Questions tagged [black-scholes]
Black-Scholes is a mathematical model used for pricing options.
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Derivatives without analytic expressions?
I was wondering if there exist options or other derivatives that do not have a known closed-form analytic expression (i.e., some sort of Black-Scholes PDE) and are usually priced using Monte Carlo ...
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Homogeneity of BS Formula
I'm reading M. S. Joshi's paper "Log-Type Models, Homogeneity of Option Prices and Convexity", and I'm having problem understanding equation 1:
$$
C(S_0, K) = \int (S - K)_+ \Phi\left(\frac{...
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Three mathematical mistakes in Black-Scholes-Merton option pricing?
In this preprint on arXiv (a revised version of the one discussed in a post here) we show that there are three mathematical mistakes in the option pricing framework of Black, Scholes and Merton. As a ...
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BS price as the first term of option price expansion
I recently saw someone write, on a generally non-technical platform, that the Black-Merton-Scholes vanilla option price is the first term of an expansion of the price of a vanilla option.
I get that ...
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Black-Scholes formula is a (probabilistic) convex combination
A call price is bounded when $\sigma\sqrt{T}$ goes to $0$ and $\infty $ by:
$$C_{inf} = e^{-rT}[F-K] \leq C \leq C_{sup}=S $$
Now a simple rearrangement of Black-Scholes formula gives:
$$ C = N_1S - ...
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Option time value is Nd1-Nd2
I can't find the below statement anywhere (rearrangement of Black-Scholes formula) :
$C(0, S) = e^{-rT}N_2[F-K] + [N_1-N_2]S$
$F$ being the forward, it reads as a straightforward decomposition to ...
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Implied Volatility Surface Interpolation for fixed moneyness and maturity on each day of the calendar
I'm new to quantitative finance and interested in performing a PCA on the implied volatility surface. However, my dataset displays certain point changes over time. As a result, I need to interpolate ...
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Effect of number of monitoring points on Asian Option Price
I want to understand conceptually the expected effect of the number of monitoring points used during the average calculation on Asian options pricing and the reason of such effect.
Asian Options ...
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How to use GARCH/ARCH/EGARCH volatility forecasts to compare the Black Scholes constant volatility assumption with GARCH/ARCH/EGARCH volatility
I should preface this by saying I am an undergraduate physics student, this is more of a side interest to me, so I apologise if I am missing something obvious. I am not following a formal class or ...
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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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Binomial tree convergence tree towards BS equation - Struggle with a limit
I am trying to prove that the Binomial tree pricing method converges towards the Black and Scholes model, but I am struggling on a specific step.
I don't understand how the limit of p*(1-p) is ...
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Solve for spot price given delta [closed]
I can use Black Scholes or Bjerksund Stensland to solve for delta given spot price, strike, expiration, vol, interest rate, etc. But is there a direct solution to solve for spot price given delta, ...
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Does skew flatten with a decline in volatility?
In Trading Volatility by Bennett, he says:
If there is a sudden decline in equity markets, it is reasonable to
assume realised volatility will jump to a level in line with the peak
of realised ...
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Get strikes from delta works with put but no with call function
From call-put parity one can derive that $\Delta_C - \Delta_P = 1$. We also know that $\Delta_C = e^{-qt}N(d_1)$. If $e^{-qt} = 0.85$ then there is no value for $d_1$ for a $\Delta_C = 0.9$ as $ 0 <...
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How does Bloomberg calculate Interest Rate Caps/Floors with Black Scholes Merton Model and Volatility set as "Normal"?
While valuing Interest Rate Caps/Floors in Bloomberg, I saw that we have an option for selecting both Model and Volatility. So, my question is how exactly does Bloomberg value the cap/floor, when we ...
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Is there a way to use normal volatility in the Black–Scholes–Merton model to value interest rate caps? [duplicate]
I am trying to understand if there is a version of the Black–Scholes–Merton model that can use Normal volatilities instead of Lognormal volatilities while valuing interest rate caps and floors?
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Pricing various classes of derivatives and replicating them
Consider the following three derivative styles and assume zero dividends for simplicity.
The "american style", "european style", and "infinite" style:
$$L_{A}(S,K,t,T)=f(...
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Question on Merton's self financing derivation
I'm reading Merton's Optimum Consumption and Portfolio Rules in a Continuous-time Model, and don't understand the step where he goes from discrete to continuous time. Specifically, my confusion is ...
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Basket option value calculation
I am reading the article, where different approximations for the pricing of basket options are presented. I have tried to reproduce the result obtained by the Gentle's method in Python.
We define the ...
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Risk free rate for currency option
I’m trying to price a call option on EUR/GBP exchange rate and it expires in 1 year. Should I use GBP Libor as foreign risk free rate in order to apply BS formula? The pricing date is 02/21/2023 but ...
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How is variance derived in BS?
The realized variance under classical Black Scholes where the stock price process follows a GBM is given as
$$V_T = \frac1T\int_0^T\sigma_s^2ds\qquad (1)$$
however, the texts I have been reading do ...
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delta-gamma-vega VaR approximation: how to calculate the delta volatility? [closed]
For an option with price C, the P&L, with respect to changes of the underlying asset price S and volatility σ, is given by
P&L=δΔS+12γ(ΔS)2+νΔσ,
where δ, γ, and ν are respectively the delta, ...
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Delta Hedging using another correlated asset
My question is about the following (from Maxime de Bellefroid, Ch. 5 The Greeks):
From my understanding $\Delta_2$ is the sensitive of the option (on the first instrument with underlying $S_1$) with ...
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What are the parallels between the Black-Scholes equation and the heat equation?
