Questions tagged [black-scholes]

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

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53 views

Can you use the SABR implied volatility in the Black Scholes formula?

The SABR implied volatility is often used as an input in Black's model to price swaptions, caps, and other interest rate derivatives. I'm wondering whether you can use the SABR closed form solution of ...
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Why a self financing portofolio equals to the value of a call option? [closed]

Why a self financing portofolio equals to the value of a call option? Actually, why we write $V(S,t) = a_tS_t+b_t\beta_t$? where $S_t$: stock price and $\beta_t$: bank account value , $V(S,t)$: value ...
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Greeks for Black Scholes model

hey where can I find the calculated greek parameters for call and put options on shares paying dividends? I would also like to find the calculations themselves
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Derivation of $u=e^{\sigma\sqrt{dt}}$ and $d=e^{-\sigma\sqrt{dt}}$

Anyone could provide me a proof of how, starting from $\frac{dS_T}{S_t}\sim \operatorname{N}(\mu dt,\sigma^2 dt)$ with $p:=\frac{e^{rdt}-d}{u-d}$, we can obtain the parameters $u$ and $d$ as from ...
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Collar Option K Term

I know that the value of a collar option on a stock (buy stock, buy put at $K_1$ and sell call at $K_2$) is given by $$Collar\ Value = K_1d(t,T)+Put\ Value-Call\ Value$$ My question is, why do we have ...
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Why am I struggling to replicate the Black-Scholes price of an option stochastically?

I am currently trying to replicate the Black-Scholes price of a call option using stochastic simulations of the price moves of the underlying. My code is as follows: ...
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84 views

Delta neutrality (derivation)

I'm confused about the math for the delta-neutral portfolio. Assume we have a short position in a European call option with price $p(t,S_t)$ and want to hedge it with the stock with price $S_t$. The ...
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Fast Monte Carlo of Local Volatility Model

I want to compute option prices via a Monte Carlo simulation. The model implemented is a Markov process, following the SDE : d X_t = alpha * dt + beta^(1/2) * d W_t ...
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Interpreting SABR calibration model output

Calibrate a SABR model? Following on from this question, I have used the same market data they attached but am unsure on interpreting the output. When I plot the SABR probabilities against strike for ...
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BS-Model not suitable for $n$-dimensional options

Anyone could explain me what the authors of this paper mean when they say that "The Black–Scholes model, in spite of its popularity, has some well-known deficiencies. Firstly, a closed form ...
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Discounted stock price under a NON risk-neutral measure

Under a risk-neutral measure $\mathbb{Q}$, the discounted stock price is a $\mathbb{Q}$-martingale. Does it mean that under the actual probability measure $\mathbb{P}$ the discounted stock price is ...
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Gamma and Gamma Hedge [closed]

I have a very basic question: Is this gamma value has something to do with the gamma hedge? In delta hedge, it's done by buying/selling delta amount of underlying. But in textbook, for a put option, ...
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How to prove that a series of random variables $Z_j = 1$ or $-1$ occurring at risk-neutral probability, converges to normal, using the CLT?

Context When pricing options with trees, it is convenient to prove that the asset value at expiry $S_t$ be of log-normal distribution: $$\log{S_t} = \log{S_0} + \mu T + \sigma \sqrt{\frac{T}{n}} \sum_{...
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Backward difference approximation (BDF-2) for Options

I am working on a project for compound options and the assignment is as following: ...
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Option Valuation With Hard To Borrow Rates

How would you include -in a simple way- high borrow rates, say 10%. Intuitively, for PUTs I'd set r as r - borrow_rate, to include the negative carry of the borrow. So If I'm selling puts, value would ...
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Why can future forward interest rates be assumed to be lognormally distributed in the standard market model?

