Questions tagged [black-scholes]

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

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Wealth process in the Black-Scholes model with discrete dividends

Good evening, The following problem is the sequel of a previous post I made here a few days ago. Consider the Black-Scholes model with discrete dividends in the interval $[0,T]$. This means that ...
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Delayed Settlement Option- how will values in Black Scholes change

If there is an option that expires a year from now, but is settled after 2 years, how would the Black Scholes formulation for such a situation look like? Will the risk free rate now be for 2 years or ...
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Expected stock price range using implied volatility calculated by Black-Scholes

What's the correct way to calculate the expected stock price range using implied volatility, without the simplifying assumption that the stock price follows a normal distribution?
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Exotics - Combination of different payoffs using Black-Scholes

I'm currently struggling with the derivation of a formula to price the following exotic option with Black-Scholes. The option has the maximum payoff of $(S_T-z)$ and $(y - S_T)$, where $S_T$ is the ...
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Black-Scholes model with discrete dividend payments

Consider the Black-Scholes model with discrete dividends in the interval $[0,T]$. This means that there's a sequence of dates such that, $$0 < t_1 < \dots < t_k < \dots < t_n < T $$ ...
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Black-Scholes option pricing [duplicate]

Consider Black-Scholes (B, S) market model. Let $r = 0$ (hence, $B_t ≡ 1$), $S_0 = 0 $. Stock price is described by $dS_t = σS_tdW_t$. Find the price of the option that pays $(S_T^3 - S_T^2 )_+ = max(...
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Why Black Scholes model needs to assume S_t follows GBM if physical probabilities do not matter?

Beginner here learning about black scholes looking for a high level/intuitive explanation. So I've learned that the physical/"true" probabilities of S_t (or whatever underlying asset) do not ...
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Numeraire explanation on currency greeks

Would it be possible to help understand the numeraire of certain currency options? Derivations from the Black Scholes models for Delta and Gamma, $Delta = e^{-r_f T} N(d_1)$ $Gamma = \frac{e^{-r_f T}}{...
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Geometric brownian motion small timesteps high volatility

I'm trying to generate some sample geometric brownian motion paths for an asset which is traded 24/7 without interruption and is highly volatile (upwards to 150% implied volatility on options markets)....
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profit opportunities from accurate forecasting of delta?

Are there any option trading strategies that can profit by modeling delta more accurately than Black-Sholes does? I'm looking at models for predicting delta, and I can clearly see how these can help ...
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Calculating the short rate from the discount curve

I'm currently looking at some code that implements the Hull-White model. As one of the inputs, the code accepts a table of discount factors at various dates. Time in Years Discount Factor 0 1 0.003 ...
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Price of barriers in black scholes

Do you know a simple way or proxy, formula to determine the price of a down and out call with strike 100, barrier 100, spot 110 in a BS world with no rate and a 10% vol ? Thanks for your help
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Why is dividend discounted stock equal to cash discounted strike?

Doing some BS problem solving and came across this beauty: $$S \cdot \text{exp}\Big(-D(T-t)\Big) \text{exp}\bigg(-\frac{d_{1}^{2}}{2}\bigg) = E \cdot \text{exp}\Big(-r(T-t)\Big) \text{exp}\bigg(-\frac{...
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Why, under Black-Scholes, do returns exceed one $\sigma$ every three days, approximately?

My book claims the following: "The other assumption of normally distributed returns would mean that returns would be between $-\sigma$ and $+\sigma$ with 68.269 per cent probability. This means ...
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Finding the PDE and replicating strategy of a european contigent claim [duplicate]

Suppose that we have the Black and Scholes model where the interest rate and the volatility are time varying: $dB(t)=r(t)B(t)dt$ and $dS(t)=S(t)b(t)dt+S(t)\sigma(t)dW(t), S(0)=s>0$ where $r,b,\...
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Derivation of the Gamma approximation formula from the Delta approximation formula

The formula of $Gamma = (Vplus + Vminus - 2V0)/(V0 * dS^2)$, where $V$ is the contract value, $S$ is the stock price. We also know that $Gamma = (Dplus-Dminus)/(Splus-Sminus)$, where $D$ is the ...
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Martingale proof: Call-prices must be increasing in maturity

I have observed that IV is increasing with time to maturity by using market prices and plotting IV (from Black-Scholes) against log-moneyness, $\log(S_t/K)$. $S_t$ being the price of the stock at time ...
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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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What is “level” and “term” in Derman figures?

Im doing my final thesis about implied volatility. In the last section I am talking (not too much deep) about volatility surface. Im using the figure below to show the differences between the real ...
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Why is implied volatility often higher for OTM/ITM european call options than ATM? [closed]

I am working on some Black-Scholes stuff and currently investigating implied volatility (IV). I understand that the typical volatility smile can be viewed as a criticism of the assumption about ...
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Strike Price Determination

Suppose you know the following: there are 2-month European call and put options on an index-like instrument with no dividends, the calculations show that the call option price is USD 10.1150, the spot ...
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Drift Term in Black-Scholes Model Martingale

How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof). For instance if Im looking at: $$dS_{t} = \mu S_{t} + \...
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64 views

European call option lower bound derivation by Black-Scholes formula [closed]

Derive the lower bound of european call options: $$C(S, t)\geq[S-e^{-r(T-t)}K]^+$$ I know how to derive it using put-call parity, but is there any way to derive from Black-Scholes formula?
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Empircal data analysis delta hedge error of Black-Scholes by Mark Davis

Regarding Mark Davis derivation of the delta-hedging error occuring in the black-scholes as a result of difference in realized volatility and implied volatily. The formula reads as follows: $$ Z_t = \...
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Multidimentional Black Scholes Formula

I need to write the Black-Scholes formula for option $V = (S_1, S_2, t)$, where: $$d S_1 = \mu_1 S_1 dt + \sigma_1 S_1 d W_1,$$ $$ d S_2 = \mu_2 S_2 dt + \sigma_2 S_2 d W_2.$$ We know that $W_1$ and $...
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Black-Scholes with two options

I have got a Black-Scholes model with portoflio with two options which bond prices are $V_1$ and $V_2$ (with different maturities or strikes). The interest rate $r$ is stochastic and given by: $$ dr = ...
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Is it a problem that there are so few stocks in the generalized Black Scholes market? [duplicate]

In the standard Black Scholes market there is only one stock. In the generealized market there can be a finite amount, but my impression is that there are few stocks in the market. The real world ...
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Volatility surface interpolation for Black-Scholes delta hedging

A general question for interpolation method for implied volatility between tenors. I've recently stumbled accross a dataset from http://www.math.ku.dk/rolf/Svend/, and I would like to interpolate the ...
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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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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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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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127 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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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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159 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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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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