Questions tagged [black-scholes]

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

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2answers
134 views

Deriving implied volatility programmatically

I'm working on a project to calculate the value of options using Python. I'm using the Black-Scholes model, and I can get accurate results by plugging in a given ...
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1answer
38 views

Arithmetic Asian Option

Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $μ$ and volatility $σ$). Let $A_T:=\frac{1}{T}...
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1answer
55 views

Black Scholes theta as function of time to maturity

I would like to understand why the Black and Scholes greek letter theta for european call option behave in the following way: as time to maturity is far away (right part of the x-axis in the the ...
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1answer
94 views

Asian Options-Change of Numeraire

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Show that ...
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0answers
33 views

Black-Scholes model - Calibration of the risk-free rate

I know there is a lot of content about this topic, but I have not seen a post which gives a satisfying answer to my problem. I am trying to hedge a European call option with real market data under ...
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0answers
61 views

Black-Scholes equation Variational / Weak form

I am having difficulty deriving the weak formulation of the Black-Scholes Equation. I have multiplied it with a test function phi and integrated over Omega. But results on the internet suggest ...
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1answer
48 views

Call Option on the Square of a Log-Nomral Asset

I'm working on a quant interview question from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors).I cannot understand its answer well and really appreciate ...
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0answers
24 views

Black-Scholes-Merton and alternatives as interpolation tools

This is a not very quantitative question, but is nevertheless related to quantitative methods in Finance. I was reading the following paragraph from Hull's Options, Futures, and other Derivatives: ...
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2answers
107 views

Proving a process is martingale under the Risk Neutral Measure

Show that for any $\lambda \in \Re$, the process $Y_{\lambda,t}$ defined as: $$Y_{\lambda,t} = (S_t/S_0)^\lambda e^{-(r\lambda-\lambda(1-\lambda)\sigma^2/2)t}$$ is a martingale under the risk ...
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0answers
57 views

Proving an Expectation

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends. Consider the perpetual American put option with payoff $(K-S_\tau)^+$ when ...
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1answer
144 views

Floating Strike Lookback Call Option

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate $r$, stock drift $\mu$ and volatility $\sigma$). If $r=\...
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1answer
100 views

Different volatility surface ( Local vol, Stochastic vol etc.)

Despite many questions about local and stochastic volatility available on this forum, i still have a few doubts left. Essentially I am seeking validation whether I am interpreting things correctly. ...
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1answer
59 views

Premium Adjusted Delta in fx market

Please explain the concept of premium Adjusted Delta in FX market. In EURUSD, why delta changes if premium currency is changed from USD to EUR and how this new delta is related to the old one with ...
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0answers
30 views

Black-Scholes delta of a barrier (knock-out or knock-in) option

I'm trying to calculate the Black-Scholes delta of a barrier option given the following information: Whether it is knock-out or knock-in Barrier price Strike price, $X$ Current stock price, $S$ ...
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0answers
63 views

Alternative derivation of Black Scholes by Merton

I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I ...
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1answer
80 views

Stochastic solution (mean, variance) to lognormal drift and normal volatility

I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model I ...
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3answers
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What causes the call and put volatility surface to differ?

I currently have a local volatility model that uses the standard Black Scholes assumptions. When calculating the volatility surface, what causes the difference between the call volatility surface, ...
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2answers
244 views

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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1answer
99 views

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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1answer
240 views

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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5answers
28k views

Why do some people claim the delta of an ATM call option is 0.5?

I am looking for a mathematical proof in terms of differentiating the BS equation to calculate Delta and then prove it that ATM delta is equal to 0.5. I have seen many books quoting delta of ATM call ...
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1answer
93 views

Linear Or nonlinear Black Scholes Equation

I have been going through the analytical solutions of black scholes equation which transforms it to a heat equation. $$u_{t}=\frac{1}{2}\sigma^{2}u_{xx}$$ Now if the volatility is constant , then its ...
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1answer
36 views

some questions about pricing an asset or nothing put option with a strike price equal to St

I am working on a homework exercise where the aim is to price an asset or nothing put with K = St, offcourse the normal formula could be used St * N(-d1), but I was wondering if pricing the asset by ...
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1answer
77 views

Derivation of Call Delta from Black Scholes Model

How is call delta mathematically derived from Black Scholes Model (without approximation) ? Please help me understand each step mathematically. And how it is approximated to say that delta is the ...
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1answer
32 views

Asset return distribution

What is the basis for assumption that asset prices follow a log normal distribution? Then how is it transformed to say that asset return follows a normal distribution? How this relationship between ...
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1answer
37 views

Security value based on futures contracts of a traded and non-traded assets

S1 - index with dividend a, S2 - non-traded asset. A security pays off $S_{1T}S_{2T}$ upon its maturity S1 and S2 are uncorrelated and follow geometric brownian motion. What is the value of ...
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1answer
81 views

Delta hedging: theoretical value vs actual price

One way to derive the Black-Scholes PDE is via the Delta-hedging argument: Suppose that $V_t = V(t, S_t)$, for some function $V: [0,T] \times \mathbb{R} \to \mathbb{R}$. We construct a portfolio by ...
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1answer
58 views

Intuitive explanation of why ITM options have low Time/Extrinsic Values?

