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

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Convert 90-Day Tbill to risk free rate on continuous basis

I am trying to use the BS formula to compute the value of a call option. To do that I need the risk free rate on a continuous basis. As far as I know, people typically use the 90 day TBill as a proxy ...
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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 dividend-...
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Solve Black scholes PDE without using any transformation

I know that one of the methods of solving the black scholes PDE given by : $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV ...
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Linear combination of Payoffs using Black-Scholes

Write the payoffs in Figure 3.8 as linear combination of call options and derive a closed form formula for the Black-Scholes price, the Delta, and the Gamma of them. All the Greeks of the option are ...
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Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
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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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Exercise probabilities in Black Scholes [duplicate]

In the Black Scholes Formula, why are the probability of an Asset or Nothing Call and Cash or Nothing Call being exercised different. The probabilities are N(d1) and N(d2) respectively.
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Applying Black-Scholes to valuing index options

I am currently attempting to use the Black-Scholes model to value index options. My issue is; what should I use as the price of the underlying? Say I want to value a call option on the German DAX with ...
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BSM Model - Actual probability

Actual probability of exercise of put option under BSM model is: PD = N(-d2(u)) (using expected return of stock, u) Risk-neutral equivalent is ...
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How to calculate Implied Volatility for out-of-the-money options?

I'm trying to calculate the implied volatility for out-of-the-money options, and to a lesser extent, in-the-money options. Most of the literature estimations I could find for implied volatility were ...
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76 views

Training data for Black Scholes

What sources of data suitable for training approximations to Black-Scholes are freely available to academics? My understanding is that the parameters to Black-Scholes are: share price strike price ...
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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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Trinomial model converges to Black-Scholes weakly

Consider risk-neutral trinomial model with $N$ periods presented by $$S_{(k+1)\delta}H_{k+1}, \ \ \text{for} \ \ k=0,\ldots,N-1$$ where $\delta:=\frac{T}{N}$ and $\{H_k\}_{1}^{N}$ is a sequence of i....
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142 views

Black-Scholes Equation with dividend

Consider a European option with payoff $$g(S_T) = S_T^{-5}e^{10S_T}$$ Assume that the interest rate is $r = .1$ and the underlying asset satisfies $S_0 = 2, \sigma = .2$, an pays dividend at ...
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106 views

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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106 views

Probability that realized volatility is larger than implied volatility

I did a test about quantitative finance. One of the question was : What is the probability, in the Black-Scholes world, that the realized volatility is larger the implied volatility ? And why ? ...
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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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85 views

Lookback option to find stock price

Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock ...
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Use of cash delta vs forward delta and the mirror image rule

There has been no mention in this text of why this formula uses forward delta not cash delta. Why should have this been obvious to the reader? How can a put be delta neutral at 30%, what does this ...
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57 views

Interpolating on the BS parameters and injecting in the BS formula vs interpolating directly on option prices

Let's consider a simple European call option. In practice, the way the Black-Scholes formula is used to price it is by injecting all of the parameters and paying special attention to the volatility ...
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62 views

Which value to use as shape parameter for Black-Scholes lognormal distribution?

When working with Scipy, lognomal distribution is defined by 3 parameters: the median (loc), the scale (standard deviation or, in our case, the implied volatility) and the shape parameter. But, which ...
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220 views

Implications of shifting the lognormal model for forward rates from a probability perspective

I have a question regarding the application of a shift to the Black-Scholes formula for negative forward rates. I am reading in the Brigo book that "increasing the shift $\alpha$ shifts the ...
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233 views

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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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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85 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 $N(d_2)$ 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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Does price of american (put) option exhibit smooth pasting in time direction under B-S model?

Let us consider the BS model and let $f(s,t)$ denote the price of an American put option with $t$ to expiry, then it is known the solution of the optimal stopping (when it is risk neutral) related to ...
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58 views

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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Are there any new Option pricing models?

Back in the mid 90's I used the Black-Scholes Model and the Cox-Ross-Rubenstein (Binomial) Model's to price Options. That was nearly 15 years ago and I was wondering if there are any new models being ...
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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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155 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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232 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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93 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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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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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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58 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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76 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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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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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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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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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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79 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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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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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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Drawbacks of Black-Scholes option pricing model

Will highly appreciate if anybody can provide logical financial proof why the Black-Scholes option pricing model overestimates the value for long-term options as described in this paper "Warren ...
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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 ...