Black-Scholes is a mathematical model used for pricing options.
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price of a “Cash-or-nothing binary call option”
I'm stuck with one homework problem here:
Assume there is a geometric Brownian motion \begin{equation} dS_t=\mu S_t dt + \sigma S_t dW_t \end{equation} Assume the stock pays
dividend, with the ...
1
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2answers
266 views
Early execise of American Call on Non-Dividend paying stock.
Let us consider an American call option with strike price K and the time to maturity be T. Assume that the underlying stock does not pay any dividend. Let the price of this call option is C$^a$ today ...
2
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0answers
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Replicating strategy in the Black-Scholes model
I have a two-asset Black-Scholes model for a financial market:
$dB_t=B_t r dt$
$dS_t=S_t(\mu dt+\sigma dW_t)$
I introduce a European claim $\xi=max(K,S_T)$ with maturity $T$, for some fixed $K$. I ...
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vote
2answers
81 views
Relationship between European, American options volatility
Suppose, if the price of a European option (say a put) can be shown to be monotone in volatility (say for any maturity), does it follow that American options has to be monotone in volatility?
...
2
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2answers
152 views
Is vega of Black-Scholes European type option always positive?
We assume we work in the risk-neural measure with a stock which pays no dividend and a continuous discount rate.
For PUT and CALL only: can someone please clarify if what I said is correct?
The ...
2
votes
2answers
142 views
Trading days or calendar days for Black-Scholes parameters?
Black-Scholes requires volatility estimated in trading days. How does this affect other parameters? Specifically, should the time-to-expiration also be in trading days? And how does this affect the ...
4
votes
5answers
367 views
In Black-Scholes, why is $\log{\frac{S_{t+\triangle t}}{S_t}} \sim \phi{((\mu - \frac{1}{2}\sigma^2)\triangle t, \sigma^2 \triangle t)}$?
Namely, I dont understand why the mean is $(\mu - \frac{1}{2}\sigma^2)\triangle t$ and not just $\mu \triangle t$. I am aware that it is supposed to represent a lognormal distribution, but I guess I'm ...
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2answers
103 views
Black Scholes Formula for Collar Option
I am wondering if there exists a Black Scholes pricing formula for a collar option?
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3answers
1k views
Is there an all Java options-pricing library (preferably open source) besides jquantlib?
I am looking for an all-java implementation of black scholes, preferably open source. I found jquantlib and quantlib (C++). Any other recommendations?
The jquantlib site seems to be down.
I'd prefer ...
6
votes
10answers
1k views
Using Black-Scholes equations to “buy” stocks
From what I understand, Black-Scholes equation in finance is used to price options which are a contract between a potential buyer and a seller. Can I use this mathematical framework to "buy" a stock? ...
27
votes
6answers
2k views
Paradoxes in quantitative finance
Everyone seems to agree that the option prices predicted by the Black-Merton-Scholes model are inconsistent with what is observed in reality. Still, many people rely on the model by using "the wrong ...
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1answer
102 views
Testing Black Scholes Analytical Options Pricer
I've written some code to calculate European option prices using the Black-Scholes analytical method. Can somebody recommend a good way to test that code? I have looked at option pricers online like ...
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2answers
200 views
Why the implied volatilities calculated are so different
I Calculated facebook option(expired in 12/4/13) Implied Volatility with the Bisection Method. The program will be attached at the end. The results for different strike prices are so different:
...
3
votes
2answers
196 views
Black-Scholes and Fundamentals
So basically
$dS_t=\mu S_tdt+\sigma S_tdWt$
and
$\mu=r-\frac12\sigma^2$
I have just been thinking about this later equation. This is very interesting because it ties together risk-free ...
0
votes
1answer
161 views
Doesn't a perpetual option contradict the Black-Scholes framework?
A standard example when learning to price American options is the perpetual American put. This is a put that has no expiry (or you can consider T = infinity). The standard solution prices this using ...
0
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1answer
67 views
volatility Table and BS formula
assume I have implied FX volatility Delta-Term table from broker. I have time noticed as 2M, 3M. what do I have to put into BS formula, is it 2/12 or "count the business days"/"daycount basis"?
I am ...
4
votes
1answer
257 views
Taylor series expansion (Volatility Trading book) explanation sought
I am currently reading Volatility Trading, I have only just started, but I am trying to understand a "derivation from first principles" of the BSM pricing model.
I understand how the value of a long ...
4
votes
5answers
700 views
How to improve the Black-Scholes framework?
Since the distribution of daily returns are obviously not lognormal, my bottom line question is has BS been reworked for a better fitting distribution?
Google searches give me nada.
The best dist ...
5
votes
5answers
2k views
How do you explain the volatility smile in the Black-Scholes framework?
Does anyone have an explanation for the currently naturally forming volatility smile (and the variations) in the market?
2
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1answer
195 views
What are $d_1$ and $d_2$ for Laplace?
What are the formulae for d1 & d2 using a Laplace distribution?
13
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1answer
1k views
Transformation from the Black-Scholes differential equation to the diffusion equation - and back
I know the derivation of the Black-Scholes differential equation and I understand (most of) the solution of the diffusion equation. What I am missing is the transformation from the Black-Scholes ...
12
votes
2answers
552 views
How do we use option price models (like Black-Scholes Model) to make money in practice?
In quantitative finance, we know we have a lot of option price models such as geometric Brownian motion model (Black-Scholes models), stochastic volatility model (Heston), jump diffusion models and so ...
