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

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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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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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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 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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149 views

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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163 views

Pricing exotic option whose payout depends on the stopping time

I am struggling with this question: Let $B$ be a standard Brownian motion. In a Black-Scholes model, at time $t$, the stock price is given by \begin{equation} S_t = \exp \{ \sigma B_t + ( r- ...
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risk-neutral valuation implies no arbitrage?

It is known that in an arbitrage-free continuous time market, the price of every asset is evaluated as the corresponding price in the replicating strategy using risk-neutral valuation. I want to ...
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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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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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Does it make sense to use upward and downward volatility in option pricing?

Historically stocks have a higher likelihood to increase in price than to fall in price. As such would it make sense to split a stocks volatility measurement into upward and downward components? For ...
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Black Scholes model: condition of payout function

Given: Consider a two-asset, continuous time model (B,S) where $$dB_t = B_t r dt, \quad dS_t = S_t ( \mu dt + \sigma dW_t)$$ Clearly, the martingale deflator is: $$Y_t = e^{(-r - ...
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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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560 views

What is the difference between market efficiency, market equilibrium, and no-arbitrage?

Aaron Brown (in the book, The Poker Face of Wall Street, p. 196), discusses four approaches to deriving the same Black-Scholes-Merton option-pricing formula: Ed Thorp, Myron Scholes, Robert ...
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Why is the rate of change of a stock price proportional to the stock price?

When deriving the Black Scholes equation, it is usually stated "we assume the change in the stock price is": $dS=\mu S(t) dt + $random term My question is why is the change in the stock price always ...
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C# - Using Black Scholes Newton returns NaN occasionally

First caveat: I'm a programmer doing this for a client, and my knowledge of options probably has holes in it. So be a little forgiving here. =) The Issue: When I run Black Scholes Newton against ...
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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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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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Uniqueness of equivalent martingale measure in Black Scholes-Model

Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
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Option pricing ? Where to get the dividend yield from?

I'm trying to apply Black & Scholes formula for a real example to price a vanilla equity option but I'm strugling a little bit whith the dividend yield. Let's assume I have a stock that trades at ...
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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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665 views

Solving Black-Scholes PDE using Laplace transform

I'm trying to obtain the Laplace transform of Call option price with repect to time to maturity under the CEV process. The well known Black scholes PDE is given by $$ ...
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Is there a good closed-form approximation for Black-Scholes implied volatility?

While the solution for IV can certainly be reached using numerical search methods, I wonder if a high precision closed-form approximation exists. For example, there is a very robust (precise within ...
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277 views

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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Black-Scholes fastest computation method

What is the fastest way to numerically compute Black-Scholes-Merton option prices? I'm trying to find fastest and still precise method. Currently I'm using numerical approximation of Normal cdf with ...
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458 views

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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Self-financing and Black-Scholes-Merton formula

Self-financing is an important concept in financial product replicating, normally used in pricing. I read about several ways to derive Black-Scholes-Merton (BSM) formula. Seems some approaches ...
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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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184 views

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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534 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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349 views

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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179 views

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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448 views

American Swaption Pricing with Monte-Carlo method

I want to price an American swaption but I am not sure about what I am doing. Tree methods and PDE discretization seem difficult to adapt to a swaption. I am trying a Monte-Carlo approach. (in ...
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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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198 views

Boundary condition for Asian Option under Black-Scholes model

I am looking at Kemna and Vorst's paper: A PRICING METHOD FOR OPTIONS BASED ON AVERAGE ASSET VALUES. see http://www.javaquant.net/papers/Kemna-Vorst.pdf Let $\text{d}S_t = S_tr\text{d}t + ...
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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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Black Scholes vs Binomial Model

I'm trying to confirm my understanding of the 2 models. It is my understanding that the black-scholes is a special case of a binomial model with infinite steps. Does this mean that if I were to start ...
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121 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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131 views

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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What exactly is the OIS Black VOL?

While poking around in Bloomberg I stumbled upon the following data set: EUR SWPT BVOL OIS for various maturities. Obviously OIS must suggest OIS-discounting but how is it related to the ...
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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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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.
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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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Black-Scholes derivation assumption contradiction

In many books and derivations of the Black-Scholes PDE one sees that $$\Pi=V-\Delta F \Rightarrow d\Pi=dV-\Delta dF$$ which implicitly assumes that $d\Delta=0$. Somewhere down the road one then ...
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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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Convexity of BS Equation for Call and Put

I have a simple question. Is the Black-Scholes Formula convex with respect to Implied volatility parameter $\sigma$ (for calls or put) ? When I say Black-Scholes I mean for a call the following one ...
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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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Can the Heston model be shown to reduce to the original Black Scholes model if appropriate parameters are chosen?

Summary For Heston model parameters that render the variance process constant, the solution should revert to plain Black-Scholes. Closed from solutions to the Heston model don't seem to do this, even ...
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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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Appropriate measure of Volatility for economic returns from an asset?

I am doing research on uncertainty analysis and risk assessment for oil field development. For doing economic forecast and valuation I use Real Options theory, which is almost similar to theory used ...
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Lower bound of ITM Calls when computing Implied Volatility

Assuming the Black Scholes model and pricing formula of a European call option. Then, if the call is ITM, i.e. if $ln(\frac{S}{K})>0$, the $d_1$-term will go towards infinity as $\sigma$ goes to ...