Questions tagged [black-scholes]

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

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1answer
1k views

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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491 views

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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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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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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1answer
104 views

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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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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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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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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1answer
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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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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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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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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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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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1answer
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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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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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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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what's the relationship between forecasted stock volatility and implied volatility?(option)

what's the relationship between forecasted stock volatility and implied volatility? I know that implied volatility is the volatility calculated by BS formula, is there any relationship between implied ...
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1answer
203 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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1answer
59 views

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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Self-financing and Black-Scholes-Merton formula

Self-financing is an important concept in financial product replicating, normally used in pricing. I read about several ways to derive Black-Scholes-Merton (BSM) formula. Seems some approaches ...
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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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1answer
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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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132 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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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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Black-Scholes equation Variational / Weak form

I am having difficulty deriving the weak formulation of the Black-Scholes Equation. I have multiplied it with a test function phi and integrated over Omega. But results on the internet suggest ...
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1answer
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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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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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1answer
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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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1answer
367 views

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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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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1answer
101 views

Stochastic solution (mean, variance) to lognormal drift and normal volatility

I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model I ...
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What causes the call and put volatility surface to differ?

I currently have a local volatility model that uses the standard Black Scholes assumptions. When calculating the volatility surface, what causes the difference between the call volatility surface, ...
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2answers
886 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.
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735 views

Where can I find a clear explanation (brief derivation) of N(d1) and N(d2)?

Where can I find a good explanation (perhaps with a brief derivation) of N(d1) and N(d2) from Black-Scholes? Just trying to understand the general idea about these 2 probability functions and how they ...
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Why do some people claim the delta of an ATM call option is 0.5?

I am looking for a mathematical proof in terms of differentiating the BS equation to calculate Delta and then prove it that ATM delta is equal to 0.5. I have seen many books quoting delta of ATM call ...
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Linear Or nonlinear Black Scholes Equation

I have been going through the analytical solutions of black scholes equation which transforms it to a heat equation. $$u_{t}=\frac{1}{2}\sigma^{2}u_{xx}$$ Now if the volatility is constant , then its ...
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some questions about pricing an asset or nothing put option with a strike price equal to St

I am working on a homework exercise where the aim is to price an asset or nothing put with K = St, offcourse the normal formula could be used St * N(-d1), but I was wondering if pricing the asset by ...
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534 views

Derivation of Call Delta from Black Scholes Model

How is call delta mathematically derived from Black Scholes Model (without approximation) ? Please help me understand each step mathematically. And how it is approximated to say that delta is the ...
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1answer
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Asset return distribution

What is the basis for assumption that asset prices follow a log normal distribution? Then how is it transformed to say that asset return follows a normal distribution? How this relationship between ...
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Security value based on futures contracts of a traded and non-traded assets

S1 - index with dividend a, S2 - non-traded asset. A security pays off $S_{1T}S_{2T}$ upon its maturity S1 and S2 are uncorrelated and follow geometric brownian motion. What is the value of ...
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Delta hedging: theoretical value vs actual price

One way to derive the Black-Scholes PDE is via the Delta-hedging argument: Suppose that $V_t = V(t, S_t)$, for some function $V: [0,T] \times \mathbb{R} \to \mathbb{R}$. We construct a portfolio by ...
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126 views

Intuitive explanation of why ITM options have low Time/Extrinsic Values?

While brushing up on my knowledge about the Greeks, I have been struggling coming up with an intuitive, probability-based explanation behind why not only Out-of-the-Money (OTM), but also In-the-Money (...
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1answer
192 views

Calculate the price at time t=0

Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Calculate the price at time $t = ...
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Develop an option pricing equation by Ornstein Uhlenbeck process

I know that Black-Scholes equation is based that the Equity price has a Geometrical Brownian movement. Can I develop from the same principles( now with transaction cost) that Black Scholes is ...

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