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

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Volatility of Multiple Stocks

According to BSM, Stock Price follows log-normal distribution s.t. $$S(t)=S(0)*\exp(\sigma\sqrt t Z-(\sigma^2t)/2)$$ where Z is standard normal variable Then volatility of this stock is $\sigma \sqrt ...
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Black Scholes Model and Dividends

My question can be summarised as such: Consider a portfolio. Say it has a price $\Pi = x$. Portfolio consists of a stock and a sequence of call options underlying on the stock. It has been announced ...
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93 views

Why is the black-scholes model arbitrage free when σ>0?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
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112 views

The source of “Cost of hedging” in the Black Scholes model

I am trying to get some intuition for the fact that a Black-Scholes price for an option is equal to the cost of replicating the option. Say the interest is 0. The option is obviously still worth ...
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72 views

Effect of massive volatility on BS formula

I am experimenting with very high volatility on the standard Black-Scholes formula. I set risk free to zero, time to expiry to 1, volatility to 1 (=100%), and underlying to 1. Then I simulate the ...
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851 views

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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216 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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543 views

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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2k views

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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577 views

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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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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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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Problems with a Black-Scholes modified equation

I haven't really studied much financial mathematics until about 2 months ago so I'm quite new to this stuff, so I'm sorry if this is a trivial question. At the moment I'm trying to work out what the ...
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Capital increase: which stock price to use as input to Black-Scholes formula?

For an exercise we have to calculate the theoretical value of a scrip / preferential right on its issue day (23 April) in the context of a capital increase. The scrips are issued on 23 April. The ...
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Black-Scholes explicit Euler implementation python

I've written some code for the explicit finite difference method to solve the BS equation. For certain sets of parameters (time-steps and asset-steps) I get a stable but wrong solution. For others, ...
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337 views

Method for finding a arbitrage opportunity when market price of call is incorrect

The solution of the Black-scholes equation is the price of a European call. And the option price assumes the underlying stock is a geometric Brownian motion with volatility $\sigma_{1}>0$. ...
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In which divisions of banking are the Greeks and Black Scholes equation applied? [closed]

I know that Black Scholes and the Greeks are important in market risk. In what other areas are they used?
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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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What is the main reasons to use Miltersen & Schartz (1998) model for commodity futures options

versus a standard Generalised Black and Scholes model (if there are any?) I have read the paper but I am not to sure about its practical implications as would people with more experience using this ...
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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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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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Why not delta of Call option is stochastic or random variable?

Delta of an option is defined as ratio of change in price of call option to change in price of underlying securities. If, $c_t$ is call option price at time $t$ and $S_t$ is the price of underlying ...
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Assuming Black-Scholes assumptions are correct, would the expected return from buying/selling options be 0?

I'm trying to solidify my understanding of options pricing and risk neutral distributions. If the assumptions of the Black-Scholes option pricing were true for an underlying (namely that the future ...
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545 views

How to get Geometric Brownian Motion's closed-form solution in Black-Scholes model?

The Black Scholes model assumes the following dynamics for the underlying, well known as the Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$ Then the solution is given: ...
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Implied Vol vs. Calibrated Vol

Consider the Black-Scholes model, in which the log stock return over a time period $\Delta t$ is given by $$ \log(S_{i+1}/S_i) = (\mu - \sigma^2/2)\Delta t + \sigma \sqrt{\Delta t} Z_i, \qquad Z_i ...
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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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Calculate volatility from call option price

Given call option price, what is the simplest formula to get the volatility value ? Test Data: ...
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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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Result linked to Black-Scholes evaluation

Why does this $$Se^{-D(T-t)}e^{-d_1^2/2} - Ee^{-r(T-t)}e^{-d_2^2/2}$$ equal to $0$? (Where $E$ is a strike)
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How to compute the volatility for the Merton's Model for Private firm?

After one day of research i did not figured how to compute the input volatility for PRIVATE COMPANY in order to calculate the PD. My goal is to compute the PD of each of my company in my portfolio, ...
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112 views

Pricing call option

Question: The price of a stock is 100. With equal probabilities, it either goes up to 130 or down to 70. What is the price of a 1 year call option with exercise price 100. Risk free rate is 5%. ...
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283 views

Implied Volatility Calculation

I want to calculate the implied volatility from the option data that I took from Bloomberg (call Option written on S&P500 index with the maturity of 19-Dec-2009 and strike of 1300), but volatility ...
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How market making in Index options is done?

I have been thinking about this one for last couple of days. With options on share we hedge on cash and the underlying equity as per Black-Scholes formulation. But I am confused on Index options. ...
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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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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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84 views

A simple question on Delta hedging

In the Black and Scholes model, when it is needed to immunize the portfolio from variations in the stock the argument given is the following. If $\alpha_t$ is the amount of invested in the stock, ...
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Why is Vega meaningful only for options which have single-signed gammas

I have been reading Wilmott Frequently Asked Question book and this was mentioned that Vega is not useful when measuring risk for options that have gammas changing signs such as Digital option or ...
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Is the Black-Scholes model price a bijection on the interval of static arbitrage free prices

Consider some stock with observed price $S$ and a call option on the stock with value $C$, time to maturity $T$ and strike $K$. Assume there is a constant, continuously compounded interest rate $r$. ...
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Delta derivation from the expectation

I'm trying to understand the following transformation leading to Delta $\frac{dC}{dx} = e^{-r\tau} \mathbb{E}[ \frac{\partial}{\partial x}\text{max}(xY-K,0)] = e^{-r\tau} \mathbb{E}[Y ...
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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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372 views

How to price this option using the Black Scholes model?

I have a question regarding regular option pricing. In the standard Black-Scholes model, with interest r and volatility $\sigma$, I have to eetermine the arbitrage free price at time $t$ of an ...
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91 views

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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627 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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388 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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50 views

Isn't Black's approximation for American options inconsistent?

I have came across a formula suggested by Fisher Black (Fact and fantasy in the use of options, FAJ, July–August 1975, pp.36) for approximating the price of an American call written on a ...
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Pricing of Black-Scholes with dividend

Consider the payoff $g(S_T)$ shown in the figure below. Consider Black-Scholes model for the price of a risky asset with $T = 1$, $r = .04$, and $\sigma = .02$ and dividends are paid quarterly with ...
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Call and Put Prices Equal at Forward Price - Why?

Consider a European call and put with values $C_t$ and $P_t$, respectively, under the Black-Scholes model. By put-call parity, $$ C_t - P_t = S_t - Ke^{-r(T-t)} $$ for expiration time $T$. Note if ...
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How do you calculate price of non-existant call option on commodity future

I've been stumped on this for awhile now. I'm trying to determine the price of a call option on a commodity futures contract that expires in the future. My issue is that while the future's contracts ...
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When valuing a vanilla option on an index, should we take dividend into account?

When valuing a vanilla option on an index (eg FTSE 100), should we take index dividend yield into account? $$ c=Se^{-q\tau}N\left(d_1\right)-Ke^{-r\tau}N\left(d_2\right) $$ $$ ...
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Option writing optimal sell time

When selling options, e.g. a straddle I read often the optimal time for selling options is 30-40 days until expiration. For me intuitively the optimal time would be around one week until expiration ...