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4answers
953 views
Methods for pricing options
I'm looking at doing some research drawing comparisons between various methods of approaching option pricing. I'm aware of the Monte Carlo simulation for option pricing, Black-Scholes, and that ...
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2answers
190 views
BS and delta hedging questions
I have two related questions concerning Black Scholes and delta hedging. I thought about this two questions, but I could not come up with an answer, so maybe you guys & girls can help me:
If an ...
4
votes
2answers
322 views
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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8answers
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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 ...
4
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1answer
261 views
Black-Scholes American Put Option
Here is my question: This is a question about Black-Scholes model, but it may be applicable to more complicated models. Throughout the discussion, the strike price $K$, interest rate $r$ and ...
2
votes
0answers
63 views
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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vote
0answers
82 views
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 ...
2
votes
0answers
261 views
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 ...
2
votes
3answers
271 views
Basic question about Black Scholes derivation
In the derivation of the Black Scholes equation, the value of the portfolio at time $t$ is given by
$$P_t = -D_t + \frac{{\partial D_t}}{{\partial S_t}}S_t $$
where $P_t$ is the value of the ...
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votes
1answer
100 views
Black Scholes equation application [closed]
I have gone through quite a few exercises using Black-Scholes equation (or formula as you wish to call it). However, I am not quite understand the following question:
A stock is currently selling at ...
6
votes
1answer
173 views
Prove or disprove “If at least 10% of an option's value is time value, it has a delta less than 90”
"If at least 10% of an option's value is time value (ie. time value >= 0.1*call price), it has a delta less than 90".
In practice and after doing many tests with an option pricing calculator, this ...
3
votes
4answers
546 views
Ways of treating time in the BS formula
The Black-scholes formula typically has time as $\sqrt{T-t}$ or some such. My questions:
What is the granularity of this? If we treat $t$ as the number of days, then logically on the day of expiry, ...
7
votes
2answers
349 views
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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vote
3answers
1k views
Why do some people claim the delta of an ATM call option is 0.5?
I am looking for a mathematical proof in terms of differentiating the BS equation to calculate Delta and then prove it that ATM delta is equal to 0.5.
I have seen many books quoting delta of ATM call ...
7
votes
1answer
219 views
When pricing options, what precision should I work with?
I'm wondering if there's any point at all in double-precision calculations, or whether it's ok to just do everything in single-precision, seeing how the difference on non-Tesla GPUs for single and ...
3
votes
1answer
338 views
Can American options with no dividends and zero risk-free rate be treated as European?
Let's say you've got American options on a future of a stock index. There are no dividends, and no risk-free rate either (assume $r=0$). Can these options then be treated as European from the ...
1
vote
1answer
373 views
What precision do I need to calculate implied volatility?
I'm developing a software to calculate the implied volatility of an option using the Black & Scholes formula and a trial-and-error method. The implied volatility values I get are correct, but I ...
5
votes
2answers
290 views
A few questions about signs of the Greek letters
Rho is the partial derivative of the value of call option, $C$, w.r.t the riskfree interest rate $r$: $$\rho \equiv \frac{\partial C}{\partial r}$$
In the standard B-S formula this term is positive, ...
4
votes
1answer
169 views
Can we explain physical similarities between Black Scholes PDE and the Mass Balance PDE (e.g. Advection-Diffusion equation)?
Both the Black-Scholes PDE and the Mass/Material Balance PDE have similar mathematical form of the PDE which is evident from the fact that on change of variables from Black-Scholes PDE we derive the ...
4
votes
3answers
300 views
Is it possible to demonstrate that one pricing model is better than another?
Take the classic GBM (geometric Brownian motion) model for equities as an example:
ds = mu * S * dt + sigma * S * dW.
It is the basis for the classic ...
7
votes
1answer
310 views
Simulating the joint dynamics of a stock and an option
I want to know the joint dynamics of a stock and it's option for a finite number of moments between now and $T$ the expiration date of the option for a number of possible paths.
Let $r_{\mathrm{s}}$ ...
6
votes
2answers
1k views
What causes the call and put volatility surface to differ?
I currently have a local volatility model that uses the standard Black Scholes assumptions.
When calculating the volatility surface, what causes the difference between the call volatility surface, ...
7
votes
2answers
658 views
Why doesn't Black-Scholes work in discrete time?
I have a question considering Financial markets in discrete Time:
One of the main theorems in discrete time is:
In finite discrete Time with trading times t={1,...,T} the following are equivallent:
...
9
votes
2answers
1k views
Why a self-financing replicating portfolio should always exist?
According to my understanding the derivation of the Black-Scholes PDE is based on the assumption that the price of the option should change in time in such a way that it should be possible to ...
6
votes
1answer
416 views
How to 'calibrate' simple pricing models for equity index options and equity options?
I am interested in doing some research on plain vanilla equity options and equity index options. I have historical data for these options. I also happen to have market maker 'fair price' (bid and ask) ...
6
votes
1answer
1k views
What is a self-financing and replicating portfolio?
I try to understand the derivation of the Black-Scholes equation based on the "constructing a replicating portfolio".
From mathematical point of view it looks simple. We assume that:
Stock prices ...
23
votes
3answers
873 views
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 ...
5
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2answers
787 views
How to extrapolate implied volatility for out of the money options?
Estimation of model-free implied volatility is highly dependent upon the extrapolation procedure for non-traded options at extreme out-of-the-money points.
Jiang and Tian (2007) propose that the ...
