Questions tagged [black-scholes]

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

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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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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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301 views

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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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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198 views

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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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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129 views

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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I just got Matlab, what are some options that I should model in a jump diffusion

Don't worry I understand mathematics: ito's calc, martingales, etc. I am just curious what options I should test, and from what indices. Is there stuff I can test from the 2008 crash to measure their ...
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108 views

Are there stocks dynamic that cannot be represented by Generalized Black Scholes model?

The generalized Black Scholes Model refers to a stock dynamic that satisfy $$ dS(t)=S(t)(\mu_t dt+ \sigma_t dW(t)) $$ By martingale representation theorem, it seems that if there is a risk neutral ...
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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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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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Why are these deep in-the-money FLEX options seemingly bought at a discount?

98% of the initial reference value is .98 x 267.88 dollars, which equals 262.52 dollars. However, the market value of each call contract they purchase is 247.42 dollars. How are they purchasing these ...
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Hedging with different volatility (Ahmad and Wilmott paper)

In their paper they show that: - if you hedge with the realised volatility, the present value of the total p&l is the difference between the option value based on the realised volatility and the ...
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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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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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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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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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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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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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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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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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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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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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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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134 views

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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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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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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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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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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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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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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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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224 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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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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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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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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