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

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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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Black-Scholes: Why the focus on volatility?

We know Black-Scholes is an imperfect model for options pricing. Why is so much of the analysis of its defects focused on implied volatility? The fact that IV varies for the same stock at the same ...
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What is the difference between the methods (listed in content) in pricing convertible bond?

To price the convertible bond, one of the models is the bond plus equity option method. That is, the value of convertible bonds is evaluated by finding the value of the straight bond and the value of ...
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Black--Scholes hedging argument

I'm trying to understand the standard hedging argument to derive the Black--Scholes PDE. There's one aspect of the derivation which I can't get passed and I'd be very grateful for some clarification ...
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Can we explain physical similarities between Black Scholes PDE and the Mass Balance PDE (e.g. Advection-Diffusion equation)?

Both the Black-Scholes PDE and the Mass/Material Balance PDE have similar mathematical form of the PDE which is evident from the fact that on change of variables from Black-Scholes PDE we derive the ...
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Physical Option Implied Distribuition

So I got risk neutral probabilities from stock option prices. How can I then map them to a physical measure?
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Question about Merton model to estimate default probability and recovery rate of the company

I recently come across Merton's model to estimate the default probability and recovery rate of the company. Here is the inputs ...
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180 views

Can option prices be characterised by an ODE?

If a stock price, $S(t)$, is governed by a geometric brownian motion. Is it possible to characterise the value of an option $V(S,t)$ as an ODE rather than a PDE (given $S$ is itself a function of ...
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Calculating the error of a Trinomial Model

I've been trying to find a formula to obtain the maximum relative error a trinomial model with n timesteps will incur given all other inputs as compared to the standard BSM model. I'm concerned mostly ...
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329 views

Question about option theta

I have a question about a option theta. When I evaluate the option theta of near expiry put option using Black-Scholes formula given the data as follow: Index Level = 20,500 Strike Price = 20,000 ...
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Black (1976) model: boundary conditions with non-convergence of spot and forward prices

Let's suppose we have a futures contract F in a market where the relation $$F(t,T)=S(t)e^{r(T−t)}$$ doesn't hold. What are the the boundary conditions for the derivation of the Black (1976) ...
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246 views

Black (1976) model: relationship between spot and forward prices

Does the Black (1976) model require the existence of the relation $F(t,T)=S(t)e^{r(T−t)}$? I studied the derivation of the Black-Scholes formula. However, although I know the Black formula, I've ...
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Black model - volatility estimation

In the Black (1976) model: We should use the settlement prices of the underlying futures contract in order to estimate the volatility, right? Or can we also use the spot prices? Because the ...
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Black-Scholes in Delphi [closed]

when trying to implement the Black-Scholes formula in Delphi, I've found this: http://www.espenhaug.com/black_scholes.html I've checked the results against option-price.com and found they are ...
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548 views

Extensions of Black-Scholes model

For the Black-Scholes model my feeling is that the volatility parameter is like sweeping stuff under the rug. Are there models which improve on the volatility aspect of Black-Scholes by adding other ...
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1answer
226 views

options pricing using vwap

This is a question about why options prices do not take volume into account. The popular option valuation formula "black-scholes" certainly does not account for this and I don't suggest that it does. ...
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731 views

How to calculate return rates with negative prices?

I'm dealing with electricity options and I'm considering the possibilty of negative prices. I want two estimate the historic volatility. However, an arithmetic mean doesn't feel appropriate and ...
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129 views

Am I reading this correctly? probability way too small with BS model

For a stock trading at $27, $28 strike, 0% interest, 15% annual vol, and one day until expiration there is about a 1 in 17000 chance of it being exercised? $d_2 = ...
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295 views

Formula for variance of European call/put in Black Scholes

I have a quite basic question, but I can't find a reference with it. Recall that we can use the Black-Scholes formula to price a European call or put for a market consisting when: the underlying ...
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208 views

Change option B&S pricing

Consider a market composed by two stocks whose prices $X$ and $Y$ are given by B&S diffusion $$dX_t= \mu X_t dt+ \sigma X_tdW_t$$ $$dY_t= \mu Y_t dt+ \sigma Y_tdB_t$$ Supposing the market is ...
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279 views

Black-Scholes fastest computation method

What is the fastest way to numerically compute Black-Scholes-Merton option prices? I'm trying to find fastest and still precise method. Currently I'm using numerical approximation of Normal cdf with ...
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4answers
2k views

Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while. We all know geometric brownian motions have drifts $\mu$: $dS / S = \mu dt + \sigma dW$ and different stocks have different drifts of $\mu$. Why would ...
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Black-Scholes PDE to heat equation, nonconstant coefficients

Can someone provide me with details or a reference on how to transform the Black-Scholes PDE with nonconstant coefficients (i.e. $r=r\left(S,t\right)$, $\sigma=\sigma\left(S,t\right)$) to the heat ...
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Easiest and most accessible derivation of Black-Scholes formula

I am preparing a QuantFinance lecture and I am looking for the easiest and most accessible derivation of the Black-Scholes formula (NB: the actual formula, not the differential equation). My favorite ...
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495 views

Replicating strategy in the Black-Scholes model

I have a two-asset Black-Scholes model for a financial market: $dB_t=B_t r dt$ $dS_t=S_t(\mu dt+\sigma dW_t)$ I introduce a European claim $\xi=max(K,S_T)$ with maturity $T$, for some fixed $K$. I ...
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price of a “Cash-or-nothing binary call option”

I'm stuck with one homework problem here: Assume there is a geometric Brownian motion \begin{equation} dS_t=\mu S_t dt + \sigma S_t dW_t \end{equation} Assume the stock pays dividend, with the ...
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Early execise of American Call on Non-Dividend paying stock.

