Questions tagged [black-scholes]

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

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Black & Scholes under stochastic interest rate (Vasicek)

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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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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Black-Scholes and solving for both $r$ and $\sigma$ ; Do I have a unique solution?

Below is a problem that I am working on. I believe that my incomplete solution is correct as far as it goes. I would like to know if my solution is incorrect. I plan to solve the system of two ...
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Optimal portfolio balancing based on past performance

Suppose you start with some money and invest for a fixed amount of time T. You have one risky asset and one risk free asset to choose from. You can change the ratio of the two assets any time ...
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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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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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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 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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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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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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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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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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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
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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
91 views

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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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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1answer
64 views

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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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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Black Scholes Separable Solutions

I want to find all the solutions of the Black Scholes PDE that are of the form f(x,t)=theta(x) or f(x,t)=phi(t). Can someone explain and help with this? I know the PDE formula is $f_{t}(t, x)=-\...
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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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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
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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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91 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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Margrabe option: change of numeraire versus conditioning and numerical integration

I am having a slight brain meltdown because I do not seem to be able to understand the following basic thing. Consider a BS economy, and two assets $X$ and $Y$ $$ dX = \sigma X dW $$ $$ dY = \nu Y dZ ...
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Implied volatility is returning infinity

I am trying to calculate implied volatility using javascript , I have following code ...
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144 views

Black and Scholes pricing

I want to price B&S with $S_t$ stock price that has payoff, $h(S_T)=(S_T^3-S_T^2)^+$. Would it be wrong if I solved as $(S_T^3-S_T^2)^+\implies (S_T^3\geq S_T^2) \implies (S_T\geq 1) \implies (S_T-...
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87 views

Historical volatility calculation to price options with the Black-Scholes formula

I'm looking for a reference algorithm for calculating historical volatility to price options. I know there are several volatility calculation models that use the time series of the underlying's ...
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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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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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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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98 views

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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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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Why is long term binary put option more expensive than call assuming driftless GBM?

Says X follows a driftless geometric brownian motion(GBM) given a volatility ($\mu = 0$). It gives the expected value of its initial spot. (Source: https://en.wikipedia.org/wiki/...
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GBM probability of hitting non constant barrier

I know there is a formula for probability of hitting a constant barrier for GBM/BM (See page 651 in Martinagle Methods in Financial Modelling). Is there a formula for non-constant barrier? The ...
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Mark Joshi uses forward price to price an option that pays $S_t^2-K$ if $S_t^2>K $ and zero otherwise? Why can we do that?

The following question is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $6.6$ Suppose a stock follows geometric Brownian motion in a Black-Scholes ...
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Show that $Ae^{rt}$ is a solution of the Black-Scholes equation. Why should this be so?

The following is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, exercise $5.6$. Question: Show that $Ae^{rt}$ is a solution of the Black-Scholes equation. ...
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Market model for european/american options on underlying paying discrete cash (and maybe proportional) dividends

Black Scholes is the market model for european and american options on an underlying paying no dividends. What is the standard market model for european or american options of underlyings paying ...
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Question regarding No Arbitrage price of a call option

I have a question regarding how to solve the NA price for a slightly modified call option. Say that I have a money account $B(T)=e^{r(T-t)}$ and a stock dynamic $\frac{dS(t)}{S(t)}=(r-\delta)dt+\...
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calibration - negative call price [closed]

Im trying to calibrate a stochastic volatility model to market. I end with an MSE of 2-3 with approximately 500 quotes. Some out of the money options with call-price under 1 dollar ends up being ...

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