This seems to be the underlying assumption that allows us to use the standard market model/Black's framework in order to value interest rate derivatives, but I haven't found any understandable ...
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how can we calculate options profit and loss using volatility and implied volatility in a span margin calculation

complete the table below, mainly we have to use black-scholes model for implied volatility calculations which I am getting as 43 % but now how to make a gain and loss table using this implied ...
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Pricing of autocallable structured product

I'm looking at this paper: https://doi.org/10.1057/jdhf.2011.25, which is on pricing autocallable structured product. The author uses the Black-Scholes equation to describe the product's dynamic value,...
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Can we use a VIX-like method to calculate implied volatility for Black Scholes model?

So I understand that the VIX is an estimate of implied volatility. Volatility can also be calculated from the Black Scholes model. My question is can we use a VIX-like method to calculate implied ...
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Determination of critical stock price in compound option pricing

Under the Black-Scholes framework, there is a closed form formula for the price of a compound options, as first derived by Geske (1979). However, the analytical formula refers to a critical stock ...
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Calibrate Stochastic Volatility Model

For stochastic volatility models, and any vol model I know, it seems the standard approach is to calibrate the model from option prices. As other user said, this seems a chicken egg problem - how do I ...
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Finding option price using intraday data [closed]

I have the option price at a rate which is much smaller than the rate at which I have tick data for the underlying. If I have option price at times $t_1, t_3, t_5$ and I have tickdata at $t_1, t_2, ...
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Black Scholes implied volatility [closed]

I am reading up on implied volatility and I encountered the term Black-Scholes implied volatility which I haven't heard before. What is the meaning of this term? Say I am looking at the Heston model ...
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Pricing of an option

I've priced a European option with payoff $\max\{S_T(S_T - K), 0\}$ and found $S_0(S_0 \exp((r + \sigma^2)T) \mathcal{N}(d_3) - K\mathcal{N}(d_1))$ where $d_i = \frac{\ln(\frac{S_0}{K} + (r + \frac{i\...
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Expected life (Fugit) of American Option

How can I use the binomial tree pricing method for American options to determine the expected time of exercise for the option (or "Fugit")? In particular, how would I modify the algirithm ...
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variance of asset returns linear for time

I am reading Wilmott's book, "Quantitative Finance" and try to understand the derivation that the variance of asset-returns, $V[\Delta S/S]$, is a linear function of the time step $\delta t$....
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Quantity of risk-free asset in Black Scholes model

When the seller of a Call option hedges themselves, we know that they should buy $\Delta(t) = \mathcal{N}(d_1(t))$ amounts of the risky asset at time $t$. But what about the riskless asset? My ...
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Why does black scholes model give lower prices for puts with further time to expiry?

Consider BS-model with parameters: Stock = 100, Strike = 100, Texp = 1 year, Vol = 13%, Rf Rate = 3%. For these parameters the BS put price is 3.76. Then consider the same parameters but with Texp = ...
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Computing the Probability Density Function (PDF) for the Heston model

I am trying to compute the PDF for the Heston model using the Breeden Litzenberger formula. I have calculated the the Heston implied volatilities for a strike range (which i have interpolated using ...
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Error in Call Option Valuation using Implicit Finite Difference implemented in Python

I am trying to valuate call option using implicit Finite difference method (Forward Marching) implemented in Python. However I am getting the error in the code. Following is the code I have developed: ...
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How do I hedge two/three zero coupon bonds with different maturity under Vasicek short rate model?

I am working on the case that I need to hedge two bonds with different maturites under Vasicek model, which is \begin{equation} dr_t=a(b-r_t)dt+\sigma dW^Q_t \end{equation} and I know how to price the ...
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Black-Scholes formula given arbitrary value of $S_{T}$

Is there a formula for Black and Scholes when we have expected payoff $\mathbb{E}[\max(se^{X}-K,0)]$ for $X$ having any normal distribution?
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Hedging costs and BS-price

I'm looking at the chapter, "The Greek Letters" in Hull's book (Options and derivatives...) and in particular the paragraph "Dynamic Aspects of Delta Hedging". He demonstrates two ...
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1answer
141 views

