Questions tagged [black-scholes]

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

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53 views

Does Black Scholes + Stochastic interest rates result in a unique price

Black-Scholes with its assumptions results in a unique price for the call. If we introduce stochastic interest rates, would this still remain the case?
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Stochastic Volatility Models - are they complete markets?

I'm reading about stochastic volatility models - the ones which resulted after Wiggins proposed in 1986/7 that $\sigma$ in Black-Scholes should be a stochastic process rather than a constant. In ...
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Is the forward price equal to the future price?

If $f^{T_1}(t)$ is the price of a forward and $F^{T_1}(t)$ is the price of a future on some stock, both maturing at date $T_1$ and with the assumptions: no dividend constant interest rates no ...
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Black-Scholes market and payoff with integrals

I am struggling with the following exercise: Prove that on Black-Scholes market, with some parameters $r, \mu, \sigma >0$, a payoff $$X=\int_{0}^{T}\ln \frac{S_t}{S_0}\mathrm{d}t+\frac{1}{\...
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Trying to measure “radius of diffusion” in the stock market

Good evening! I'm quite new to quantitative finance (coming from the math world!), so please excuse me if I'm not familiar with every concept! I am currently studying the Black-Scholes equation, ...
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Prove that it is possible to make a self-financing portfolio

I would like to prove that if there exists $(X_1,\ldots,X_n)$ satisfying $\mathbb E[\int |X(s)|^2 d[Y]_s]<\infty$ (for a standard filtered probability space $(\Omega, F,( F_t)_{t\ge 0},\mathbb P)$ ...
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What is the relationship between the Black-Scholes Model and Geometric Brownian Motion [closed]

Im just a beginner in finance and understand a bit of mathematics. Im just wondering in layman terms. Please explain very simply, in addition, I don't understand the log-normal aspect.
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In BS model, is there a way to show that the risk-neutral Q is unique without using MRT nor the fact that the market is complete?

In Black-Scholes model, is there a way to show that the risk-neutral probability measure is unique without using the martingale representation theorem nor the fact that the market (in BS model) is ...
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Convert option inputs to standard Brownian motion

I want to know the probability that the strike price of an option is touched. My input values are: P = price S = strike v = vol t = time to expiration According ...
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What are the relation between the risk neutral measures in binomial tree and in Black Scholes model?

I appreciate that both are the direct result of constricting a replicate portfolio using stock and bonds. Are there deeper relationship between the two?
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Barrier option with zero-strike

Good morning, everybody, I would like to know whether an up-and-out call option with a zero strike has a special name in the list of exotic options or is still a special case of a barrier option.. ...
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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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Can you explain the Black-Scholes fair option equation with RND?

I am trying to learn Black-Scholes risk-neutral densities with only prior knowledge of fundamental B-S equations (not the derivation). Sorry if this was asked already or if I sound completely clueless....
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BS model without volatility

Maybe it is a naive question, I simply can't understand how the industry is using the BS model to price options, as the option pricing formula requires implied volatility as an input, which itself is ...
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yield curve basics

Suppose we observe the following term structure (of annualised spot rates): 0-3 Months $\rightarrow$ 4.0%. 0-6 Months $\rightarrow$ 4.2%. 0-9 Months $\rightarrow$ 4.4%. Question1) How can we ...
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Why are the risk neutral probabilities constant in the Cox Rubinstein model when delta needs to be changed at each time step

Consider the Cox Rubinstein binomial pricing model with N steps, with stock price change given by parameters u and d so that at step $i$ we have $S_{i+1} = uS_{i}$ or $S_{i+1} = dS_{i}$ with $0\leq i \...
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Black and Scholes equation for portfolio **with** arbitrage

I am well aware of how the ordinary Black and Scholes equation is derived, under the assumption of an arbitrage free portfolio, $V=G-hS$. Here $S$ is the price of the underlying and $G$ is the option ...
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Insured Portfolio via call + cash: how much cash?

I am unsure about the quantities to keep in the risky asset, S, and the non-risky asset, M, when constructing an insured portfolio via Call + Cash (rather than Stock + Put). My understanding so far is ...
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Quick Question: Error in Theta Put Option for Black Scholes VBA

I have started an analyst role and I am trying to familiarize myself with the Black-Scholes formula in VBA to gauge option prices. However, I cannot seem to get the Put Theta to work properly. I have ...
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Which process is the most commonly used for modeling stock prices?

I'm thinking of writing a master's thesis about pricing options using Levy processes, but I wonder if these processes are actually used for modeling stock prices or not (and which specifically)? And ...
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Black Scholes PDE boundary conditions

So I'm trying to solve the black scholes equation using a finite difference model, but I'm getting a answer that's off and I'm having trouble understanding why. This is the result for a option with K ...
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48 views

Why does Implied volatility fall when the options market shows an upward trend?

While reading How does implied volatility affect option pricing by Investopedia, it states the following in key takeaways When options markets experience a downtrend, implied volatility ...
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Question about Forward Price + Constant Interest Rate Approximations

Sorry if this is an obvious question, but I'm reading the following paper An Explicit Implied Volatility Formula, Dan Stefanica, Rados Radoicic, International Journal of Theoretical and Applied ...
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Calculate historical Value at Risk (VaR) of European Vanilla option

To calculate historical VaR at 95% confidence level of on a stock following Geometric Brownian Motion distribution, we do a large number of Monte Carlo simulation, say 10,000 samples of stock prices, ...
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Fourier transform of price function

If the expiry value is given by $f(x,T) = e^{-c x}$ for $x \ge a$ and 0 otherwise and c is a +ve constant, prove that in the Fourier domain: $$ (c + j \omega) F(\omega, 0) = e^{-rT} e^{-a(c+j\omega)}...
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What is the source of gamma risk?

