculate the premium for a put option that has an arbitrary strike. ...
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In the paper "By Implication" by Jaeckel, he says that put-call parity should never be used in practic
In this paper by Jackel (2006), on page 2, he writes:
The normalised option price $b$ is a positively monotic function in $\sigma \in[0, \infty)$ with the limits
$$
h(\theta x) \cdot \theta \cdot\left(...
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Delta hedge for derivative in Black-Scholes market
Consider a derivative in the Black-Scholes market with the price formula $\Pi_t = F(t,S_t)$. I want to find a self-financing portfolio consisting of the stock and the bank account that hedges the ...
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Change of measure when the underlying dynamic is Ornstein-Uhlenbeck
Let the $r$ riskless rate to be constant. Let's consider the following underlying dynamic under the $\mathbf{P}$ “physical measure”
$$dS_{t}=\mu_{t}S_{t}dt+\sigma_{t}S_{t}dW_{t}^{\mathbf{P}},$$
where $...
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Is it possible to price a call option given a daily underlying returns distribution?
Apologies in advance if this problem is somewhat ill-posed. But I was thinking given the price of a call option can be formulated in terms of a implied probability density function at time $T$, would ...
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Decomposing option payoffs [closed]
Suppose an option payoff function $$max(min(S-1, 2-S), 0)$$ To value such an option, one would decompose this function, for example, as follows: $$max(S-1, 0) - max(2S-3, 0) + max(S-2, 0)$$ Now, it ...
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Practical use of Dual Delta?
I am wondering what the practical use of the Black-Scholes Dual-Delta is?
I know it is the first derivative wrt the strike price:
$$
\frac{\partial V}{\partial K} = -\omega e^{-r T} \Phi(\omega d_2)
$$...
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Increasing or decreasing BS-formula respect their parameters [closed]
The Black-Scholes formula depends of many parameters, is easy to note that it is increasing respect to the parameter $S_t$, $\sigma$ and $r$, it means to fix a all parameters and vary only one. Is ...
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Commodity forward curve Monte-Carlo
I need to value an Asian commodity option using Monte Carlo and a log-normal model. The inputs are the commodity forward curve and the volatility surface for futures/options expiry. Unfortunately, all ...
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Clarification on a Claim in Black-Scholes-Merton Derivation
In these notes: https://johnthickstun.com/docs/bscrr.pdf, towards the end of the proof of Proposition 5.2 on page 6, the author claims:
$$
\log
\lim_{n \to \infty} \Bbb{E}_\pi \left[\frac {S^*_n} S \...
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Options market making process (step-by-step)
What are the steps involved in options market making?
Does it roughly follow this procedure:
Choose a pricing model, e.g. Black-Scholes.
Calibrate the model, e.g. Volatility.
Quote a bid-ask spread ...
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Is there a Black Scholes PDE for a GBM with path-dependent volatility?
Question: Is there a known path-dependent Black-Scholes PDE?
To be a little more precise, let $S$ be a stock price under a risk-neutral measure such that $S$ satisfies the SDE with path-dependent ...
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Vanna Volga Price of an Up and In Put
In the Vanna-Volga approach to pricing first generation exotics, such as single barriers, as I understand it the pricing is as follows:
Let $K,S_t < B$. I'll choose the ATM IV $I_{ATM}$ as the ...
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Calendar spreads under black scholes world
If IV skew is flat (all strikes with the same IV ss ATM) as in the black-scholes world for all maturities, would calendar spreads be considered as pure arbitrage?
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Forward Black Implied Volatility For Within Risk Neutral European Option Pricing
Going to preface this question with an acknowledgement with how silly the ask is, but alas that is the working world; if anyone can share any ideas I'm all ears.
We're pricing an exotic option in risk ...
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Delta on dividend paying equity index
we calculate the delta as change in NPV for 1% change in spot * 100. Would bumping up the forward price by same 1% produce the same results? I'm assuming F = S *exp(... ) or it's too simplifying ...
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Financial software: academia vs. real world [closed]
I am looking for resources (if they exist) that explain the differences between quant finance software in academia and the real world, or explain how quant software is implemented in practice.
For ...