Let us consider an American call option with strike price K and the time to maturity be T. Assume that the underlying stock does not pay any dividend. Let the price of this call option is C$^a$ today ...
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452 views

Relationship between European, American options volatility

Suppose, if the price of a European option (say a put) can be shown to be monotone in volatility (say for any maturity), does it follow that American options has to be monotone in volatility? ...
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481 views

Is vega of Black-Scholes European type option always positive?

We assume we work in the risk-neural measure with a stock which pays no dividend and a continuous discount rate. For PUT and CALL only: can someone please clarify if what I said is correct? The ...
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532 views

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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In Black-Scholes, why is $\log{\frac{S_{t+\triangle t}}{S_t}} \sim \phi{((\mu - \frac{1}{2}\sigma^2)\triangle t, \sigma^2 \triangle t)}$?

Namely, I dont understand why the mean is $(\mu - \frac{1}{2}\sigma^2)\triangle t$ and not just $\mu \triangle t$. I am aware that it is supposed to represent a lognormal distribution, but I guess I'm ...
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Black Scholes Formula for Collar Option

I am wondering if there exists a Black Scholes pricing formula for a collar option?
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Is there an all Java options-pricing library (preferably open source) besides jquantlib?

I am looking for an all-java implementation of black scholes, preferably open source. I found jquantlib and quantlib (C++). Any other recommendations? The jquantlib site seems to be down. I'd prefer ...
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Using Black-Scholes equations to “buy” stocks

From what I understand, Black-Scholes equation in finance is used to price options which are a contract between a potential buyer and a seller. Can I use this mathematical framework to "buy" a stock? ...
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Testing Black Scholes Analytical Options Pricer

I've written some code to calculate European option prices using the Black-Scholes analytical method. Can somebody recommend a good way to test that code? I have looked at option pricers online like ...
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Why the implied volatilities calculated are so different

I Calculated facebook option(expired in 12/4/13) Implied Volatility with the Bisection Method. The program will be attached at the end. The results for different strike prices are so different: ...
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Black-Scholes and Fundamentals

So basically $dS_t=\mu S_tdt+\sigma S_tdWt$ and $\mu=r-\frac12\sigma^2$ I have just been thinking about this later equation. This is very interesting because it ties together risk-free ...
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285 views

Doesn't a perpetual option contradict the Black-Scholes framework?

A standard example when learning to price American options is the perpetual American put. This is a put that has no expiry (or you can consider T = infinity). The standard solution prices this using ...
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volatility Table and BS formula

assume I have implied FX volatility Delta-Term table from broker. I have time noticed as 2M, 3M. what do I have to put into BS formula, is it 2/12 or "count the business days"/"daycount basis"? I am ...
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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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817 views

How to improve the Black-Scholes framework?

Since the distribution of daily returns are obviously not lognormal, my bottom line question is has BS been reworked for a better fitting distribution? Google searches give me nada. The best dist ...
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221 views

What are $d_1$ and $d_2$ for Laplace?

What are the formulae for d1 & d2 using a Laplace distribution?
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How do we use option price models (like Black-Scholes Model) to make money in practice?

In quantitative finance, we know we have a lot of option price models such as geometric Brownian motion model (Black-Scholes models), stochastic volatility model (Heston), jump diffusion models and so ...
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Methods for pricing options

I'm looking at doing some research drawing comparisons between various methods of approaching option pricing. I'm aware of the Monte Carlo simulation for option pricing, Black-Scholes, and that ...
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BS and delta hedging questions

I have two related questions concerning Black Scholes and delta hedging. I thought about this two questions, but I could not come up with an answer, so maybe you guys & girls can help me: If an ...
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2answers
991 views

Basket option pricing: step by step tutorial for beginners

I would like to learn how to price options written on basket of several underlyings. I've never tried to do it and I would appreciate if you can provide some documents, papers, web sites and so on in ...
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Option pricing before Black-Scholes

According to the Wikipedia article, Contracts similar to options are believed to have been used since ancient times. In London, puts and "refusals" (calls) first became well-known trading ...
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Black-Scholes American Put Option

Here is my question: This is a question about Black-Scholes model, but it may be applicable to more complicated models. Throughout the discussion, the strike price $K$, interest rate $r$ and ...
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Changes to option valuation for dollar-pegged underlying

In Russia, options on futures on the RTS index are priced in points instead of currency, with points being directly related to the value of the US dollar such that, for example, if the dollar rises, ...
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Pricing a Power Contract derivative security

I'm trying to price a "power contract" and would appreciate guidance on the next step. The payoff at time $T$ is $(S(T)/K)^\alpha$, where $K > 0$, $\alpha \in \mathbb{N}$, $T > 0$. $S$ is ...