While brushing up on my knowledge about the Greeks, I have been struggling coming up with an intuitive, probability-based explanation behind why not only Out-of-the-Money (OTM), but also In-the-Money (...
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1answer
155 views

Calculate the price at time t=0

Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Calculate the price at time $t = ...
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0answers
49 views

Develop an option pricing equation by Ornstein Uhlenbeck process

I know that Black-Scholes equation is based that the Equity price has a Geometrical Brownian movement. Can I develop from the same principles( now with transaction cost) that Black Scholes is ...
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0answers
73 views

Does an option need to be tradable for Black Scholes pricing formula to hold?

Given the classic Black-Scholes model, e.g. $dS(t)/S(t)=rdt+\sigma dW^{\mathbb{Q}}(t)$ with $S(0)=S_0$ and $dB(t)=rB(t)dt$ with $B(0)=1$, whereby $r$ and $\sigma$ are constants and $\mathbb{Q}$ ...
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1answer
69 views

Björks second $S$ process when introducing martingale measures

When Björk presents the Black-Scholes model and martingale measures he starts off with a process modeling the stock price calling it $S$ with some given dynamics w.r.t some measure $P$. Then he ...
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1answer
37 views

How to derive Balck Scholes from the Binomial Model?

The book gives the following recipe, but no further details: Do a Taylor series expansion of $$V = V(S,t)$$ Do a Taylor series expansion of $$V^{+} = V(u \cdot S, t + dt) \hspace{5mm}:\hspace{5 mm} u ...
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2answers
137 views

Understanding $N(d_1)$ and $N(d_2)$

Firstly, if the solution to geometric Brownian motion is $S_t = S_0 \exp((r-\sigma^2)t + \sigma W_t$ then if I have a payment that is not necessarily a full call option e.g. if the exercise price $K$ ...
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2answers
106 views

How to derive Black-Scholes equation with dividend?

Question: The Black-Scholes equation without dividend is given by $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} -rV = ...
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2answers
91 views

Rate of return in Black-Scholes model

The rate of return of a stock is denoted $\frac{dS}{S dt}$ where $S$ is the solution to the SDE modeling the price of a stock. Can someone give an explanation of the rate of return and what it is ...
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36 views

Is my derivation of Black-Scholes equation correct or am I missing something (eg assumption)?

Question: The following is my derivation of the Black-Scholes equation. Is it correct or am I missing some details (eg assumption)? Let $V$ be value of an option. Suppose value $\Pi$ of a portfolio ...
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1answer
2k views

Black-Scholes formula producing a negative number for a Call Option

I would expect that the Black Scholes model should always give a value for a call option, $c$, to be at least $0$. However, I am seeing some cases where that is not the case. Here is the Black-Scholes ...
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1answer
57 views

If the volatility is zero (i.e. σ=0), what is the call worth? After valuing the call, how to hedge the call (assuming you sold it)

Question: All Black-Scholes assumptions hold. Assume no dividends. The stock price is $100. The riskless interest rate is 5% per annum. Consider a one-year European call option struck at-the-money (i....
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1answer
109 views

Arbitrage free in a Black-Scholes/Poisson model

I am trying to solve the following exercise from Bjork's Arbitrage Theory in Continuous Time: Consider a model for the stock market where the short rate of interest $r$ is a deterministic ...
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0answers
35 views

Option pricing with definite integral

I would like to consider a slight generalisation of this question, which I recall here: At date of maturity $T_2$ the holder of a financial contract will obtain the amount: $$ \frac{1}{T_2−T_1}\...
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2answers
166 views

Black-Scholes-Merton formula and option pricing

If the distribution is skewed to the right,Black-Scholes overprices out-of-the-money puts and in-the-money calls. It underprices in-the-money puts and out-of-the-money calls. How? Stock price log-...
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1answer
68 views

Why and how is Implied volatility directly related to stock price but inversely related to strike price?

I know that in equity markets there is a volatility smirk which results in higher IV for lower strike price options because of crashophobia and leverage related factors but I can't wrap my head around ...
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6answers
18k views

How do you explain the volatility smile in the Black-Scholes framework?

Does anyone have an explanation for the currently naturally forming volatility smile (and the variations) in the market?
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2answers
1k views

Arbirtage free price process question in Bjork's Arbitrage Theory in Continuous Time

I am currently working through questions in Bjork's Arbitrage Theory in Continuous Time. However, I am unable to solve the following question, 7.2 in the book. A solution would be greatly appreciated. ...
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1answer
61 views

Exercise on arbitrage-free process

Consider the following problem, from Bjork's Arbitrage Theory in Continuous Time: Consider the standard Black-Scholes model. Derive the arbitrage free price process for the $T$-claim $\mathcal{X}$ ...
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2answers
194 views

Why does expected price of OTM option not equal to BS price?

If I assume that stock returns follow normal distribution with drift = 0% and S.D. = 10%. In the long, if I keep investing in this stock for a year with the same capital every year for a consecutive ...
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1answer
58 views

In literature, is IV constantly adjusted during option delta hedging?

In a lot of literature, they like to compare the performance of buying an option, and then delta hedging either at that options implied volatility (IV) or the true future volatility. This is under ...
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1answer
74 views

When is a numerical solution the only way to obtain a solution to BS?

I am only now reading into Mathematical Finance, I understand the derivation of the BS equation with vanilla European options. On the next page of my book it starts to delve into obtaining exact ...
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1answer
303 views

Option pricing with negative short-term interest rates

In countries with negative short-term risk-free interest rates, do you just use a negative "r" in the Black-Scholes formula, or do adjustments need to be made?