Questions tagged [black-scholes]

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

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109 views

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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35 views

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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85 views

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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1answer
57 views

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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1answer
88 views

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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32 views

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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145 views

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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2answers
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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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143 views

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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104 views

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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1answer
74 views

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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153 views

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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518 views

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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1answer
99 views

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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149 views

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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1answer
169 views

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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1answer
82 views

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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3answers
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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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Change of variables - Black-Scholes [closed]

While manipulating the Black-Scholes equation, Paul Wilmott (Paul Wilmott on Quantitative Finance, chapter 7, page 110) makes the following change of variables and I am having trouble understanding ...
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FX American call option optimal exercise and holding region

Problem I am considering an American call option which gives a domestic investor the right to buy a unit of foreign currency at a strike of $K$ units of domestic currency. I have an exchange rate $S_t$...
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115 views

delta neutral option cost

I am trying to understand how an delta neutral profile is generated. I sell a call for strike of 50$ and the delat of this call is 0.5. I buy 0.5*100 stocks to ...
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116 views

Black Scholes price of exotic claim

Given a time horizon N, I want to know the time-$t$ Black-Scholes fair price of $$\int_0^T S_u du$$ where $S_u$ denotes the time-$u$ stock price. I have used the formula I have been given as follows: $...
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1answer
137 views

Pricing of Asian-like option

I am considering an option which has payoff function $\max\{S_T-\frac1\tau\int_0^\tau S_t\mathrm{d}t,0\}$ for a fixed $\tau$ in the risk-neutral measure $\mathrm{d}S_t/S_t=r_t\mathrm{d}t+\sigma_t\...
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Trading desk assumes zero percent discount rate?

All the swaption and option models I have encountered at my employer's trading desks have assumed a zero percent discount rate. I have proposed using the LIBOR curve, but management responded that &...
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Hull's book - Futures option's rho

In Hull's book (9th edition), on page 420, in table 19.6, it says rho of a European call on an asset with yield $q$ is $$KTe^{-rT}N(d_2)$$ Below it says we can compute greeks of European options on ...
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Forward starting options concepts

Consider $t_0<t<T$, with $t_0=0$ (today date) and the standard payoff of a vanilla forward starting call option, $F_{t,T} = (S_T - S_t\cdot K)^+$, with strike $K$. If the price of this option is ...
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How to derive put option from Black-Scholes equation?

The Question is as follows: The diffusion equation is: I have tried attempting this question by making some change of variables and separating the cumulative distributive function but I get stuck ...
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Boundary conditions Heston's stochastic volatility model

I'm trying to derive the following boundary conditions for heston's stochastic volatility model. This is p. 289 of Shreve's Stochastic calculus for finance \begin{align} c(T, s, v) &=(s-K)^{+} \...
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Which volatility should I use in a long-term futures swaption?

Consider an option expiring in 12/31/2023 on an hourly swap from 2024 through 2029 such that: a) I pay the floating price of electricity and b) receive $20 in return. Using shaped monthly futures and ...

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