Questions tagged [black-scholes]
Black-Scholes is a mathematical model used for pricing options.
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Can you use the SABR implied volatility in the Black Scholes formula?
The SABR implied volatility is often used as an input in Black's model to price swaptions, caps, and
other interest rate derivatives.
I'm wondering whether you can use the SABR closed form solution of ...
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25 views
Why a self financing portofolio equals to the value of a call option? [closed]
Why a self financing portofolio equals to the value of a call option?
Actually, why we write $V(S,t) = a_tS_t+b_t\beta_t$?
where $S_t$: stock price and $\beta_t$: bank account value , $V(S,t)$: value ...
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51 views
Greeks for Black Scholes model
hey where can I find the calculated greek parameters for call and put options on shares paying dividends? I would also like to find the calculations themselves
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70 views
Derivation of $u=e^{\sigma\sqrt{dt}}$ and $d=e^{-\sigma\sqrt{dt}}$
Anyone could provide me a proof of how, starting from $\frac{dS_T}{S_t}\sim \operatorname{N}(\mu dt,\sigma^2 dt)$ with $p:=\frac{e^{rdt}-d}{u-d}$, we can obtain the parameters $u$ and $d$ as from ...
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1answer
29 views
Collar Option K Term
I know that the value of a collar option on a stock (buy stock, buy put at $K_1$ and sell call at $K_2$) is given by
$$Collar\ Value = K_1d(t,T)+Put\ Value-Call\ Value$$
My question is, why do we have ...
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1answer
110 views
Why am I struggling to replicate the Black-Scholes price of an option stochastically?
I am currently trying to replicate the Black-Scholes price of a call option using stochastic simulations of the price moves of the underlying. My code is as follows:
...
2
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1answer
84 views
Delta neutrality (derivation)
I'm confused about the math for the delta-neutral portfolio.
Assume we have a short position in a European call option with price $p(t,S_t)$ and want to hedge it with the stock with price $S_t$. The ...
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66 views
Fast Monte Carlo of Local Volatility Model
I want to compute option prices via a Monte Carlo simulation. The model implemented is a Markov process, following the SDE :
d X_t = alpha * dt + beta^(1/2) * d W_t
...
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1answer
83 views
Interpreting SABR calibration model output
Calibrate a SABR model?
Following on from this question, I have used the same market data they attached but am unsure on interpreting the output.
When I plot the SABR probabilities against strike for ...
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1answer
87 views
BS-Model not suitable for $n$-dimensional options
Anyone could explain me what the authors of this paper mean when they say that "The BlackāScholes model, in spite of its popularity, has some
well-known deficiencies. Firstly, a closed form ...
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48 views
Discounted stock price under a NON risk-neutral measure
Under a risk-neutral measure $\mathbb{Q}$, the discounted stock price is a $\mathbb{Q}$-martingale. Does it mean that under the actual probability measure $\mathbb{P}$ the discounted stock price is ...
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1answer
72 views
Gamma and Gamma Hedge [closed]
I have a very basic question: Is this gamma value has something to do with the gamma hedge? In delta hedge, it's done by buying/selling delta amount of underlying. But in textbook, for a put option, ...
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73 views
How to prove that a series of random variables $Z_j = 1$ or $-1$ occurring at risk-neutral probability, converges to normal, using the CLT?
Context
When pricing options with trees, it is convenient to prove that the asset value at expiry $S_t$ be of log-normal distribution:
$$\log{S_t} = \log{S_0} + \mu T + \sigma \sqrt{\frac{T}{n}} \sum_{...
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0answers
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Backward difference approximation (BDF-2) for Options
I am working on a project for compound options and the assignment is as following:
...
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28 views
Option Valuation With Hard To Borrow Rates
How would you include -in a simple way- high borrow rates, say 10%.
Intuitively, for PUTs I'd set r as r - borrow_rate, to include the negative carry of the borrow. So If I'm selling puts, value would ...
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2answers
209 views
Why can future forward interest rates be assumed to be lognormally distributed in the standard market model?
This seems to be the underlying assumption that allows us to use the standard market model/Black's framework in order to value interest rate derivatives, but I haven't found any understandable ...
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how can we calculate options profit and loss using volatility and implied volatility in a span margin calculation
complete the table below, mainly we have to use black-scholes model for implied volatility calculations which I am getting as 43 % but now how to make a gain and loss table using this implied ...
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0answers
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Pricing of autocallable structured product
I'm looking at this paper: https://doi.org/10.1057/jdhf.2011.25, which is on pricing autocallable structured product. The author uses the Black-Scholes equation to describe the product's dynamic value,...
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1answer
63 views
Can we use a VIX-like method to calculate implied volatility for Black Scholes model?
So I understand that the VIX is an estimate of implied volatility. Volatility can also be calculated from the Black Scholes model. My question is can we use a VIX-like method to calculate implied ...
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0answers
38 views
Determination of critical stock price in compound option pricing
Under the Black-Scholes framework, there is a closed form formula for the price of a compound options, as first derived by Geske (1979). However, the analytical formula refers to a critical stock ...
2
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1answer
109 views
Calibrate Stochastic Volatility Model
For stochastic volatility models, and any vol model I know, it seems the standard approach is to calibrate the model from option prices. As other user said, this seems a chicken egg problem - how do I ...
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1answer
45 views
Finding option price using intraday data [closed]
I have the option price at a rate which is much smaller than the rate at which I have tick data for the underlying. If I have option price at times $t_1, t_3, t_5$ and I have tickdata at $t_1, t_2, ...
