Questions tagged [black-scholes]

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

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How does one transform the Black Scholes equation (u_t +0.5A^2 x^2 u_{tt} +Bxu_x - Cu= 0) to the heat equation [duplicate]

Given that A, B and C are constants, how does one transform (u_t +0.5A^2 x^2 u_{tt} +Bxu_x - Cu= 0) to the heat equation.
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Correlated Wiener Process

I am in trouble with a task: I have a portfolio of 5 assets, and I Have the correlation among them, with a 5x5 matrix. Since each asset follows the BS formula: , I need to perform a montecarlo ...
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No-arbitrage bounds on Implied Volatility under Black-Scholes

Suppose the overnight (1-day) at-the-money implied volatility is X% and the two week (14-day) at-the-money implied volatility is also X%. How would I go about finding the upper and lower no-arbitrage ...
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How smooth is Black-Scholes?

For each variable $(S,T,K,r,q,\sigma)$ in the Black-Scholes formula, how many times can you take a partial derivative? Adjacently, is the nth order greek for some variable a constant? Thanks
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Early Exercise of American Options on dividend-stock

I am reading the chapter 15 of Options, futures, and other derivatives by John Hull. Specifically, 15.12 Dividends-American Call Options. I am stuck while proving the fact that exercising an American ...
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Formula for a Black-Scholes option [closed]

In a Black-Scholes market of two assets, we have the riskless asset $B_t = e^{rt}$ and the following stock: $$ S_{t} = S_{0} \exp \left( \left( r - \frac{\sigma^2}{2} \right) + \sigma W_{t} \right) $$ ...
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Black 76 and Asian Style Options on Shaped Power Futures

I am attempting to price a monthly lookback option on the gen-weighted average price of power at a particular solar plant over a given month. If the option settles at hub H, am I right to shape the ...
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Black-Scholes - Security value with two sources of risk

Consider the Black-Scholes economy with two sources of risk. A security pays off $S_{1T} S_{2T}$ upon its maturity at time T, where $S_1$ is the level of the S&P500 index and $S_2$ is the price of ...
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Solving the Black-Scholes for any arbitrary payoff

Good evening, I'm currently working on the following problem and I would like an opinion on it, Let's consider the Black-Scholes model with (time-varying) volatility, $\sigma = \sigma(t)$, and (time ...
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Spot the mistake in final step of BS solution via PDE approach!

Doing last step -- un-change of variable, where in my case I have $$k = -\frac{2r}{\sigma^{2}},$$ $$v(\tau, x) = u(\tau, x) \cdot \exp\left(-\frac{1}{4}(k+1)^{2} \tau - \frac{1}{2}(k-1)x\right),$$ $$x ...
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Proving $\frac{\Pi(t)}{B(t)}$ is a martingale

Consider the stanadard Black-Scholes model and a T-claim $\mathcal{X}$ of the form $\mathcal{X}=\Phi(S(T))$. Denote the corresponding arbitrage free price process by $\Pi(t)$. Show that, under the ...
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What is wrong in my Heston model's code

I am trying to code a heston model pricer.However,it seems correct at the beginning but when inserting extreme data I retrieve myself with negative probabilities or negative prices. There is the code :...
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What is the Radon-Nikodym derivative in the Heston model?

It is clear to me that $$ \frac{dQ}{dP} = e^{-\lambda W_T-\frac{\lambda^2}{2}T}$$ is the Radon-Nikodym derivative that defines the change of measure in the framework described by Black and Sholes. But ...
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Question on boundary conditions when using Finite Difference

I have two questions appearing to me (they are not related directly to each other). My first question is about boundary conditions when using Finite difference methods. There are two ways to do it: a)...
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Relationship between Vega and Gamma in Black-Scholes model

my question is the following one: I don't manage to prove that, in Black-Scholes model, single-signed Gamma options have values that are monotonic in the volatility. I am looking for an exhaustive and ...
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Delta hedge error black-scholes by Mark Davis

