Questions tagged [black-scholes]

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

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Does the Black-Scholes formula work when unit of time is in hours?

In the Black-Scholes formula, the unit of time is usually in years from what I understand. An online calculator I found allows the users to input the time in days and years. Would the formula still ...
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Black-Scholes Delta value at maturity?

Having to implement a replication strategy for European options, I encounter the following problem: Delta tells me how many shares to hold at time t in my replication strategy. To do so, I simply ...
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Option pricing before Black-Scholes

According to the Wikipedia article, Contracts similar to options are believed to have been used since ancient times. In London, puts and "refusals" (calls) first became well-known trading ...
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Options delta as a percentage of option price

I'm dissatisfied with the usefulness of delta and would like to get your feedback on a slight tweak on it. Example Consider two options for a made-up stock at \$5 with IVs around 120%. Option A: ITM ...
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Black-Scholes under stochastic interest rates

I'm trying to implement the Black-Scholes formula to price a call option under stochastic interest rates. Following the book of McLeish (2005), the formula is given by (assuming interest rates are ...
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1answer
62 views

Black & Scholes under stochastic interest rate (Vasicek) [closed]

I'm a beginner in Quantitative finance and I'd like to ask you for help about this exercise. I have to price a put option on a risky asset by working under stochastic interest rate, so I have to ...
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1answer
70 views

Implied volatility and greeks of options

When we are calculating deltas or vegas for different strikes should we use the underlying asset's volatility or should we use the implied volatility for the specific strikes at a fixed maturity? ...
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1answer
60 views

lognormal assumption of Black Scholes

I have recently started learning about option pricing and the Black Scholes formula, where stock prices are assumed to be lognormally distributed and returns normally distributed. While trying to do ...
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Connecting the dots: Black Scholes, Volatility and Implied Volatility

I am a first year Management & Finance undergrad preparing for my second year Finance courses, given that term 3 and exams have pretty much been cancelled for all British first years. During that ...
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3answers
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Why is it so rare for finance theory to depart from the normal distribution?

I understand almost all of the theory that has been built upon in quantitative finance is based on the normal distribution, and obviously you wouldn't want to throw all of it out the window on a whim ...
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Pricing In Real Life vs Theory

When selling/buying vanilla call options, do one price them according to some pricing formula (i.e Black-Scholes)? Or is the only point using pricing formulas to find the implied volatility and then ...
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What is the delta of an at-the-money European call option with respect to volatility?

Question: What is the delta of an at-the-money European call option with respect to volatility? Note that $$\frac{\partial\Delta}{\partial\sigma} = N'(d_1) \frac{\partial d_1}{\partial\sigma} = N'(...
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Pricing an interest rate floor

I am trying to estimate the value of a 0% interest rate floor by pricing each individual floorlet. Since BS won't work for this problem, I am trying to use normal volatility in a Bachelier model like ...
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1answer
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Can a delta hedge be negative for all values at one time, and positive for all values at another time?

I have a problem that states there was a formula for the hedge $\delta(t, S_t)$ for a contingent claim whose value depends on only the stock value when $T=20$. In this hedge, $\delta(t, S_t)<0$ at $...
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why does monte carlo simulation become less accurate as volatility increases? [closed]

I simulated sample paths to approximate the price of a vanilla European call and then plotted a graph comparing this to the value achieved from the Black Scholes. Why do these values diverge as the ...
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Is gamma always positive for American call/put options under Black-Scholes framework?

Most reference I could find only consider European options, but I would like to know whether this also holds for American options in general (with continuous dividend yield and/or discrete dividends)?
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Pricing call option using risk-neutral martingale approach with squared stock price boundary?

I have to use the risk-neutral martingale 5 step approach under BS pricing framework to price the following call option at time 0: $$X = \begin{cases}1, &{if} &S_T^2\geq K,\\0, & {...
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Martingale Property {Proof [closed]

Can someone assist with this proof? I apologize for such a vague post. I have no idea where to begin. I am in a class a little above my level with this stuff. I have added a picture of the proof he is ...
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Black-Scholes Theory vs Actual Market Price

I have a question of which I am uncertain on how to answer, that is: Assume the Black and Scholes differential equation for option pricing with constant risk free rate, $ r $ and constant volatility $...
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Derivation of Black-Scholes for a derivative on a stock that pays continuous dividends, and the derivative pays continuous cashflows

I need help with the derivation of Black-Scholes PDE. The condition is that the derivative is written on a stock that pays dividends continuously (dividend yield D). Additionally, the derivative pays ...
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1answer
359 views

What is the formula to calculate Implied Volatility Percentile [closed]

I googled and I am unable to find any formular . Can some one give me the formula to calculate IVP , based on sets of IV's given. Thanks.
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Calculating the risk free interest rate, or the continuously compounded yield on a T-bill, at any given time

I'm working on a program using the Black-Scholes model to price options over time. I need to be able to derive the risk free interest rate, and found this while researching: In theory, r is a ...
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1answer
610 views

Longstaff Schwartz Algrorithm in R

I recently discovered the LSMonteCarlo library in R which basically determines the price of American options via Longstaff Schwartz method. I tried the ...
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Black-Scholes formula and implied vol

Is the Black-Scholes formula the only way "implied volatility" is calculated/defined in markets?
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Vanilla option pricing at different points in time

Let $C(t) = C(t; S,K,T)$ the price at time $t$ of a plain vanilla call option with maturity $T$ and strike $K$ on an underlying $S$; if for $t_1<t_2$ we have $C(t_1) > C(t_2)$, it could not be ...
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Average Strike Option with bounds

I'm looking to price a call option with an exotic feature. The price I'm trying to calculate at time $t=0$ is \begin{equation} C = E^\mathbb{Q}[(S_T-K_T)^+] \end{equation} where $S_t$ is the stock ...
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No arbitrage conditions for normal implied volatility

usually the term implied volatility refers to Black-Scholes implied volatility (also Log-Normal volatility): it is defined as a quantity which when plugged in the Black-Scholes formula returns the ...
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Cash-or-Nothing Call Option

I am trying to price a cash or nothing call option and I know know that the Cash or Nothing formula for a call option is $C(t,s)=Xe^{-r(T-t)}*N(d)$ If I have payoff X=100 r=0.03 T=2 $\sigma=0.3$ I ...
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Why the volatility of log-returns and not the volatility of the absolute level of the underlying is used in the Black-Scholes model?

