Questions tagged [black-scholes]

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

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2k views

Is there any public data to get OIS for differal time (1d, 1W, 1M, …, 10Y)?

I want to get data of Overnight Index Swap, also known as OIS rate, there is any public why to get this always from yesterday? For example, I want to get EFFR(Effective Federal Funds Rate), I can get ...
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2answers
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Implicit finite difference method always guarantees positive and stable price of derivative?

For the following black scholes pde $$ f_t + rSf_S+\frac{1}{2}\sigma^2S^2f_{SS} = rf $$ By denoting $f_{i}^{n} = $ Price of derivative at price node $i$ and time node $n$ and assume uniform grid, the ...
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Arbitrage free pricing of option to trade stocks

Consider Black-Scholes model with constant interest rate r and stocks with prices $S_t^A$ and $S_t^B$ that satisfy the SDE's $dS_t^A = S_t^A(\mu^A dt + \sigma^A dB_t)$ and $dS_t^B = S_t^B(\mu^B dt + \...
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59 views

Summary of Pricing Options of Log-Normal Claims Using Black's Formula

Cross posted from here. Let $B$ be a $Q$-Brownian motion and $X^{s,x}$ given by $$dX_t = X_t(\mu_t dt + \sigma_t dB_t),\quad X_s = x$$ for $\mu, \sigma$ deterministic. Let $\mu_{s,t}=\int_s^t \mu_u du$...
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165 views

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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Changing numeraire in Margrabes formula

Consider a Black Scholes market with constant coefficients, a bond and two risky assets: $$dB_{t}=r B_{t}dt \\ dS_{t}^{i}=S_{t}^{i}(b_{i}dt+\sigma_{i,1}dW_{t}^{1}+\sigma_{i,2}dW_{t}^{2})$$ where $i=1,...
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Higher moments of a straddle

Following the logic of Ben-Meir and Schiff (2012) and this question the first, second, third and fourth raw moments of a put are: Similarity, for a call it is as follows: where and ...
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4answers
206 views

Price of Call Option with or without jumps

Suppose two assets in the Black Scholes world have the same volatility, but different drifts and that one has downward jumps at random times. How does this affect the option prices? I would have ...
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1answer
143 views

Black Scholes Separable Solutions

I want to find all the solutions of the Black Scholes PDE that are of the form f(x,t)=theta(x) or f(x,t)=phi(t). Can someone explain and help with this? I know the PDE formula is $f_{t}(t, x)=-\...
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Compute the price of a derivative which pays $\log(S_T)S_T$ in the Black Scholes world

Compute the price of a derivative which has pays $\log(S_T)S_T$, you can assume that the Black Scholes model is valid. Using the stock measure we can write the expectation as $$D(0) = S_0 \mathbb{E}...
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What is the meaning that Geometric Brownian motion is leptokurtic? [closed]

Does this have any relation to the symmetry of the normal distribution?
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Realizing profit with Gamma Trading doubt

Lets suppose we have a delta-neutral portfolio and that we want to trade the gamma. If we are long gamma, we can profit from every rebalancing to keep the portfolio delta-neutral. Lets suppose the ...
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1answer
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Numerical Solution to 3 Dimensional Backward BS PDE

I have a three dimensional backward BS PDE. $$ \frac{\partial V}{\partial t} + a(t) S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma(t, S)^2 \frac{\partial^2 V}{\partial S^2} + b(t, M) \frac{\...
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1answer
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Black-Scholes Implied Volatility

I'm working my way through the following paper: Malz. A. M. (2014). A Simple and Reliable Way to Compute Option-Based Risk-Neutral Distributions I am completely stuck on the following derivation. The ...
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1answer
703 views

Classic dynamic delta-gamma hedging in Python

I am trying to run a delta-gamma hedge for a Black-Scholes model in Python.The Euler disceretizatioin of the paths is the simplest possible. I wrote the code below but the PnL looks undesirable and ...
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What are some alternatives to Geometric Brownian motion that can be used in the Black-Scholes? [closed]

I hear that there are many extensions to the black scholes model to make it more realistic, however, GBM does not account for volatile swings. Is there any sort of alternative approach to use instead?
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Historical volatility - Black Scholes

How do you best incorporate the weekends in the calculation of the Black Scholes historical volatility? (Of course historical volatility serves as approximation, if the market price of the options is ...
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Gamma PnL from Itô's Lemma derivation

The change in a call portfolio ($f$), derived from Itô's Lemma, is: \begin{align*} \left( \frac{\partial f}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2}\right)\mathrm{d}t &=...
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0answers
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Can option chain data be used as a quick and dirty substitute for proper pricing calculations for non-traded options?

I wonder if the option chain data that is published everyday by websites like Yahoo finance can be used to quickly price options that are not yet traded? So in other words, to estimate option prices ...
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Convenience yield

I need to price an option on gold in some local currency. If I use the Black Scholes formula, then I need to input convenience yield for spot gold. Given generally available market data, how can I ...
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1answer
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How to best predict option prices using Brownian motion and compare it to the Black and Scholes model?

