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

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Black-Scholes: If exercise probability is 0.5, should $D_2$=0?

Let's say we have option strike price equal to current stock price. And we have zero risk-free rate. In this case I assume that probability of exercise is 0.5 because chances that price will go up or ...
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50 views

Why does expected price of OTM option not equal to BS price?

If I assume that stock returns follow normal distribution with drift = 0% and S.D. = 10%. In the long, if I keep investing in this stock for a year with the same capital every year for a consecutive ...
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generalized black scholes

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
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Option greeks: sensitivity to 1% move

In a Black&Scholes framework how can I compute the following sensitivities: to 1% move in the underlying price to 1% move in implied volatility I would like the greeks to tell me how many ...
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40 views

Implied Volatility Calculation for Deep In The Money Calls, Numerical Issues

I have two implementations for finding the implied volatility under Black-Scholes formula. One is bisection and the other is brent's method. (I know Newton-Raphson is popular due to speed and will ...
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Black-Scholes formula with deterministic discrete dividend (Musiela approach)

For deterministic discrete dividend, there are two approach Musiela approach, works when every dividend are paid at maturity of the option. Hull approach, works when every dividend are paid ...
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128 views

Continuity of Black-Scholes formula

How to proof B&S pricing formula is continuous in time $t$ (or it is not?). The general pricing formula is $$ C_t = e^{-r(T-t)} \mathbb{E}^*[(S_T-K)^+ | \mathcal{F}_t] \hspace{1cm} 0\leq t\leq T ...
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Black Scholes formula with continuous dividend paying stock

I am reading the part of constructing B&S price for stock paying dividends. The simplest model used continuous yield dividend. But I can not see that rigorous in term of formulations. Firstly, ...
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111 views

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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105 views

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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57 views

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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100 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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135 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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122 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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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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48 views

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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145 views

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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41 views

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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146 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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70 views

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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140 views

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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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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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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118 views

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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137 views

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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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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How to get Black Scholes' Geometric Brownian Motion differential form form the closed form?

My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that: $$S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu dt + ...
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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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84 views

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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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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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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508 views

Why should we expect geometric Brownian motion to model asset prices?

Disclaimer: I am a complete ignoramus about finance, so this may be an inappropriate forum for me to ask a question in. I am a mathematician who knows nothing about finance. I heard from a popular ...
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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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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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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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118 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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306 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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133 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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164 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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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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130 views

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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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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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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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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392 views

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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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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Transform the American cash-or-nothing call into a linear complementarity problem for the diffusion equation

Transform the American cash-or-nothing call into a linear complementarity problem for the diffusion equation and show that the transformed payoff is g(x,τ) = be^[(1/2)((k+1)^2)τ+(1/2)(k−1)x]H(x),  ...
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America option early exercice boundary via Monte Carlo simulation

I am trying to calculate an american option price via the simulation of the early exercise boundary using the method presented in this document: Monte Carlo Method For pricing a put Option. I have ...
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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 ...