If I want to price an option with the B-S model, why do I have to use the standard deviation of the log-returns of the underlying for the sigma parameter and not just the standard deviation of the ...
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3answers
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On which model is based the Finite Differences method for implied volatility computations?

I am very new to finance, so I don't know if my question makes sense but I have seen that there are different methods to estimate the implied volatility of an American Option. One of them is the ...
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Delta hedging an option with earlier expiry

The answer here states: For instance a volatility product that would expire at 10:42 am on a random day would be off term. One that expires at the same time than a major listed contract would ...
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2answers
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Black Scholes on Eurodollar Options

I am trying to replicate the Black Scholes results of CME option calculator for options on Eurodollar Options. (link) I am trying to replicate the implied volatility result by unaltering the spot and ...
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Risk-neutral Simple Return Moment Log-return Moment

I am trying to find a way to link Risk-neutral moment of simple return to risk-neutral moment of log-returns. Specifically, by making the same standard assumptions of the Black-Scholes model with the ...
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Explicit formula replication of variance swap using vanilla option under black and scholes model with nonzero risk-free rate and nonzero dividend [duplicate]

I didn't find the formula for the following portfolio (variance swap replication) with nonzero risk-free rate and nonzero dividend under black and scholes model : (1) I found formula and proof only ...
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1answer
96 views

Replication of variance swap using vanilla option under black and scholes model with nonzero risk-free rate and nonzero dividend [duplicate]

I didn't find the formula for the following portfolio (variance swap replication) with nonzero risk-free rate and nonzero dividend under black and scholes model : I found formula and proof only with ...
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Risk-neutral price of $H=e^{X_T^1+X_T^3}$

Let $B=(B_t^1,B_t^2,B_t^3)$ a $\mathbb R^3$-valued Brownian motion. Let $r_t$ (risk free rate) be bounded and deterministic. Let consider the DISCOUNTED market $$d\overline X_t^1=\frac52dt+2dB_t^1-...
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Computing implied volatilities of ITM and OTM options

For an ATM call the implied volatility can be computed by using the Newton-Raphson method: ...
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1answer
115 views

Why do we perform change of variable for Black Scholes equation

As an entry level financial engineer, I'm studying the Black Scholes equation, which looks like follows: $${\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\...
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3answers
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Merton model riskless self-financing derivation

Suppose $dA_t = A_t[\mu dt+\sigma dW_t]$ (assets' value) under the physical measure, plus the other assumptions of the Merton model. Suppose further that debt and equity are tradeable assets that ...
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How to mathematically calculate the probability of GBM generating difference of less than some value

I have a custom index that follows Geometric Brownian Motion (GBM) with volatility v. I started this index at 10k with 4 decimal places i.e the starting price of ...
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1answer
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In literature, is IV constantly adjusted during option delta hedging?

In a lot of literature, they like to compare the performance of buying an option, and then delta hedging either at that options implied volatility (IV) or the true future volatility. This is under ...
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Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while. We all know geometric brownian motions have drifts $\mu$: $dS / S = \mu dt + \sigma dW$ and different stocks have different drifts of $\mu$. Why would ...
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1answer
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Greeks, European puts

I'm trying to solve this question but i have a lot of problems with it. European puts with maturity 6 months are written on an asset with current price $S_0=150.$ The annual interest rate is $r=16\%$ ...
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1answer
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how does stochastic volatility models generate smiles?

When calibrating call price with the BS-model, we achieve some parameters and especielly we achieve $\sigma^*$. Now, lets say I will price call options using these parameters. Then we achieve, lets ...
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1answer
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Question About Converting Black Scholes Differential Equation to Heat Equation

I'm reading a book about converting Black Scholes equation to heat equation and I highlighted in bold for those I have doubts, and really appreciate your advice on it. Let $S$,$T$,$V$ denote ...
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Why the Inconsistency in the Derivation of BS for Dividend-Paying Underlying?

The basic idea is that we get two expressions for $\Delta \Pi = ...$ and equate them. The thing that does not make sense is that in one we take into account the dividend $$\Delta \Pi = \frac{d}{dS}V ...
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Nonlinear Black-Scholes model Vs linear Black-Scholes

I am working on a project related to Nonlinear BS partial differential equation, with terms for transaction costs and/or discrete hedging. I have two questions: Is there any exact solution to the ...
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What's the intuition behind the transformation of Black-Scholes into Heat equation?

A sequence of transformations can be used to turn the Black-Scholes PDE into the heat equation. Let $C(S, t)$ be the price of a vanilla European option at time $t$, maturing at time $T$, where the ...
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Repo rate in GBM [closed]

I have seen people use $\mu = r_f - repo$ in GBM. 1, Why do we subtract repo from risk free rate? 2, Is the stock price still a martingale?
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Monte Carlo option pricing with R

I am trying to implement a vanilla European option pricer with Monte Carlo using R. In the following there is my code for pricing an European plain vanilla call option on non dividend paying stock, ...

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