Questions tagged [black-scholes]
Black-Scholes is a mathematical model used for pricing options.
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Black-Scholes: Volatility Smile "sharpens" with time to expiry
I have tried to calculate IV and log-moneyness (=log(S/K)) for different times to expiry
(M = less than 1 month, Q = less than 1 quarter, S = less than 1/2 of an year, Y = less than 1 year, Y (+) = ...
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implied volatility and strike price
Assume for simplicity that the expiration time of an option is $1$ the initial stock price is $1$ and there is no dividend yield and the risk free return is $0$.
How is it possible to show that the ...
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Deriving the Black-Scholes formula as the expected value on the payout of an option
My question concerns the Black-Scholes formula for the value of a European option, namely
\begin{align}
C(S_t, t) &= N(d_1)S_t - N(d_2) Ke^{-r(T - t)} \\
d_1 &= \frac{1}{\sigma\sqrt{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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Where can I find a clear explanation (brief derivation) of N(d1) and N(d2)?
Where can I find a good explanation (perhaps with a brief derivation) of N(d1) and N(d2) from Black-Scholes? Just trying to understand the general idea about these 2 probability functions and how they ...
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How to get the probability of exercise call option in Black-Scholes model?
From Black-Scholes model, I'm trying to prove:
$p(S_t>K) = N(d_2)$
No luck yet!
Can anyone suggest a reference showing that how to obtain this equation?
All I get is:
$S_t = S_0e^{ (\mu-0.5 \...
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Conceptual explanation of the relationship between gamma and vega plotted against delta for a European call option
I recently plotted Gamma and Vega against Delta for a European call option and found that the graphs look very similar. This makes sense to me mathematically since the two formulas are pretty much the ...
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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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Relationship between forward and option prices
Do forward prices factor into option prices at all? It seems to me from Black-Scholes that you just need a spot price and interest rate r.
I understand that $F_t = S_0 e^{r t}$, but I don't know if ...
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Using Black-Scholes to price a geometric average price call
Sorry if this is the wrong exchange for this question. It seems to be the most relevant, anyway.
I'm trying to learn and understand the Black-Scholes framework, with a focus on the stochastic ...
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Lower bound of ITM Calls when computing Implied Volatility
Assuming the Black Scholes model and pricing formula of a European call option. Then, if the call is ITM, i.e. if $ln(\frac{S}{K})>0$, the $d_1$-term will go towards infinity as $\sigma$ goes to ...
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Garman-Kohlhagen (Black-Scholes) Formula vs. Bloomberg OVML Calculator
I'm trying to price a European call option on USDJPY. We have that $S = 112.79, K = 112.24, \sigma = 6.887\%, r_d = 1.422\%, r_f = -0.519\%, T = 0.25$. My model, based on Black-Scholes, returns the ...
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Derive vega for Black-Scholes call from this formula?
Is it possible to get the right formula for vega of a call option under the black scholes model from this formula?
$$\frac{\partial{C}}{\partial{\sigma}}=\frac{S_0}{\sqrt{2\pi}}{e^\frac{-d_+^2}{2}}(\...
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Boundary condition for Asian Option under Black-Scholes model
I am looking at Kemna and Vorst's paper:
A PRICING METHOD FOR OPTIONS BASED ON AVERAGE ASSET VALUES. see http://www.javaquant.net/papers/Kemna-Vorst.pdf
Let $\text{d}S_t = S_tr\text{d}t + S_t\sigma\...
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Trading days or calendar days for Black-Scholes parameters?
Black-Scholes requires volatility estimated in trading days. How does this affect other parameters? Specifically, should the time-to-expiration also be in trading days? And how does this affect the ...
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Pricing call option using risk-neutral martingale approach with squared stock price boundary?
I have to use the risk-neutral martingale 5 step approach under BS pricing framework to price the following call option at time 0:
$$X = \begin{cases}1, &{if} &S_T^2\geq K,\\0, & {...
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volatility input for black scholes formula
I am not a mathematician but want to try and understand the BS model for option pricing. I get the intuitive sense of it but am unable to figure out calculation of volatility (as an input). Some ...
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Trouble arriving at Black-Scholes Formula
I am attempting to arrive at the Black-Scholes formula for my own understanding. I can accept one can use the risk-free distribution & rate, so I am attempting to use the distrution to arrive at ...
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Interpertation of delta hedge error in Black Scholes
I have spent some time to prove the delta hedge error as described in this paper paper page 16-17 by Davis. The proof is discussed here Deriving Delta Hedge error in the B-S setup (part 2) (a post by ...
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Taylor series expansion (Volatility Trading book) explanation sought
I am currently reading Volatility Trading, I have only just started, but I am trying to understand a "derivation from first principles" of the BSM pricing model.
I understand how the value of a long ...
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The choice of portfolio in the proof of the Black-Scholes formula
Consider a stock whose price $S$ satisfies $$dS_t=\mu S_tdt+\sigma S_tdW_t$$ for constants $\mu,\sigma$ and where $W$ is a $\mathbb{P}$-Brownian motion. Further assume that the stock pays out ...
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How should option prices differ when using the Heston versus the Black-Scholes model?
I am running Monte Carlo simulations for a European Call using Heston Model and I am trying to compare them with prices calculated using Black-Scholes formula. I am not quite sure if the prices I get ...
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good R package for vectorized option pricing
I am using for now the package fOptions but it doesn't allow for vectorized computation of black76 prices and delta.