I'm currently reading a paper by Mark Davis in which he talks about a delta hedging error in the Black-Scholes formula. The delta hedging error is given expressed as $Z_t$ with the formula: $$Z_t = \...
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Calculate error at all spatial indices for a given time step between BS equation and its numerical solution using explicit method

I am using the explicit finite backward difference scheme to discretize and calculate the price of an European call option in a discretization stencil. My goal is to find the error at a given time ...
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Theoretical returns of Short Straddle in an efficient Options Market

Assumptions: Market is efficient All assumptions of BS Model apply Implied Volatility predicted using BS model is same as actual volatility in future. Needless to say that the volatility is constant ...
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Black Scholes Stochastic Taylor expansion question [closed]

I am currently deriving Black-Scholes formula, and i get the following equation when Im doing the Tayler expansion: $dG=\frac{\partial G}{\partial S}dS+\frac{\partial G}{\partial t}dt+\frac{1}{2}\frac{...
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American options and stopping times

The price of an American put option can be written as the following optimal stopping problem: $V(0) = \mathop {\sup }\limits_{\tau \in \mathcal{T}} {\mathbb{E}^\mathbb{Q}}\left[ {{e^{ - r\tau }}\max [...
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Replicate a claim in a complete market

Consider the Black-Scholes market wher $\sigma > 0$, and a claim paying $S_T^{\gamma}$ at time $T$, where $\gamma$ is some positive constant. How do I find the replicating portfolio of such a claim?...
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Log Moneyness vs Log Strike

In How to calibrate a volatility surface using SVI, is said: "(log-moneyness would be more accurate) ". First, why do we talk about "moneyness", is it a reference of "being in ...
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Sensitivities under Bachelier process

The sensitivity profile like (delta, vega, gamma etc.) of an option contract is quite established if the valuation model follow ...
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What's the intuition behind there being a perfect linear relationship between option value and expected volatility?

I modelled option prices using the BS model at different levels of volatility. Surprisingly, I came out with a perfectly linear relationship. As volatility rises, so does the option value, which is ...
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Hedge error - Willmot and Ahmad

I'm currently reading the paper: Willmot and Ahmad: Which free lunch would you like today, Sir? Delta Heding, volatility arbitrage. In case 1: They delta hedge with the actual volatility, by going ...
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Boundary Conditions for Call Option in Black Scholes Model

Let $C(t,S)$ be the value function of a call option. I want to price that option using (explicit) finite differences and the Black Scholes PDE. I consider the grid $0=t_0<t_1<...<t_{N-1}<...
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How do you finance theta decay when replicating an option?

When constructing a replicating portfolio for a short position in a call option under Black Scholes, I am not able to pinpoint the source of gains from theta decay. When theta decay materializes, I ...
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Generalized Black Scholes PDE in a Two Factor model

I'm reading the book of Clewlow and Strickland on Energy derivatives. In the section about the two-factor model, an equation, similar to B&S PDE is presented, but the proof is not presented. Spot ...
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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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Arbitrage portfolio example

Can you give me a concrete example of a self financing portfolio which gives arbitrage opportunity in the two-dimensional Black-Scholes model? By the two-dimensional Black-Scholes model I mean $$dS_{1}...
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Method of comparing two option pricing models?

I am currently writing a small paper comparing the Black-Scholes formula to the Bachelier model. However I am wondering how exactly I should compare the two models? Obviously I am comparing the prices ...
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Martingale pricing with time-dependent risk-free rate

I want to find the price of a European call-option under the assumption that the risk-free rate $r$ is time-dependent, i.e. $$ d\beta = r(t)\beta dt \leftrightarrow \beta(T) = e^{\int_0^T r(u)du} $$ I ...
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Why do the prices of deep in-the-money options increase with volatility in the Black Scholes framework? [closed]

I can understand that volatility increases the value of an option when a stock is out/at the money. Then more volatility means the stock's distribution gets more upside without suffering a greater ...
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Calculation of market price for option at underlying strike price at some point in future

Would appreciate clarification on the below scenario. If a put option was sold at the start of the week, when the broker (Interactive Brokers) calculates the cost basis (the premium collected) are the ...
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How to prove no-arbitrage when a long butterfly is strictly positive?

