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Questions tagged [black-scholes]

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

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572 views

Lookback option explicit formula using Black Scholes

I would like to compute the time-0-price for a lookback option using Black Scholes formula, the explicit formula is given by $$S_0[(\frac{2r+\sigma^2}{2r})\Phi((\frac{2r+\sigma^2}{2\sigma/\sqrt{T}}))-...
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What are the underlying events that the random variables map to the real line in the derivation of the Black-Scholes PDE?

When we first try and set up a model for the evolution of S, the value of the underlying stock, I have seen in a lot of textbooks that they model the evolution by the formula $$\frac{dS_t}{S_t}=\mu dt+...
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572 views

Garman-Kohlhagen (Black-Scholes) Formula vs. Bloomberg OVML Calculator

I'm trying to price a European call option on USDJPY. We have that $S = 112.79, K = 112.24, \sigma = 6.887\%, r_d = 1.422\%, r_f = -0.519\%, T = 0.25$. My model, based on Black-Scholes, returns the ...
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800 views

Analytical solution for a modified Black-Scholes equation

Recently, a modified Black-Scholes equation was proposed (Zheng), namely Please consider the case when $$\sigma \left( S,t \right) =\sigma\,{S}^{k/2}$$ and with the European put option Using ...
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569 views

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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1k 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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3k views

How to prove price of Asian option under geometric averaging is cheaper than a European call?

This was an exam question at Cambridge University. Let $S_t = S_0 \exp \left(\sigma W_t + (r-\dfrac{1}{2}\sigma^2) \right)$ and a bank account returns a continuously-compounded rate of interest $r$....
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285 views

What are $d_1$ and $d_2$ for Laplace?

What are the formulae for d1 & d2 using a Laplace distribution?
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167 views

Value of a European Call option with Infinite maturity

It is a job interview question. So, what's the value of a vanilla European call option of infinite maturity, and a given strike, vol, interest rate, spot price. I think, the answer should be "zero". ...
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1answer
147 views

Dependence of implied volatility on spot-vol correlation

I have the following general SV model: $$ dS = \sigma S dW_S $$ $$ d\sigma = a(\sigma,t) dt + b (\sigma, t) dW_\sigma $$ $$ dW_S dW_\sigma = \rho dt $$ where $a , b$ are deterministic functions of $\...
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What is the Brownian motion in the model for the return of a stock price trying to capture?

I have read that in the derivation of the Black-Scholes PDE, we assume that the return of a stock $S$ is given by $$\frac{dS}{S}=\mu dt+\sigma dB$$ where $\mu$ is the average growth of $S$, $\sigma$ ...
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150 views

Interpretation of IV and its use in stock movement prediction

I would like to validate my understanding of IV as a prediction tool. Black-Scholes model is based on the assumption that rate of return of a stock is a Wiener process: $$ \frac{dS_t}{S_t} =\mu \,...
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535 views

Pricing Corridor Variance Spreads

Recently in the equity derivatives market there have been some trades on what are known as "Corridor Variance Spreads." The large equity derivative dealers and investment banks have been promoting it ...
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1answer
160 views

Expectation of N(d2)?

I am trying to find out the Pricing Equation for certain type of Options under Risk-Neutral pricing. This is the equation I am getting, but I am not sure if this can be solved or not. Any help is ...
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1answer
286 views

Parametric estimation of risk-neutral density/implied distribution

since a long time I'm struggling with a particular question regarding the parametric estimation of the risk-neutral density (or implied probability) from option prices. I want to pursue the ...
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1answer
133 views

Barrier Derivative Pricing

Assume constant interest rate $r$ and a stock with current price at $S_0$ that pays no dividend (assume $S_t\ge0$). When the stock price hits the barrier $B$ (where $B<S_0$) you receive \$$1$ and ...
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163 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 ? I ...
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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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1answer
177 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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1answer
263 views

Distribution of Black Scholes call option price at time 0<t <T

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $? (Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t ...
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637 views

The source of “Cost of hedging” in the Black Scholes model

I am trying to get some intuition for the fact that a Black-Scholes price for an option is equal to the cost of replicating the option. Say the interest is 0. The option is obviously still worth ...
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613 views

Why is the volatility smile important

One thing I can't understand clearly is why there is so much focus on the volatility smile. Given my knowledge of the Black and Scholes model, this is what I get: People use the volatility smile as a ...
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3k 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: http://www.treasury.gov/...
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Black Scholes and Monte Carlo implementations in Java [duplicate]

Possible Duplicate: Is there an all Java options-pricing library (preferably open source) besides jquantlib? Can anyone recommend a library with an implementation of Black Scholes and Monte Carlo ...
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Deriving the black-scholes formula for the European asset-or-nothing call option

I would like to find out what boundary/final conditions i should be using to find the formula for a European asset-or-nothing call option, as i feel that is where I'm making my mistake. I've read ...
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Two barrier options puzzle

I come across an interesting question about barrier option as shown below. Two barrier options are given with the same parameters including the barrier level. The first one is knocked out when it ...
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How to Compute the payoff of Var Swaps, which I have replicated

I used Derman(1999) method, to calculate the fixed Kvar for Variance Swaps using actual option price data. The first Pic Shows the outcome. (ignore the 0s). Now the profit and loss of short var swaps ...
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How to interpret CDF($d_1$)/PDF($d_1$) from BS model ?

