# Tagged Questions

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

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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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### 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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### Hedging - calculating option prices using implied volatility surface

To hedge a strategy is it accurate "enough" to price an option using an implied vol curve vs moneyness (strike/spot) assuming sticky delta? The moneyness can be read off the chart, its corresponding ...
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### Difference between Deterministic Volatility Function approach and Ad Hoc Black Scholes?

I am thoroughly confused after reading Dumas, Fleming & Whaley (1998) "Implied Volatility Functions: Empirical Tests". Both the Ad Hoc BS Model and the Deterministic Volatility Function ...
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### Which distribution do I get?

Let's assume the stock moves according to a classic Black-Scholes model, and makes a proportional jump with an unknown proportion. Say, it is either +1% or -3% of the stock value, and we know for sure ...
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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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### Finding the dynamics of a dividend paying asset under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
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### Range options in BS

I know how barrier options are priced in Black-Scholes scheme. I'm wondering if an analytical formula exists also for range (corridor) digital options i.e. options paying only if the price remains ...
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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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### 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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### 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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### $\mathbb{P}$ and $\mathbb{Q}$ probability measure/distribution interpretations

I'm trying to understand probability distributions implied from market prices and was reading through this reference explaining the interpretation of $N(d_1)$ and $N(d_2)$ in the log-normal vol Black-...
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### Pricing with-profit/smoothed bonus annuity using Black-Scholes

Would this be possible? Subsequently, would the pricing of such an annuity be somewhat similar to pricing a lookback option?
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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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### 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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### 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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### 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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### Black-Scholes formula with deterministic interest rate and dividend yield

Does any one have the Black-Scholes formula for a European call with time-dependent but deterministic interest rate and dividend yield ?
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### Black Scholes with Dilution

I've seen two ways to account for dilution when valuing a European option using Black Scholes. I'm not sure which is the correct way and why these methods differ. The two ways I've seen are: 1) ...
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### B-S Put Option Formula: Derivation using expected value under Q

I have been working on an old problem in one of my finance classes and, since no solution has been provided and I won't be able to contact my teacher anytime soon, I was hoping I could ask you guys to ...
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### Can someone try this Boundary Condition for the Black-Scholes PDE out for me?

I have a bit of a favor to ask and if anyone could help me out with this I'd really appreciate it. At the moment I'm trying to use the triangle wave formula as the payoff for the Black-Scholes PDE i.e....
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### Estimate Option Price Given X% Move N Days in the Future

I was wondering if someone could recommend a method to estimate the price of an option N days from now given an X% move in the underlying. I have fitted a volatility surface but where I am running ...
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### Intuitive way of calculating Option Prices

I am trying to figure out a way to price options without using the black scholes model(at-least remove some dependency from it). I want to approximate the price of options in the Black Scholes world, ...
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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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### 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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### 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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### 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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### 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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### 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?
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) formula?...