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0answers
47 views
Black Scholes implied volatility [closed]
I am reading up on implied volatility and I encountered the term Black-Scholes implied volatility which I haven't heard before. What is the meaning of this term?
Say I am looking at the Heston model ...
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0answers
72 views
Pricing of an option
I've priced a European option with payoff $\max\{S_T(S_T - K), 0\}$ and found
$S_0(S_0 \exp((r + \sigma^2)T) \mathcal{N}(d_3) - K\mathcal{N}(d_1))$
where $d_i = \frac{\ln(\frac{S_0}{K} + (r + \frac{i\...
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0answers
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Expected life (Fugit) of American Option
How can I use the binomial tree pricing method for American options to determine the expected time of exercise for the option (or "Fugit")? In particular, how would I modify the algirithm ...
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0answers
33 views
variance of asset returns linear for time
I am reading Wilmott's book, "Quantitative Finance" and try to understand the derivation that the variance of asset-returns, $V[\Delta S/S]$, is a linear function of the time step $\delta t$....
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52 views
Quantity of risk-free asset in Black Scholes model
When the seller of a Call option hedges themselves, we know that they should buy $\Delta(t) = \mathcal{N}(d_1(t))$ amounts of the risky asset at time $t$.
But what about the riskless asset? My ...
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1answer
85 views
Why does black scholes model give lower prices for puts with further time to expiry?
Consider BS-model with parameters: Stock = 100, Strike = 100, Texp = 1 year, Vol = 13%, Rf Rate = 3%. For these parameters the BS put price is 3.76. Then consider the same parameters but with Texp = ...
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1answer
70 views
Computing the Probability Density Function (PDF) for the Heston model
I am trying to compute the PDF for the Heston model using the Breeden Litzenberger formula.
I have calculated the the Heston implied volatilities for a strike range (which i have interpolated using ...
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0answers
43 views
Error in Call Option Valuation using Implicit Finite Difference implemented in Python
I am trying to valuate call option using implicit Finite difference method (Forward Marching) implemented in Python. However I am getting the error in the code. Following is the code I have developed:
...
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32 views
How do I hedge two/three zero coupon bonds with different maturity under Vasicek short rate model?
I am working on the case that I need to hedge two bonds with different maturites under Vasicek model, which is
\begin{equation}
dr_t=a(b-r_t)dt+\sigma dW^Q_t
\end{equation}
and I know how to price the ...
0
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1answer
101 views
Black-Scholes formula given arbitrary value of $S_{T}$
Is there a formula for Black and Scholes when we have expected payoff $\mathbb{E}[\max(se^{X}-K,0)]$ for $X$ having any normal distribution?
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36 views
Hedging costs and BS-price
I'm looking at the chapter, "The Greek Letters" in Hull's book (Options and derivatives...) and in particular the paragraph "Dynamic Aspects of Delta Hedging". He demonstrates two ...
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1answer
141 views
Black Scholes to Heat Equation
Equation (2) was derived by setting r=0 in the Black-Scholes equation for the Bachelier model (1).
Can someone please help me understand all the steps for how we get from the heat equation under time ...
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29 views
Breakdown of Wilmott's Binomial Tree derivation of Black-Scholes equation
Hi guys,
I tried to follow the chapter of PWIQF on binomial model and got stuck when it derived the Black-Scholes (please see image). I tried to backtrack the said equations but couldn't trace back ...
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Value in time of the bond in delta-hedging
I am trying to implement a simple delta-hedging strategy. The idea is that I want to verify that the covered position "1 option long + delta stocks short" is actually evolving as $e^{rt}$, ...
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Estimating dividend yield & risk-free rate from Futures prices
I would like to work with the dividend-adjusted Black Scholes formula and need to estimate the dividend yield and risk-free rate. I know that I could compute both rates exogenously. But I am working ...
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Using EOINA, ā¬STR for option valuation
for an assignment I have to value options using an OIS rate using the Black-Scholes model. Since my options are traded in Germany I was looking at the EONIA or newly ā¬STR. (Since Hull 2012 recommends ...
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0answers
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Option that never expires
I have been struggling with the problem below for quite some time now. I really don't know how to approach it. All I could think of is to use the Black-Scholes formula with $T \rightarrow \infty$, ...
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2answers
68 views
Volatility estimation based on a 60 days range
In Hutchinson et al: A Nonparametric Approach to Pricing and Hedging Derivative Securities Via Learning Network (1994) paper (link), to estimate $\sigma$ for the Black-Scholes formula, it says (p. 881)...
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0answers
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Stocks with same volatility but different drifts
In the book Quant Job Interview Questions & Answers, in section 2, question 2.4 says suppose two assets in a Black-Scholes world have the same volatility but different drifts. How will the price ...
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1answer
87 views
Asian option sensitivity
I am looking for some materials for profiling all options sensitivities for Asian options with both geometric averaging and arithmetic averaging . The underlying price $S_t$ follows a standard GBM.
Is ...
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1answer
108 views
How to calculate dividend yield - option pricing
Hey how do you calculate the dividend rate if you want to price your stock options eg apple? Just take the dividends paid last year and divide by today's share price? This page reports 0.85% (https://...
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1answer
68 views
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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1answer
48 views
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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0answers
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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 ...
2
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2answers
280 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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3answers
193 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):
$$\...
2
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1answer
105 views
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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1answer
125 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 ...