In my research on put options, I come across the ratio: $\frac{(1-\mathcal{N}(d_1))}{\mathcal{N'}(d_1)}$ where $d_1=\frac{\log(S/X)+(r+\sigma^2/2)t}{\sigma \sqrt{t}}$ and $\mathcal{N}(.)$ is the ...
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Which areas of statistical physics do not get enough attention in quantitative finance?

It seems that over the past few decades many ideas from statistical physics have been successfully incorporated into economics and finance to form the sub-discipline of econophysics. However, it is ...
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474 views

Black-Scholes PDE - Change of Variables

In the derivation below, I cannot figure out how to solve for "Step 3". Can anyone help me walk through the steps in detail? Derivation:
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Pricing Options on Fixed Income ETFs

The market for trading options on fixed income ETFs like HYG has become increasingly prominent in the past couple years, but I've been unable to find any discussion related to the pricing methodology ...
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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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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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266 views

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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Black Scholes diffusion well coded in 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 in Delphi [closed]

when trying to implement the Black-Scholes formula in Delphi, I've found this: http://www.espenhaug.com/black_scholes.html I've checked the results against option-price.com and found they are ...
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Pricing a Power Contract derivative security

I'm trying to price a "power contract" and would appreciate guidance on the next step. The payoff at time $T$ is $(S(T)/K)^\alpha$, where $K > 0$, $\alpha \in \mathbb{N}$, $T > 0$. $S$ is ...
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Why “ Even if the underlying asset price remains unchanged, the option delta for an in-the-money option increases as expiration nears”

I saw this line on some website but can not understand it. Can anyone explain it? "Even if the underlying asset price remains unchanged, the option delta for an in-the-money option increases as ...
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3answers
347 views

A paradox about the American Put option price

Suppose a put option on a stock $S(t)$ following a Geometric Brownian motion is given, with strike $K$ and maturity $T$. Let us denote its price at time $t$ by $p(t,S(t))$. Now, by no-arbitrage ...
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420 views

Stochastic volatility

Suppose we have : $\frac{dS_{t}}{S_{t}}= \sigma dW_{t}$ with $\sigma_{t}$ a stochastic volatility process. How to compute $\mathbb{E}^{Q}[(S_{T}-K)+]$ ? Is there a BS alike formula : "$S_{0}N(d+)-Ke^{-...
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1answer
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Why does Black Scholes formula give inconsistent dimensional analysis result?

For example, distance = speed * time, m = m/s * s. But this technique gives wrong answer on the Black Scholes formula. The square root in the denominator gives wrong unit inside of the culumulative ...
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4k views

Calculate strike from Black Scholes delta

I have a list of deltas and their corresponding volatilities in an FX market but I want to go from delta to strike price. In this Question similar problem is being discussed How can I calculate the ...
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3answers
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Implied Vol vs. Calibrated Vol

Consider the Black-Scholes model, in which the log stock return over a time period $\Delta t$ is given by $$ \log(S_{i+1}/S_i) = (\mu - \sigma^2/2)\Delta t + \sigma \sqrt{\Delta t} Z_i, \qquad Z_i \...
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Approximation of CRR as Black Scholes PDE

I have a formula for intermediate european option price calculated at, say, m-th possible tree value. $S_n^{(m)}$ is a price at node after going up $n$ times and down $n - m$ times $V(S_n^{(m)}, t + ...
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1answer
122 views

How to get the probability of exercise call option in Black-Scholes model?

From Black-Scholes model, I'm trying to prove: $p(S_t>K) = N(d_2)$ No luck yet! Can anyone suggest a reference showing that how to obtain this equation? All I get is: $S_t = S_0e^{ (\mu-0.5 \...
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1answer
487 views

1y10y vs. 10y1y Swaption

Say you have two identical payer swaptions, exception for their terms and tenors. In other words, suppose you have two payer swaptions: $1y10y$ and $10y1y$. All other things being equal, according ...
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2answers
334 views

Black-Scholes call option formula, which probability measure

The stock and bond under the Black-Scholes framework, no dividends: $$S_t=S_0e^{\sigma W_t+\mu t}=S_0e^{\sigma \tilde{W}_t +(r-\frac{1}{2}\sigma^2)t}$$ $$B_t=e^{rt}$$ where $\tilde{W}_t$ is $\mathbb{Q}...
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2answers
111 views

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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1answer
2k 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: $$S_t=S_0\,e^{\...
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Black Scholes Formula for Collar Option

I am wondering if there exists a Black Scholes pricing formula for a collar option?