I am trying to use Brownian motion to predict option prices and compare the outcomes to Black and Scholes. For this purpose, I would like to calculate the average returns (mu) and volatility (sigma) ...
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1answer
82 views

Convert option inputs to standard Brownian motion

I want to know the probability that the strike price of an option is touched. My input values are: P = price S = strike v = vol t = time to expiration According ...
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3answers
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Implied Volatility, annualized quantity ? And Total Implied volatility

so Implied Volatility is computed by equalizing the value of the call option given by the black and scholes model with the one observed. Then, by inversing $C_{BS}$, one gets "$\sigma_{IMP}$"...
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1answer
67 views

Kurtosis of a straddle

I want to determine the kurtosis of a straddle. My question is closely related with the following topic here. According to the following paper of Ben-Meir and Schiff (2012) the expected value of a ...
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6answers
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A simple formula for calculating implied volatility?

We all know if you back out of the Black Scholes option pricing model you can derive what the option is "implying" about the underlyings future expected volatility. Is there a simple, closed form, ...
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Different distributions for option pricing

So the classic BS assumption of lognormal prices imply that the stock price can not be negative. Now since recently also oil prices were negative I was wondering, whether it would be possible to ...
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R: How do i finish the tails in the risk neutral density, obtained from option prices

Im currently working on constructing the risk neutral probability distribution of a stock, based on the option prices. In doing so, i calculate the implied volatilities from the option prices, and ...
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2answers
102 views

May someone please explain the intuition behind the Black-Scholes Equation?

Consider the Black-Scholes equation for a European Call Option, \begin{equation} \begin{cases}\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r\frac{\...
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2answers
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Binary Option Valuation With Skew

In searching for methods of valuation of Binary options with skew, I have found two formulas which are at odds. I cannot find any other references to this valuation formula. Should Vega be positive ...
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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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1answer
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FX Call under stochastic rates and deterministic volatility

Lets denote $S_t$, $r^d_t$,$r^f_t$ respectively the FX spot, the domestic rate and the foreign rate at time $t$. Lets $\mathbb{Q}^d$ , $\mathbb{Q}^f$ respectively be the domestic and foreign mesures,...
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1answer
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Black & Scholes formula derivation from a Binomial Tree - John C. Hull

I am reading "Option, Futures and other Derivatives" by John C. Hull, and on Appendix chapter 13, he derives BSM formula from a Binomial Tree. When he builds U2, I just don't understood how to get ...
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Which process is the most commonly used for modeling stock prices?

I'm thinking of writing a master's thesis about pricing options using Levy processes, but I wonder if these processes are actually used for modeling stock prices or not (and which specifically)? And ...
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70 views

How are option values in real life calculated without volatility?

Implied volatility is the volatility that when inputted in the Black-Scholes model, it returns the theoretical market price of a European option value. I understand that implied volatility is not ...
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1answer
219 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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2answers
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Calculating the 0.50 delta strike

According to most books the ATM option is the option with a delta of 0.50. However, this is only the case when the distribution is normal. The more positively skewed the distribution, the further the ...
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About the log return in the Black&Scholes model

I'm currently studying the Black&Scholes model and I'm not sure about the following thing: the log return, say r, doesn't evolve in time? I mean, dr/dt = 0, its derivative is zero? Does only its ...
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1answer
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Trying to measure “radius of diffusion” in the stock market

Good evening! I'm quite new to quantitative finance (coming from the math world!), so please excuse me if I'm not familiar with every concept! I am currently studying the Black-Scholes equation, ...
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1answer
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Does Black Scholes + Stochastic interest rates result in a unique price

Black-Scholes with its assumptions results in a unique price for the call. If we introduce stochastic interest rates, would this still remain the case?
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Black-Scholes market and payoff with integrals

I am struggling with the following exercise: Prove that on Black-Scholes market, with some parameters $r, \mu, \sigma >0$, a payoff $$X=\int_{0}^{T}\ln \frac{S_t}{S_0}\mathrm{d}t+\frac{1}{\sigma}\...
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Stochastic Volatility Models - are they complete markets?

I'm reading about stochastic volatility models - the ones which resulted after Wiggins proposed in 1986/7 that $\sigma$ in Black-Scholes should be a stochastic process rather than a constant. In ...
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1answer
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Is the forward price equal to the future price?

If $f^{T_1}(t)$ is the price of a forward and $F^{T_1}(t)$ is the price of a future on some stock, both maturing at date $T_1$ and with the assumptions: no dividend constant interest rates no ...
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Prove that it is possible to make a self-financing portfolio

I would like to prove that if there exists $(X_1,\ldots,X_n)$ satisfying $\mathbb E[\int |X(s)|^2 d[Y]_s]<\infty$ (for a standard filtered probability space $(\Omega, F,( F_t)_{t\ge 0},\mathbb P)$ ...
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In BS model, is there a way to show that the risk-neutral Q is unique without using MRT nor the fact that the market is complete?

In Black-Scholes model, is there a way to show that the risk-neutral probability measure is unique without using the martingale representation theorem nor the fact that the market (in BS model) is ...
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What are the relation between the risk neutral measures in binomial tree and in Black Scholes model?

I appreciate that both are the direct result of constricting a replicate portfolio using stock and bonds. Are there deeper relationship between the two?
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Should a normal distribution be used for valuing options on assets that can potentially have negative prices?

The Black-Scholes-Merton model assumes that the prices of the underlying asset at maturity are log-normally distributed. I understand that this assumes that the prices can never go below zero. ...
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Barrier option with zero-strike

Good morning, everybody, I would like to know whether an up-and-out call option with a zero strike has a special name in the list of exotic options or is still a special case of a barrier option.. ...
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What does it mean to “calibrate vols”

As a beginner, it can sometimes be hard to discern what different terms and phrases mean in QF. I've heard multiple people such as academics and market-makers say things like "calibrate vols" or "...
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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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