Questions tagged [black-scholes]

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

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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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62 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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106 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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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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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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95 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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Relationship between Calendar Spread Arbitrage and Probability Density Function (pdf)

We all know that the butterfly spread no-arbitrage condition can be expressed as an inequality restriction on the second-order derivative $\partial ^2C/\partial K^2 \geq 0$, which also means the ...
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40 views

How to determine the no arbitrage price of following claim? (change of numeraire)

How do I determine the no arbitrage price for claims such as $min(S_1(T),S_2(T))$ or $max(S_1(T),S_2(T))$? We can consider a standard Black Scholes model. Hence $S_i(T)=S_i(t)e^{(r-\sigma_i^2/2)(T-t)+\...
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Determining the No Arbitrage price of max[B(T), S(T)]

Following is given, $dB(t)=rB(t)dt$ $dS(t)= (r-\delta)S(t)dt+\sigma S(t)dW(t)$ where, $r$ is the risk-free interest rate, $\delta$ the continous dividend yield $\sigma$ is the stock asset ...
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Option on Futures vs. Stocks

The Black-Scholes call on a Futures is valued as: $$ C_t=e^{-r(T-t)}[F_tN(d_1)-KN(d_2)] $$ It holds: $F_t=S_te^{r(T-t)}$. If I plug this back in, I get the Black-Scholes call on a stock: $$ C_t=S_tN(...
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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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Black Scholes PDE

I seen two variations of the Black-Scholes PDE with either $+{\frac {\partial V}{\partial t}}$ or $-{\frac {\partial V}{\partial t}}$, and wanted to ask why that is? a) https://en.wikipedia.org/wiki/...
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66 views

Delta of an option which is approaching expiration when stock price decreases

The following is an interview question. It is 10 months since you sold a one-year European call option to a customer. You have been delta-hedging your exposure to the written call since it was sold....
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how to calculate implied volatility

I have some options prices I found using the Heston Model. How do I calculate the implied volatility? In Matlab there exist a blsimpv function, but is this the right tool for me since I'm working with ...
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Assumptions in using risk-neutral pricing formula

The well-known risk-neutral pricing formula goes as follows (extracted from Shreve's Volume 2, section $5.2.4$ (Pricing Under the Risk-Neutral Measure)): Given any $T>0$ and any $t\in[0,T],$ if $V(...
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Estimation of volatility into Black-76 formula

I am trying to estimate the (annualized) volatility that should go into an European Swaption (such as 2y5y). Given we take the black76-formula, where the discounting is the term outside the ...
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Gamma for ATM options with low spots

I'm trying to compute gamma for a vanilla call with spot and strike equal to 0.001. BLACK & SCHOLES formula gave me a value of 554.761 for gamma which is a very high. I have then two questions: ...
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Will OTM Vanilla Put equal to OTM Vanilla Call with different relative strike?

I wanted to test if my strike moves 0.01 away from my current spot for OTM Call and Put. Say I have the following parameters: For put: $Spot = 1, \sigma = 1, K_p = 0.99 , r = 0, q = 0, $ For call: $...
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106 views

Pricing European call with Feynman-Kac

I am trying to calculate the solution to the Black-Scholes (BS) equation using the Feynman-Kac (FK) formula for a simple European call. According to FK, the solution to BS is the discounted average of ...
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66 views

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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63 views

Arithmetic Asian Option

Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $μ$ and volatility $σ$). Let $A_T:=\frac{1}{T}...
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Black Scholes theta as function of time to maturity

I would like to understand why the Black and Scholes greek letter theta for european call option behave in the following way: as time to maturity is far away (right part of the x-axis in the the ...
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Black-Scholes model - Calibration of the risk-free rate

I know there is a lot of content about this topic, but I have not seen a post which gives a satisfying answer to my problem. I am trying to hedge a European call option with real market data under ...
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108 views

Asian Options-Change of Numeraire

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Show that ...
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63 views

Call Option on the Square of a Log-Nomral Asset

I'm working on a quant interview question from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors).I cannot understand its answer well and really appreciate ...
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188 views

Deriving implied volatility programmatically

I'm working on a project to calculate the value of options using Python. I'm using the Black-Scholes model, and I can get accurate results by plugging in a given ...
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Black-Scholes-Merton and alternatives as interpolation tools

This is a not very quantitative question, but is nevertheless related to quantitative methods in Finance. I was reading the following paragraph from Hull's Options, Futures, and other Derivatives: ...
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Proving an Expectation

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends. Consider the perpetual American put option with payoff $(K-S_\tau)^+$ when ...
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Proving a process is martingale under the Risk Neutral Measure

Show that for any $\lambda \in \Re$, the process $Y_{\lambda,t}$ defined as: $$Y_{\lambda,t} = (S_t/S_0)^\lambda e^{-(r\lambda-\lambda(1-\lambda)\sigma^2/2)t}$$ is a martingale under the risk ...
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Floating Strike Lookback Call Option

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate $r$, stock drift $\mu$ and volatility $\sigma$). If $r=\...
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Different volatility surface ( Local vol, Stochastic vol etc.)

Despite many questions about local and stochastic volatility available on this forum, i still have a few doubts left. Essentially I am seeking validation whether I am interpreting things correctly. ...
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Black-Scholes delta of a barrier (knock-out or knock-in) option

I'm trying to calculate the Black-Scholes delta of a barrier option given the following information: Whether it is knock-out or knock-in Barrier price Strike price, $X$ Current stock price, $S$ ...
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Alternative derivation of Black Scholes by Merton

I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I ...
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Premium Adjusted Delta in fx market

Please explain the concept of premium Adjusted Delta in FX market. In EURUSD, why delta changes if premium currency is changed from USD to EUR and how this new delta is related to the old one with ...
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305 views

Possibility of delta greater than 1 [closed]

Can delta of an option be greater than 1? Please illustrate it with an example.
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Stochastic Volatility and Sticky Delta

"Stochastic volatility models can be thought of as sticky delta model. And Local volatility model as sticky Strike." Please help me understand how the author has reached this conclusion.