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

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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 $N(d_2)$ not needed for hedging?

I'm trying to understand delta hedging. If I sell a plain vanilla call option, in order to delta hedge it, I have to buy delta amount of stocks. What I don't understand is that the BS price of the ...
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How to derive Black's formula for the valuation of an option on a future?

I've got a question about 1976 Black Model and Bachelier model. I know that a geometric brownian motion in the P measure $dS_{t}=\mu S_{t}dt+\sigma S_{t} dW_{t}^{P}$ for a stock price $S_{t}$ leads (...
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Normal Black&Schole model for swaptions isn't working properly

I just wrote two functions in Matlab which calculates the swaption prices based on the Lognormal model and on the Normal model, although I have the idea that the Normal model is wrong because the ...
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149 views

How to use the Black-Scholes formula with LIBOR rates?

I want to price an FX option using the Black-Scholes model, but I don't know the risk free rate, nor the volatility. I only know the LIBOR rates, the strike, and that the expiration day is 87 days ...
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185 views

Find call and put volatilities using ATM, Risk reversal and Butterflies volatilities

I have to plot the implied volatility surface for EUR/USD. So, my goal is to produce something like that, from put delta 10 to call delta 10: Searching for informations, I found that I could find ...
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What is the theoretical expected growth in an option's value over a given period of time?

Say an option with five years left before maturity has a value of $x$ today. Theoretically, under the B/S framework, what is its expected value in five years (upon maturity)? Do we assume it will ...
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Vega in a “constant volatility” Black-Scholes world?

A little confused, I consulted the Wilmott forums for guidance on how I can interpret vega/vomma. Another user's post reminded me that the Black-Scholes model assumes that the underlying has constant ...
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156 views

Discrepancy between binomial model, Black-Scholes and Monte-Carlo Simulation

I try to use Monte-Carlo Simulation to price a 10-year call option. Based on below parameter, S = 1, X = 1, volatility = 80%, T = 10, risk-free rate = 0.22% The option value based on Monte-Carlo ...
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56 views

For a call option, what is the real-world probability of expiring in-the-money?

In the Black-Scholes world, the risk-neutral probability of expiring in-the-money is given by N(d2). Can I just replace the risk-free rate by the drift rate to obtain real world probabilities? Thank ...
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Determining the implied volatility for options with bid/ask prices below the intrinsic value

I need some help in understanding the Black-Scholes option pricing model. In my data there are several deep itm European index put options that have an ask price below the intrinsic value. ...
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91 views

Black_scholes formula for a butterfly option

Im wondering if I can apply Black-Scholes formula to valorate a butterfly option, i.e: $$B(T)=Vcall(S(T)-K,0)+Vcall(S(T)-K',0)-2Vcall(S(T)-K'',0)$$ with $K<K''<K'$, just evaluating each call ...
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School project about Black Scholes with stochastic volatility

In a university project I am looking at Black Scholes model with a stochastic volatility. I’m still not quite sure about my focus (I am in the beginning 'Idea phase'). I want to explain the theory ...
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75 views

Ratio of gaussian CDFs in Black-scholes option pricing formula

What is meant by $\frac {\Phi (d_2)}{\Phi (d_1)}$ in the Black Scholes call option price? I found it in a solution as $\frac{\text{short position in cash}}{(\text{number of shares})(\text{strike ...
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28 views

How to adjust Black-Scholes price in function of liquidity?

Black-Scholes pricing formula assume a lot of thing, included perfect liquidity : One can buy/sell any fraction of Stock at any time and buy/sell prices are equal. The cost of the option reflect the ...
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94 views

Analytical solution to the Black-Scholes equation with time-dependent volatility

I am stuck with the following exercise and I would appreciate any help with it. I have to calculate the analytical function for the price of a call option given the following process for the ...
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63 views

proven implementation of Black scholes formula

We are writing our own implementation of the Back Scholes model. What on-line, well-known implementation do you recommend to test against? I have found several including the one below, but it doesn’t ...
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158 views

Black-Scholes formula proof, without stochastic integration

I've looked into many books at my academic library, and very often it goes like this: Brownian motion Then, stochastic integration (Itô's formula etc.) Application: Black-Scholes formula for price ...
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78 views

Leveraged ETF calculation - dropping below zero?

I'm running some simulations with a leveraged ETF to investigate that notorious leveraged-ETF decay effect I keep hearing about. When I put in a typical Black-Scholes lognormal model of returns on the ...
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45 views

Leland's model of non-linear Black-Scholes equation

Leland's model of non-linear Black-Scholes equation: (http://i.stack.imgur.com/EkqQb.png) where с is round-trip transaction costs and S is price of stock. c is said to be constant, c=2(Sask-Sbid)/(...
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self financing property vs. unlimited borrowing

How the self financing property of a portfolio should be understood in the problems where the unlimited access to the borrowing is assumed?
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How should option prices differ when using the Heston versus the Black-Scholes model?