I have two quasi definitions or interpretations of gamma risk in the context of the BSM model (please correct me if these don't make sense): 1) it is the option's sensitivity to jumps in the ...
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How to extrapolate shorter tenor from volatility surface?

Overnight(ON) volatility is the first input of a volatility surface, 1 weeks, 2 weeks and so on... Say I have a volatility surface with ON expiry of 1 day, is there anyway to extrapolate volatility ...
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Price adjustment of Black-Scholes delta and gamma for a quanto option

A quanto option is a derivative with the underlying and strike price denominated in one currency, but the instrument itself is settled in another currency. This has consequences for the calculation of ...
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Annualised Risk Free Rate with Black and Scholes

In regards to the Black and Scholes model, I am considering two months of call and put options and have a two zero rate. For the model specification, am I correct in needing an annualised risk free ...
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Calculate Third Order Greeks Options

Hope you're doing great! I'm struggling to develop the code for the Third Order Greeks. In all places I have searched, the development is missing. For example: But I don't know how to develop it, ...
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Simple application of the fundamental theorem of asset pricing

From what I understood the fundamental theorem of asset pricing (FTAP) details that discounted asset prices are martingales under the risk neutral mesure. As an example: We consider an ATM call ...
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Should a normal distribution be used for valuing options on assets that can potentially have negative prices?

The Black-Scholes-Merton model assumes that the prices of the underlying asset at maturity are log-normally distributed. I understand that this assumes that the prices can never go below zero. ...
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Idea of using logarithm for solving SDE in Black-Scholes model

In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$ dS = \mu S dt + \sigma S\ dW $$ I was wondering, was it common at the time they work on this ...
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Call Probability of European callable IRS

When pricing a callable IRS (say only one call date) with a diffusion model (e.g. HW 1F) with a Montecarlo resolution, one can get the call probability on the call date versus maturing the date (which ...
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Is implied volatility model specific in the context of options on stocks and indices?

I just wanted to clarify a few things around implied volatility which I've read in the Hull/Natenberg books, which I find quite confusing. Both books refer to implied volatility in the context of a ...
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Black-Scholes Model with Cost of Carry

Good day, I was going over some exercises for my course and found this rather long one but I dont know how to start. Consider a commodity whose unit price at time t is St. Ownership of a unit of ...
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How to use black scholes for spot trading?

Since I am very new to this topic, i am struggling to connect the option pricing with Spot Trading. Is there a way to use the Black-Scholes model to derive entry and exit Events? Any blogs or papers?...
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Does the Black-Scholes formula work when unit of time is in hours?

In the Black-Scholes formula, the unit of time is usually in years from what I understand. An online calculator I found allows the users to input the time in days and years. Would the formula still ...
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Black-Scholes Delta value at maturity?

Having to implement a replication strategy for European options, I encounter the following problem: Delta tells me how many shares to hold at time t in my replication strategy. To do so, I simply ...
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Options delta as a percentage of option price

I'm dissatisfied with the usefulness of delta and would like to get your feedback on a slight tweak on it. Example Consider two options for a made-up stock at \$5 with IVs around 120%. Option A: ...
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Black & Scholes under stochastic interest rate (Vasicek) [closed]

I'm a beginner in Quantitative finance and I'd like to ask you for help about this exercise. I have to price a put option on a risky asset by working under stochastic interest rate, so I have to ...
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63 views

Implied volatility and greeks of options

When we are calculating deltas or vegas for different strikes should we use the underlying asset's volatility or should we use the implied volatility for the specific strikes at a fixed maturity? ...
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Black-Scholes and solving for both $r$ and $\sigma$ ; Do I have a unique solution?

Below is a problem that I am working on. I believe that my incomplete solution is correct as far as it goes. I would like to know if my solution is incorrect. I plan to solve the system of two ...
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Connecting the dots: Black Scholes, Volatility and Implied Volatility

I am a first year Management & Finance undergrad preparing for my second year Finance courses, given that term 3 and exams have pretty much been cancelled for all British first years. During that ...
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lognormal assumption of Black Scholes

I have recently started learning about option pricing and the Black Scholes formula, where stock prices are assumed to be lognormally distributed and returns normally distributed. While trying to do ...
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Why is it so rare for finance theory to depart from the normal distribution?

I understand almost all of the theory that has been built upon in quantitative finance is based on the normal distribution, and obviously you wouldn't want to throw all of it out the window on a whim ...
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Pricing an interest rate floor

I am trying to estimate the value of a 0% interest rate floor by pricing each individual floorlet. Since BS won't work for this problem, I am trying to use normal volatility in a Bachelier model like ...
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Pricing In Real Life vs Theory

When selling/buying vanilla call options, do one price them according to some pricing formula (i.e Black-Scholes)? Or is the only point using pricing formulas to find the implied volatility and then ...
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What is the delta of an at-the-money European call option with respect to volatility?

Question: What is the delta of an at-the-money European call option with respect to volatility? Note that $$\frac{\partial\Delta}{\partial\sigma} = N'(d_1) \frac{\partial d_1}{\partial\sigma} = N'(...
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Can a delta hedge be negative for all values at one time, and positive for all values at another time?

I have a problem that states there was a formula for the hedge $\delta(t, S_t)$ for a contingent claim whose value depends on only the stock value when $T=20$. In this hedge, $\delta(t, S_t)<0$ at $...

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