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Uncertain Volatility Model - Option Pricing R code help
I am trying to price the following call option using the UVM method in R.
The code I wrote below keeps producing the same price for the min and max volatilities, which is wrong, however, I can't seem ...
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Volatility Time and Interest Rate Time
In Sheldon Natenberg's book "Option Volatiliy & Pricing (2nd)", he mentioned that (on page 65), only trading days (roughly 252 in a year) are counted when computing vol time and all ...
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Bisection method for implied volatility not working for European Put Options
I am trying to implement a Bisection method for implied volatility calculation. I use an algorithm from Haug (page 455).
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Cash-Or-Nothing Option pricing formula when volatility and interest rate go zero
How does Cash-Or-Nothing Call Option (Fix payoff if S > K) pricing formula behaves when volatility and interest rate both go zero? Say for the scenario of deep out of money (eg. S = 0.5 x K), or ...
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Difference Between Option market price and Theoretical price? [closed]
So I am working on strategies that depends on the difference between Actual market price of option and price derived using black and scholes model.
For eg: Spot 19000 , strike 19200 . It is OTM call ...
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Modelling the Implied volatility of an asset
I want to estimate the implied volatility of an asset which has not historical implied volatility data. I do have the historical realized volatility ( I have the historical prices). What would be some ...
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Option Pricing for Illiquid case
I am currently studying crypto options trading and have observed that there is often a lack of liquidity for options (such as BTC Options) on various exchanges, including Binance. In many cases, there ...
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asian geometric option valuation-- unable to get monte carlo simulation to converge to analytic value
I'm trying to price asian put options in which the averaging window begins immediately (T=0). currently, I'm trying to match up geometric averaging between my Monte Carlo simulations and my attempt at ...
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Closed-form equation for geometric asian call option
I'm looking to use the geometric asian option as a control variable for a monte carlo simulation. However, I have an issue with the closed-form equation to get the geometric price.
I'm using the ...
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Analytical formula for discounted exposure of a European Put on a stock in Real-World measure
Is there an analytical formula to approximate the discounted exposure for a European Put on a Stock in the Real-World measure? This is just an initial phase to be able to assess the accuracy of using ...
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Convergence in the CRR model
Under certain conditions, the option price of the CRR (Cox-Ross-Rubinstein) Binomial model converges to the Black-Scholes price as the maximal step size of the partition converges to zero (i.e. a ...
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How to apply put-call parity in volatility surface construction?
How to make the volatility surface free of put-call parity arbitrage? If I bootstrapped the implied vol from a call price and plugged it into the BS model to have a put price, what if it violates the ...
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Why is the stochastic process of the volatility of a stock price square integrable?
I am taking a course in financial mathematics(Ito-Integrals, Black-Scholes,...) and there is something that is not immediately clear to me. When constructing our stock price model, the integral $\...
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m-of-n Soft Call Trigger
I'm trying to do some convertible bond pricing. Typically, there's a soft call provision which is triggered if the value of the underlying equity is above x% of par for m of the last n days. (Usually ...
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Black-Scholes implied volatility using a GARCH model
Why I'm not getting the same Black-Scholes implied volatility values as the ones given in the paper "Asset pricing with second-order Esscher transforms" (2012) by Monfort and Pegoraro?
The ...
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Valuation via decomposition or via simulation of the underlying?
My question might be very straight forward but I have seen both approaches being followed in practice so I am curious to see if there are arguments in favor or against each one. I am explaining my ...
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Is my solution on Black Scholes correct? [closed]
Question: Consider in the context of the Black Scholes model with stock price dynamics $d S_{t}=\mu S_{t} d t+\sigma S_{t} d W_{t}$, initial stock price $S_{0}>0$ and with money market account $B_{...
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Can the PDE of Black and Scholes really be derived from the CAPM?
Black and Scholes (1973) argue that their option pricing formula can directly be derived from the CAPM. Apparently, this was the original approach through which Fischer Black derived the PDE, although ...
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Delta-hedge experiment of American Put option
I am trying to run a delta-hedge experiment for an American Put option but there's a (systematic) hedge error which I cannot seem to understand or fix.
My implementation is found in the bottom of this ...
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