I'm trying to understand the analogy between the Black-Scholes equation (1) and the heat partial differential equation (2). I understand that (1) can be written in the form of (2) mathematically, but ...
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What is the dynamic of the forward price process under $\mathbf{Q}$?
Let me define the Spot price process of an underlying as follows: $$dS_{t}=\mu_{S}S_{t}dt+\sigma_{S}S_{t}dW_{t},$$
where $\left(W_{t}\right)_{t\geq0}$ is an appropriate Wiener-process, so $\left(S_{t}\...
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What does implied volatility say about the underlying?
Here's a question that's been on my mind on-and-off for some time now.
It's well known that Black-Scholes is an unsuitable model for pricing in the current (post 80s) market as it fails to capture the ...
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Calculating the PnL of a delta-hedged option at a point in time
In a BS world (constant volatility, no transaction costs, continuous hedging) If I buy or sell an option and continuously delta-hedge, I know how to calculate the final expected PnL based on implied ...
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Pricing Option with Payoff of $S_T$ units of a European Digital Call [duplicate]
I am in a Black Scholes market with the usual riskless asset $B$ and risky asset $S$ with dynamics given by
\begin{align*}
dB_t &= rB_t dt, B_0 = 1, \\
dS_t &= rS_t dt + \sigma S_t dW_t^{\...
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Call option on forward [closed]
What is the trade description behind a call option on a forward? How can it be described with words and not with mathematical formulas?
So what is the intuition behind the following payoff:
$$Payoff_{...
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Right risk free rate to price an Option using BS formula
I understand this is very basic question but I still scramble to determine what would be right risk free rate to price a simple European call option using Black-scholes formula, with maturity is 5 ...
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Rough approximation of option delta (or binary option) given option price wihtout implied volatility?
Under the Black-Scholes framework, is there any approximation of option delta (i.e., $N(d_1)$) or binary options price(i.e., $N(d_2)$) given the option price $C$?
Of course, you can calculate those ...
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Price of financial assets at $t=0$ in Black-Scholes framework
Given the share price equation
$$
dS_t=rS_tdt+\sigma S_tdW_t
$$
working in the framework of Black-Scholes model, find the price at $t=0$ of the following two financial assets:
(a) The asset pays at $t=...
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Find strike of an option based on a delta without option price
I would like to use the Black Scholes model to get the strike based on delta, without having the option price.
So I read this: From Delta to moneyness or strike
Which show that we can get a strike ...
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Access expired options data [duplicate]
I would like to access expired options data, for example, I would like to know the evolution of the call price for a certain option, during the month before the expiry date, which would for example be ...
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Derivation of Call Theta from Black Scholes Model [closed]
How is call theta mathematically derived from Black Scholes Model (without approximation) ? Please help me understand each step mathematically.
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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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Delta of a forward ATM option
Reading:
What are some useful approximations to the Black-Scholes formula?
I understand that a ATM Call option can be approximated to $$ C(S,t)≈0.4Se^{−r(T−t)}σ \sqrt{T−t}$$
Also, I often hear that an ...
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University problem about Bond option [closed]
Good morning,
Next week I'll have Derivates Final test and I've a doubt about Bond Option.
If I have one ZCB, price 90 with 3 month maturity and strike 100, and I want a minimum yield of 2%, what type ...
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What happens trying to price derivatives starting from a non-geometric brownian motion?
To get a better understanding, I tried going through BSM-model starting from a non-geometric brownian motion. However, during the derivation I got stuck, which led me to a specific question.
The set-...
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Black and Scholes PDE in terms of Future Price [closed]
I was trying to understand why the Black and Scholes PDE for the value of an option, $V (F , t)$, with the forward price, $F$, as underlying is
$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2F^2\...
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University Problem about interpolation Implied volatility BS Model (volatility smile)
Good morning,
this is my first question on this forum, I'm writing from Milan (Italy) and I have a question about a University Problem.
The problem is about entering in a Long Range Forward (buy a ...
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Finite Difference Application
We all know that the traditional BS equation is:
$$\frac{\partial \mathrm V}{ \partial \mathrm t } + \frac{1}{2}\sigma^{2} \mathrm S^{2} \frac{\partial^{2} \mathrm V}{\partial \mathrm S^2}
+ \...
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Limit of BSM Gamma as stock price goes to 0 [closed]
BSM gives the following formula for option gamma
$$
\Gamma = \frac{e^{-qT-\frac{d_1^2}{2}}}{S\sigma\sqrt{2\pi T}}
$$
where
$$
d_1=\frac{\ln\frac{S}{K}+(r-q+\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}
$$
...
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Since $S = e^{(\mu-\frac{\sigma^2}{2})t+\sigma W_t}$, why treat it as a constant when calculating the greek Theta (dC/dt) for a European call option?
In a nutshell, if S is dependent on 't', why treat it as a constant when calculating the partial derivative $\frac{dC}{dt}$?
The equation for $\frac{dC}{dt}$ in a European call option is:
$\frac{SN'(...
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Expected value of option spreads
I'm attempting to write a tool that will automatically calculate the expected value of arbitrary options positions, and I need to clarify my understanding. I am neither a statistician nor a ...
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Simple Black-Scholes alternatives
I work at an accountancy firm and we use Black-Scholes to value equity in private companies that has option like features. The equity we typically value is akin to deeply out of the money European ...
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Feymann Kac for multidimensional pde
I Have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{\...
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Proving lognormality of security in Black-Scholes market
Can someone prove that for some security $S_t$ with drift $\mu$ and volatility $\sigma^2$ in a Black-Scholes market we have that $Y_t = (S(t))^{1/3} \sim \text{Lognormal}$, w.r.t. the risk-neutral ...
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