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

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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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What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?

The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price S: $dS=\mu S dt + \sigma S dX$ Clearly $S$ ...
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Black-Scholes Equation - Riskless portfolio derivation

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
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Is this arbitrage?

Assume the stockprice as in the Black-Scholes model (Geometric Brownian Motion): $$S_t=S_0e^{(\mu-\sigma^2/2)\cdot t+\sigma W_t}$$ Wouldn't there be an immediate arbitrage opportunity, to just buy ...
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what is the vol in the BS formula?

I need to compute the delta of an option for which I know a) the time to maturity, b) the price of the option, c) the price of the underlying asset. what is the formula to get this delta It seems ...
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313 views

Price of a down-and-out call in terms of European call

If $EC(S_0, K, \sigma, r, T)$ represents the price of a European call option with strike $K$, expiry $T$, initial price $S_0$, volatility $\sigma$ and where the constant interest rate is $r$, then I ...
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56 views

Which risk free rate is assumed by market when pricing american options?

I'm just started with finance, so maybe my question is dumb or answered elsewhere. Please guide me to relevant materials. According to put-call parity more time to expiration means more difference ...
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133 views

Bachelier model: number of stocks in replicating strategy

Given: Consider a two-asset, continuous time model (B,S) where \begin{equation} dB_t = B_t r dt, \quad dS_t = \mu dt + \sigma dW_t. \end{equation} The question is: Show that there exists a ...
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Black scholes text book

I am looking for an easy and well presented introduction to Black-Scholes theory and stochastic calculus aimed at undergraduate mathematics students. Please can you recommend a book? How about Paul ...
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Selling an American call option early

I understand it is never optimal to exercise an American call option early. [1] [2] However, here are my two contradictory thoughts about selling an American call option early. Assumptions I can ...
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In a Black-Scholes world, why must volatility be strictly increasing in time-to-expiration?

This question is from Rebonato's Volatility and Correlation 2nd Edition. Rebonato states that if $\sigma_T^2T$ is not strictly increasing, it would be simple to set up an arbitrage. Unfortunately ...
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107 views

Volatility of Option

I hope I'm asking this at the right place. This pertains to actuarial exam MFE/3F on Financial Economics. If $\sigma$ is "volatility" and $\Omega$ the elasticity of the stock, one formula that is ...
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Forward rates diffusion

I used a simple market model (Black 76) to price an american swaption. It's a formula similar to B&S, with another numeraire and forward rate as underlying. I used the SDE: $$ dF = \sigma * ...
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145 views

How does Vega of a call/put behave under the Black-Scholes model?

I have two questions. I would prefer a reference if possible. Is the value of vega bounded for $\sigma\in [0,\infty)$? (I assume so, I imagine it goes to 0 as $\sigma$ go to infinity.) Are there any ...
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172 views

Error message in calculation Implied Volatility

I am unsuccessfully trying to find the Implied Volatilities for the SPX on a given date using information of the CBOE, as well as Open Interest, but as I run the code I am getting and error message ...
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302 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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How to prove price of Asian option under geometric averaging is cheaper than a European call?

This was an exam question at Cambridge University. Let $S_t = S_0\exp(\sigma W_t + (r-\dfrac{1}{2}\sigma^2)$ and a bank account returns a continuously-compounded rate of interest $r$. Consider the ...
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Black 76 for Options on Interest Rate Futures

This is my first time using Black76 to value options on IR futures and I have a question on $F$ and $K$. I understand the price for an IR future is usually quoted as $100 - r$. Do I use this price ...
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What precision do I need to calculate implied volatility?

I'm developing a software to calculate the implied volatility of an option using the Black & Scholes formula and a trial-and-error method. The implied volatility values I get are correct, but I ...
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Black Scholes and Monte Carlo implementations in Java [duplicate]

Possible Duplicate: Is there an all Java options-pricing library (preferably open source) besides jquantlib? Can anyone recommend a library with an implementation of Black Scholes and Monte ...
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Black Scholes - how to calculate delta with a vol skew

I am trying to calculate the delta of an option at different strike prices where the underlying has a pronounced implied volatility skew in order to correctly hedge an options strategy. Researching ...
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PDE and Black Scholes problem

Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
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Does a delta hedged short option guarantee profit of extrinsic value at expiration?

