# Questions tagged [black-scholes]

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

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### 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: ...
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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### 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}{...