Which package can be used to do that?
As noted ...
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Why gamma and theta have opposite signs?
I saw some textbooks use B-S equation to explain why gamma and theta have opposite signs in most of the cases. For example, John Hull's classic book.
The explanation is, first write B-S equation in ...
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what's the relationship between forecasted stock volatility and implied volatility?(option)
what's the relationship between forecasted stock volatility and implied volatility? I know that implied volatility is the volatility calculated by BS formula, is there any relationship between implied ...
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Which volatility as input in Black Scholes formula?
I am trying to price an option on an Index using Black Scholes formula. I estimated the daily volatility $\sigma_{day}$.
My question is should I use an annual volatility based on the business days of ...
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Why do I get a curved line when I plot "implied interest rate" on the strike price?
Currently, I am working on my thesis (MSc. Finance) and I run into an interesting “phenomenon”. I have option data for a non-dividend paying stock. In class I have learned, how to calculate the ...
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risk-neutral valuation implies no arbitrage?
It is known that in an arbitrage-free continuous time market, the price of every asset is evaluated as the corresponding price in the replicating strategy using risk-neutral valuation.
I want to ...
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Delta-hedge experiment of American Put option
I am trying to run a delta-hedge experiment for an American Put option but there's a (systematic) hedge error which I cannot seem to understand or fix.
My implementation is found in the bottom of this ...
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Black Scholes differential
I'm studying a BS derivation and I don't understand one part .We have a portfolio consisting of $\Delta(t)S(t)+B(t)$ where the first term is risky and the second is a riskless bond. The part i don't ...
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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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Calculate strike from Black Scholes delta
I have a list of deltas and their corresponding volatilities in an FX market but I want to go from delta to strike price.
In this Question similar problem is being discussed
How can I calculate the ...
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Implied Vol vs. Calibrated Vol
Consider the Black-Scholes model, in which the log stock return over a time period $\Delta t$ is given by
$$
\log(S_{i+1}/S_i) = (\mu - \sigma^2/2)\Delta t + \sigma \sqrt{\Delta t} Z_i, \qquad Z_i \...
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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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Gamma for ATM options with low spots
I'm trying to compute gamma for a vanilla call with spot and strike equal to 0.001.
BLACK & SCHOLES formula gave me a value of 554.761 for gamma which is a very high.
I have then two questions:
...
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Possibility of delta greater than 1 [closed]
Can delta of an option be greater than 1? Please illustrate it with an example.
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Black-Scholes pricing of binary options
I'm trying understand something basic about Black-Scholes pricing of binary options. In my example above, the current price is over the strike price. The volatility is extreme but I'm still having ...
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Option Valuation
Can Black-Scholes option values be derived via the Capital Asset Pricing Model, without resort to the use of a risk-free portfolio being created from the option and a Delta determined quantity of the ...
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A simple question: Cost of delta hedging when a call option is sold
Consider a vanilla European call option C, with underlying asset S, strike price K and time to maturity T. Assume that S follows a geometric Brownian motion with mean growth rate of μ and volatility σ....
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Equivalent form of Black-Scholes Equation (to transform to heat equation)
I am trying to understand the transformation of the Black-Scholes equation to the one-dimensional heat equation from Joshi, M. (2011). The Concepts and practice of mathematical finance. 2nd ed. ...
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Stochastic Volatility and Sticky Delta
"Stochastic volatility models can be thought of as sticky delta model. And Local volatility model as sticky Strike." Please help me understand how the author has reached this conclusion.
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Risk-neutral expectation equation with collateral and funding costs
I am looking at a paper by V. Piterbarg, Funding beyond discounting: collateral agreements and derivatives pricing, that you can download on the following link, in which the author adapts the Black-...
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Value options when the currency’s risk free rate is negative?
How would you handle a negative interest rate in index/equity options valuation?
An example would be negative rates for short term maturities for Swiss Frank (CHF).
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Expected Growth
The model assumption of the Black-Scholes formula has two parameters for the geometric Brownian motion, the volatility $\sigma$ and the expected growth $\mu$ (which disappears in the option formulae). ...
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Pricing 'Down and In' claims
I came across this question in a sheet of practice problems which has me a bit stumped.
A down-and-out call option with maturity T, strike K = 100 and
barrier L = K coinciding with the strike, ...
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is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?
I've read an answer here that say if your security has vega, then it has gamma and theta.
is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?
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A few questions about signs of the Greek letters
Rho is the partial derivative of the value of call option, $C$, w.r.t the riskfree interest rate $r$: $$\rho \equiv \frac{\partial C}{\partial r}$$
In the standard B-S formula this term is positive, ...
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Where does 1/2 in Fourier Transform method of pricing options come from?
I am reading Jianwe Zhu's Applications of Fourier Transform to Smile Modeling. On page 26, the author is describing how to use the Fourier tranform to price vanilla European call options. If $f_j$ is ...
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Black Scholes in Practice: Delta Hedging
From the Wikipedia page, we know call option as an example is price through delta hedging.
$$\Pi=-V+V_SS$$
and over $[t,t+\triangle t]$
$$\triangle\Pi=-\triangle V+V_S\triangle S$$
My questions ...
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Find call and put volatilities using ATM, Risk reversal and Butterflies volatilities
I have to plot the implied volatility surface for EUR/USD.
So, my goal is to produce something like that, from put delta 10 to call delta 10:
Searching for informations, I found that I could find ...
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