If a trader shorts an option and dynamically delta hedges to ensure the delta is equal to 0 if that option expires out of the money does the trader profit that options extrinsic value at the time of ...
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Black Scholes well coded Python

I have some trouble with the following code. Some jump and a decentered path are present but it's not the case, normally for Black Scholes diffusion ! Is anyone see a problem in my code ? ...
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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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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 ...
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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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Briefly stated, why does the function N(x) appear in the European call option pricing model?

I'm aware of the the mathematical formula for the price of a European call option on a stock however I'd like to think about it in an intuitive way.
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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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209 views

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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185 views

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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Skew in Black Scholes model

We are modeling Foreign exchange rates using Black Scholes model given below: $$F_{t}=F_{t−1} + (r_d−r_f)F_{t−1}dt + \sigma F_{t−1}dW_t$$ Where: $F_t$ and $F_{t−1}$ are FX rates at time $t$ and ...
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Black scholes OTC

Let's say you want to find the fair price of a call option. One way is to use the black scholes formula. This assumes you can delta-hedge the underlying asset and the option to eliminate risk, and ...
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Does Implied Volatility always exist?

I am considering a simple Heston Model Market with one risky and one riskless asset. The dynamics of the riskless asset is simply $dB_t=r*B_t*dt$ The dynamics of the risky asset is as follows, $ ...
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Intuitive understanding of Black-Scholes pricing

The Black-Scholes formula entails market completeness, so the price of an option is only the cost associated with dynamically hedging the option. Where does this cost come from? I don't see how ...
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366 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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878 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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314 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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Which interest rates to use for options pricing?

I am looking at the historical treasury interest rates and am uncertain which rates would be best to use for options pricing. Should I use 1 month, 6 month, 2 year? See: ...
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Any Simple Way to Prove Black Scholes Type Identies?

A certain complicated option pricing formula results in products of Black Scholes $N$ components like this: $-p_1N(d_1)N(d_6)+p_sN(d_2)N(d_5)>?0$ where $p_s>p_1$ Trying to find a simple way ...
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Transform the American cash-or-nothing call into a linear complementarity problem for the diffusion equation

Transform the American cash-or-nothing call into a linear complementarity problem for the diffusion equation and show that the transformed payoff is g(x,τ) = be^[(1/2)((k+1)^2)τ+(1/2)(k−1)x]H(x),  ...
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America option early exercice boundary via Monte Carlo simulation

I am trying to calculate an american option price via the simulation of the early exercise boundary using the method presented in this document: Monte Carlo Method For pricing a put Option. I have ...
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232 views

Java Implied Volatility Solving

After using RQuantLib and RCaller from Java I am desiring a bit more speed on my implied volatility calculations (for anyone who has used this knows it is quite slow). I need to price a large number ...
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Explain $1Gamma vs %1 Gamma

What is $1 Gamma and what is 1% Gamma? please describe the difference? I understand Gamma but cant make the diff between the two.
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BS Implied Volatility under Normal returns

If I use theoretical prices under a normal valuation model, and I estimate their implied volatility using BLACK SCHOLES implied volatility, do I'll get corresponding log normal volatility?
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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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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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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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Option pricing within the Black Scholes model

Have a question regarding regular option pricing. In the standard Black-Scholes model, with interest r and volatility $\sigma$. Determine the arbitrage free price at t of an option which at $T>t$ ...
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Can one use the Greeks (delta,gamma,theta) to show that the Black-Scholes call formula satisfies the Black-Scholes PDE?

If so, is there a derivation anywhere that shows this? I was told that this could be done in a class but I don't see how it's possible.