I want to prove why there are no arbitrage opportunities when a long butterfly is strictly positive. I know there is a similar topic out there, but it seems it doesn't solve my question: Prove that ...
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Risk free rate in black-scholes model

Currently reading A. Damodaran‘s book Investment Valuation. In chapter 5 in order to value an option using black-scholes model he adjusts risk free rate using the following formula: $1-e^{-r}$ I. E. ...
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Expected return on Black-Scholes priced option?

Suppose we have a European-style call option on some stock, and it was priced according to Black-Scholes. Everybody agrees on the stock's volatility and expected return. What's the expected return (...
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What is the expected return of a “correctly” priced option if volatility stays constant?

For example, if we take a call option on some stock priced using an option pricing model such as Black-Scholes and assume that volatility stays constant and the underlying stock moves according to the ...
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Intepreting European call option when expiration approaches to infinity

Assume that dividend = 0, then the price of call option is $$ C = S\cdot P_{s}[S(T) > K] - e^{-rT}K\cdot P_F[S(T) > K] = SN(d_1)-e^{-rT}KN(d_2) $$ where $P_s[S(T) > K]$ = Probability of ITM ...
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Heston model vs. GARCH

Heston model is a stochastic volatility extension of the Black-Scholes model. On the other hand, there is also closed-form expression for option pricing that uses GARCH stochastic volatility model. ...
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Confusion about terminology : Finite difference for option pricing

Consider the following initial-boundary value problem for $u = u(x,t),$ $$u_t - a u _{xx} = f(x,t) \text { for } 0 < x < L \text { and } 0 < t< T$$ along with bunch of initial and boundary ...
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FX Option Price Quotation

I'm trying to replicate the following FX vanilla option pricing exercise (and the conversion between the quote types), taken from Wystup (2006). A call's value today is well-known given by BS / ...
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How to price barrier options under Black-Scholes?

I am looking for a rigorous proof for the closed form of the price of a barrier option (up-in/up-out) under Black-Scholes model, that is a step by step solution of the solution of the heat equation ...
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Pricing of strange Asian lookback option with European-style payoff $\max\{ \max_{u\in[0,T]}S_u-\frac1T\sqrt{\int_0^TS_t^2\mathrm{d}t},0\}$

I am trying to price the Asian lookback option at time $t$ with time-$T$ (European) payoff $\max\{M_T-A_T,0\}$, where $$M_t=\max_{u\in[0,t]}S_u,\quad A_t=\frac1t\sqrt{\int_0^tS_u^2\mathrm{d}u},$$ and $...
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Exchange option dynamic replication strategy

I am looking for references to better understand the dynamic replication strategy of exchange options. For example, for an European call option without dividends, I know I need to hold $\Delta_t=\Phi(...
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Implied volatility model-free

I know that $\operatorname{IV-model \space free}=2 \int_{0}^{+\infty}\frac{c_0(T,Ke^{r(T-t)})-c_0(t,Ke^{r(T-t)})}{K^2}\operatorname{d}K$ is calculated using an iterative procedure, i.e. setting a ...
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Dividends reinvested at the risk free rate

I am just learning the black scholes model and came across continuous dividends. I understand that when it comes to pricing, the dividends need to be reinvested in the risky asset, but what if they ...
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Maximum norm stability for implicit Black-Scholes equation

I am trying to prove maximum norm stability for the following implicit approximation to the Black-Scholes equation $$\frac1{\Delta t}\left(U_j^{(n+1)}-U_j^{(n)}\right)+\frac{rS_j}{\Delta S}\left(U_{j+...
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Intraday “Time to expiration” for Black-Scholes on the expiration day

In Black-Scholes, T is the % of year, how do we calculate T intraday on the expiration day? Does the expiration happen at the exact moment of that trading session? For example, for SPXW options that ...
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Are there really closed-form pricing formulas? [closed]

Good morning to all, I wanted to post this question here hoping to have more details. The concern, in my opinion, comes from the fact that the concept of "closed-form" is not clear. Because, ...

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