I am running Monte Carlo simulations for a European Call using Heston Model and I am trying to compare them with prices calculated using Black-Scholes formula. I am not quite sure if the prices I get ...
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Is the European call option delta an increasing function of the spot?

In the Black-Scholes' setting, the delta hedge ratio of a European call option is given by $N(d_1)$, which is an increasing function of the underlying equity spot $S_0$. Does this property hold ...
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Black-Scholes PDE: what is the form of the boundary conditions

I'm working on the Black-Scholes equation, but I'm pretty new to financial modeling. Right now, I am trying to understand the Black-Scholes PDE. I understand that the Black-Scholes equation is given ...
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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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Tradable information from BS Implied volatility

These are two follow up questions to: Implied volatility as price transform I understand that the BS model is used as a 'Blackbox' that takes a market price and maps it in a 1to1 fashion to a 'BS ...
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33 views

Solving Black Scholes PDE using Laplace transform with barrier up and in, up and out call option

I tried to finish the option pricing in european barrier up and in, up and out call option using Laplace transform. The barrier option there is a boundary condition. Can you explain step by step ...
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79 views

Calculating the volatility for Black Scholes

The following problem is from the book by Hull. I did it but I am not sure it is right. I am hoping that somebody here can tell me if I did it right and if not where I went wrong. Thanks Bob Problem:...
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209 views

Formula behind pandas.Options() implied volatility

I noted that implied volatility (IV field) from pandas.Options class is very different (especially, for out of money options) than what I compute with Black-Scholes model. (risk free rate is pulled ...
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80 views

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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124 views

What should be the sign of greek letter $\rho$?

I'm reading the book Risk Management and Shareholders Value in Banking by Resti & Sironi. I quote a paragraph from the book (Chapter 5, appendix): The derivative of an option’s value with ...
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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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249 views

Why is the time value of an option mathematically always positive?

Let's consider a simple European option in the Black-Scholes framework. What is it about the maths of $SN(d_1) - KN(d_2)$ that makes its value always greater than $S-K$, when $S>K$? (I assume zero ...
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Verifying an identity of an equation for Black Scholes formula

I just started working on the Black Scholes formula with help of the book Financial option valuation by Higham. Apparently you are possible to derive the following function: $\log(\frac{SN'(d_1)}{e^{-...
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Can I get Black-Scholes option price from greeks?

I am unpleased with current Interactive Brokers risk graph for option strategies, so I'm planning on writing an application myself to plot it. My initial idea is to get the option greek values from ...
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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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92 views

Derivation of Magrabe formula

I'm going through the following note by Davis, link. In chapter 3 he derives the Magrabe formula. I got stuck at equation $(3.16)$. We have two assets: $$dS_i(t)...
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104 views

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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Gil-Palaez Inversion Formula in Black Scholes world

I am trying to calculate numerically the price of a plain vanilla call through Fourier Transform, by applying the Gil-Pelaez formula. More precisely, we have that C(K)=S0*Π1-Kexp(-rT)Π2 where Π1=1/2+...
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Which option pricing models agree best with the market, given the asset price is known?

Assuming you can somewhat forecast the underling asset price movement, and you want to translate this value into the corresponding option price. In practice, which are the better models for this task? ...
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Initial holdings of bonds with delta hedging (Black Scholes model)

Consider the Black Scholes model so $$dS_t = \mu S_t dt + \sigma S_t dW_t, \;\;\; dB_t = rB_t dt$$ I want to delta hedge an European call option with strike price $K$ and strike time $T$. It is known ...
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derivation of the hedging error in a black scholes setup

I'm reading the following short paper by Davis. In section 2.6 he wants to derive an expression for the hedging error. Assume we have Black scholes setup: $$ dS_t = S_t(r dt + \sigma dW_t)$$ $$ dB_t =...
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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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Kurtosis in asset logarithmic returns

Assets such as stocks usually display kurtosis in their logarithmic returns. However, their logarithmic returns in a time interval $n$ are the sum of smaller logarithmic returns in $1/n$ time ...
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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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Volatility smile risk (negative effect) on dynamically hedged portfolio?

About last week you can see MSFT call & put option appears to be resembling volatility smile. And then I open trade positions on a 4 MSFT long call option contract (all 4 contract with fixed/...
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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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is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?

I've read an answer here that say if your security has vega, then it has gamma and theta. is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?
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Why t (time) in Black Scholes & Binomial defined as year?

What's the logical/scientific explanation for Black Scholes & Binomial using year rather than second (SI standard for time) ?
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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? $$ c=Se^{-q\tau}N\left(d_1\right)-Ke^{-r\tau}N\left(d_2\right) $$ $$ d_1=\frac{\ln\left(\...