# Questions tagged [black-scholes]

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

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### Merton's portfolio problem with constraints

Suppose the investor can invest in a Black-Scholes market with one risky asset $S$ with drift $\alpha$ and volatility $\sigma$ and a riskless asset $B$ with a riskless rate of return $r$, and the ...
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### Delta-hedge experiment of American Put option

I am trying to run a delta-hedge experiment for an American Put option but there's a (systematic) hedge error which I cannot seem to understand or fix. My implementation is found in the bottom of this ...
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### Pricing and hedging of vanilla options based on non-tradable underlying

Consider a non-tradable stock index $S$ which satisfies: $dS_t=\mu S_tdt+\sigma S_tdW_t$ and a risk-free asset $B$. I want to price an European Call option with the payoff $C_T=max(S_T-K,0)$. The ...
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### Wealth process in the Black-Scholes model with discrete dividends

Good evening, The following problem is the sequel of a previous post I made here a few days ago. Consider the Black-Scholes model with discrete dividends in the interval $[0,T]$. This means that ...
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### Are Black-Scholes Greeks bounded?

For time to maturity greater than zero, has it been proved somewhere that the Black-Scholes greeks $$\frac{\partial^n BS}{\partial x^n}$$ are bounded, where $x := \log S$ and $S$ is the current spot ...
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### how does margin affect the Option Price when Selling an Option

Currently I'm thinking the effect of margin. When selling an option, you need to pay margin everyday and mark to market. In most exchanges, margin is overcollateralized. But when buying a option, you ...
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### Why does it hold true that $\theta_{t} d\overline{X}_{t}$ is a local $Q$ martingale if $\overline{X}$ is a local $Q$ martingale

I am learning from Bernt Oksendal's Stochastic Differential Equations and on page 276 Lemma 12.1.6, it is stated that: The existence of an equivalent martingale measure $Q$ on the discounted price ...
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### Operator splitting method on three assets black scholes equation

Currently I am studying finite difference method on derivatives with three (or more) underlyings and little bit confused on operator splitting method because two papers have different result. For the ...
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### Black and Scholes equation for portfolio **with** arbitrage

I am well aware of how the ordinary Black and Scholes equation is derived, under the assumption of an arbitrage free portfolio, $V=G-hS$. Here $S$ is the price of the underlying and $G$ is the option ...
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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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### 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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### Fast implied volatility for american options

Peter Jäckel has developped a method to compute implied volatilites from option prices, called "by implication", see the papers : By Implication Let's be Rational on its website -- as well as a ...
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### What are the main problems for calculating the implied volatility of in the money American put options?

As stated in the question I have a problem with calculating the implied volatility for in the money put options I have a data set of 2.6 million American style plain-vanilla call and put options. For ...
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### Deriving the black-scholes formula for the European asset-or-nothing call option

I would like to find out what boundary/final conditions i should be using to find the formula for a European asset-or-nothing call option, as i feel that is where I'm making my mistake. I've read ...
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### How to Compute the payoff of Var Swaps, which I have replicated

I used Derman(1999) method, to calculate the fixed Kvar for Variance Swaps using actual option price data. The first Pic Shows the outcome. (ignore the 0s). Now the profit and loss of short var swaps ...
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### Black-76 Model for Swaption Price and Greeks

I'm in the early stages of developing a swaption pricing model. Suppose $t_1$ is the tenor of the swap rate in years, $F$ is the forward rate of the underlying swap, $X$ is the strke rate of the ...
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### Understanding put-call parity

I'm a person with math background trying to break into quantitative finance, and there's something about put-call parity that is not making sense to me. Below I'll detail my understanding of the ...
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### Pricing Options on Fixed Income ETFs

The market for trading options on fixed income ETFs like HYG has become increasingly prominent in the past couple years, but I've been unable to find any discussion related to the pricing methodology ...
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### Range options in BS

I know how barrier options are priced in Black-Scholes scheme. I'm wondering if an analytical formula exists also for range (corridor) digital options i.e. options paying only if the price remains ...
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### PDE and Black Scholes problem

Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
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### Black Scholes diffusion well coded in Python

I have some trouble with the following code. Some jump and a decentered path are present but it's not the case, normally for Black Scholes diffusion! Is anyone see a problem in my code ? ...
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### Pricing a Power Contract derivative security

I'm trying to price a "power contract" and would appreciate guidance on the next step. The payoff at time $T$ is $(S(T)/K)^\alpha$, where $K > 0$, $\alpha \in \mathbb{N}$, $T > 0$. $S$ is ...
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### Practical use of Dual Delta?

I am wondering what the practical use of the Black-Scholes Dual-Delta is? I know it is the first derivative wrt the strike price: $$\frac{\partial V}{\partial K} = -\omega e^{-r T} \Phi(\omega d_2)$$...
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### Volatility Time and Interest Rate Time

In Sheldon Natenberg's book "Option Volatiliy & Pricing (2nd)", he mentioned that (on page 65), only trading days (roughly 252 in a year) are counted when computing vol time and all ...
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### How to apply put-call parity in volatility surface construction?

How to make the volatility surface free of put-call parity arbitrage? If I bootstrapped the implied vol from a call price and plugged it into the BS model to have a put price, what if it violates the ...
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### Problem matching prices of Black-Scholes vs. GARCH(1,1) in Duan (1995)

In the paper of Duan (1995) the author compare European call option prices using Black-Scholes model vs. GARCH(1,1)-M model (GARCH-in-mean). To be brief, the author fits the following GARCH(1,1)-M ...
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### Is there anyway to compute the CEV-implied volatility from option prices?

Under Black-Scholes, there exists a solution for the option price for a volatility. The volatility can then be backed out from the option price using numeric methods. For the constant-elasticity of ...
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### Exact delta-hedging for endogenous payoffs

I would like to derive the exact delta-hedging strategy in the Black-Scholes market to replicate the following non-standard endogenous payoff. The particularity is that the payoff does not only depend ...
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### First known reference using martingale theory to derive BS formula

What is the first known paper which derives the Black-Scholes valuation formula for an option (1973) using martingale machinery - instead of PDEs?
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### Perpetual Option Paying Chooser Option

A perpetual option solves the ODE $$rSV_S+\frac{1}{2}\sigma^2S^2V_{SS}-rV=0$$ The general solution is $$V(S)=aS+bS^{\gamma}$$ where $\gamma=-\frac{2r}{\sigma^2}<0$. For an American put option with ...
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Assuming B&S world, is it possible to price an (European) option on a general transformation $f(\cdot)$ of $X$? What kind of assumptions should we make on $f$? Is convexity sufficient to find some ...
Suppose I have a model with 2 primary assets, a stock $S$ and a short rate. The stock will be driven by a Brownian motion $W_1$. The short rate will be random and will be driven by a Brownian motion \$...