Black Scholes to Heat Equation

Equation (2) was derived by setting r=0 in the Black-Scholes equation for the Bachelier model (1). Can someone please help me understand all the steps for how we get from the heat equation under time ...
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Breakdown of Wilmott's Binomial Tree derivation of Black-Scholes equation

Hi guys, I tried to follow the chapter of PWIQF on binomial model and got stuck when it derived the Black-Scholes (please see image). I tried to backtrack the said equations but couldn't trace back ...
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Value in time of the bond in delta-hedging

I am trying to implement a simple delta-hedging strategy. The idea is that I want to verify that the covered position "1 option long + delta stocks short" is actually evolving as $e^{rt}$, ...
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Estimating dividend yield & risk-free rate from Futures prices

I would like to work with the dividend-adjusted Black Scholes formula and need to estimate the dividend yield and risk-free rate. I know that I could compute both rates exogenously. But I am working ...
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Using EOINA, €STR for option valuation

for an assignment I have to value options using an OIS rate using the Black-Scholes model. Since my options are traded in Germany I was looking at the EONIA or newly €STR. (Since Hull 2012 recommends ...
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Option that never expires

I have been struggling with the problem below for quite some time now. I really don't know how to approach it. All I could think of is to use the Black-Scholes formula with $T \rightarrow \infty$, ...
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2answers
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Volatility estimation based on a 60 days range

In Hutchinson et al: A Nonparametric Approach to Pricing and Hedging Derivative Securities Via Learning Network (1994) paper (link), to estimate $\sigma$ for the Black-Scholes formula, it says (p. 881)...
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Stocks with same volatility but different drifts

In the book Quant Job Interview Questions & Answers, in section 2, question 2.4 says suppose two assets in a Black-Scholes world have the same volatility but different drifts. How will the price ...
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Asian option sensitivity

I am looking for some materials for profiling all options sensitivities for Asian options with both geometric averaging and arithmetic averaging . The underlying price $S_t$ follows a standard GBM. Is ...
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1answer
108 views

How to calculate dividend yield - option pricing

Hey how do you calculate the dividend rate if you want to price your stock options eg apple? Just take the dividends paid last year and divide by today's share price? This page reports 0.85% (https://...
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1answer
68 views

Over-night Black-Scholes

I have a question for Black-Scholes. It is a continuous approach, but the real market closes every day. So for the Black-Scholes, how do we count the time effect of during the time when the market is ...
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Can a down-and-out barrier call option be priced using the Black & Scholes formula or should it be approximated?

I am trying to price of a Down-and-Out Barrier call option with leverage. When the price of the underlying asset hits a certain barrier (B), the option becomes worthless. The issuer of these options ...
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Black Scholes model calibration

the only parameter in the Black Scholes model that needs to be estimated is the volatility. Which approach is correct: Estimation of volatility from daily log returns Estimating volatility by ...
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Market price of risk on two assets

Under the assumptions of the Black--Scholes model, I read that the market price of risk of two assets $S_1$ and $S_2$ are the same, if they both follow Geometric Brownian motion driven by the same ...
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Arbitrage Condition and Identity in Black-Scholes

After I went through the derivation to get the skew in Backus et al., I had two questions: In the proof, it mentioned the application of the arbitrage condition and then obtained equation (31): $$\...
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1answer
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Black-Scholes Formula under $T$-forward measure

The Black-Scholes price of a European call option is given by $$ C_0^{BS}(T, K) = \mathbb{E}_Q[e^{-rT}(S_T - K)_+] = S_0 \Phi(d_1) - Ke^{-rT}\Phi(d_2) ,$$ where $$ d_{1,2} = \frac{\log\big(\frac{S_0}{...
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Can “Turbo warrants” be priced using the Black & Scholes model?

I am trying to model the pricing of an asset called a "Turbo warrant", which to me looks a lot like a Down-and-Out Barrier option with leverage. When the price of the underlying asset hits a ...

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