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

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Volatility of Multiple Stocks

According to BSM, Stock Price follows log-normal distribution s.t. $$S(t)=S(0)*\exp(\sigma\sqrt t Z-(\sigma^2t)/2)$$ where Z is standard normal variable Then volatility of this stock is $\sigma \sqrt ...
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Kurtosis in asset logarithmic returns

Assets such as stocks usually display kurtosis in their logarithmic returns. However, their logarithmic returns in a time interval $n$ are the sum of smaller logarithmic returns in $1/n$ time ...
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How do you calculate price of non-existant call option on commodity future

I've been stumped on this for awhile now. I'm trying to determine the price of a call option on a commodity futures contract that expires in the future. My issue is that while the future's contracts ...
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3answers
213 views

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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159 views

How to compute the volatility for the Merton's Model for Private firm?

After one day of research i did not figured how to compute the input volatility for PRIVATE COMPANY in order to calculate the PD. My goal is to compute the PD of each of my company in my portfolio, ...
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is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?

I've read an answer here that say if your security has vega, then it has gamma and theta. is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?
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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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3answers
119 views

When valuing a vanilla option on an index, should we take dividend into account?

When valuing a vanilla option on an index (eg FTSE 100), should we take index dividend yield into account? $$ c=Se^{-q\tau}N\left(d_1\right)-Ke^{-r\tau}N\left(d_2\right) $$ $$ d_1=\frac{\ln\left(\...
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408 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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1answer
104 views

Option writing optimal sell time

When selling options, e.g. a straddle I read often the optimal time for selling options is 30-40 days until expiration. For me intuitively the optimal time would be around one week until expiration ...
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257 views

Why $N(d_1)$ and $N(d_2)$ are different in Black & Scholes

I'm struggling to understand the meaning of $d_1$ and $d_2$ in Black & Scholes formula and why they're different from each other. As per the formula, $$C = SN(d_1) - e^{-rT}XN(d_2)$$ which ...
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73 views

Connection between implied volatily and implied probability

I am reading some lecture notes about Black-Scholes (BS) option pricing. Since the BS-formula is not supported by observed data because of the dependence of the implied volatility on the strik and ...
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475 views

Understanding $N(d_1)$ and how to use the stock itself as the numeraire?

Assume the stock price follows a geometric Brownian motion Then in Black-Scholes pricing model, $N(d_2)$ is the risk-neutral probability that the option expires in-the-money. However, it is said that $...
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1answer
421 views

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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1answer
92 views

Is the Black-Scholes model price a bijection on the interval of static arbitrage free prices

Consider some stock with observed price $S$ and a call option on the stock with value $C$, time to maturity $T$ and strike $K$. Assume there is a constant, continuously compounded interest rate $r$. ...
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122 views

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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1answer
155 views

Beta between stock and option

In Black Scholes model I would like to compute $$ \beta_K = \frac{\mathrm{cov}(C_{K,T},S_T)}{\mathrm{cov}(S_T,S_T)} = \frac{\mathrm{cov}((S_T - K)^+,S_T)}{\mathrm{cov}(S_T,S_T)} $$ with respect to say ...
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59 views

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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669 views

good R package for vectorized option pricing

I am using for now the package fOptions but it doesn't allow for vectorized computation of black76 prices and delta. Which package can be used to do that? As noted ...
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1answer
249 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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651 views

The greeks: where do they come from?

I’m studying the BSM model and having a look at the greeks. I was reading Derivatives, by Paul Wilmott, and he gives the closed form solutions without making the reader see where these solutions come ...
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1answer
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Does price of american (put) option exhibit smooth pasting in time direction under B-S model?

Let us consider the BS model and let $f(s,t)$ denote the price of an American put option with $t$ to expiry, then it is known the solution of the optimal stopping (when it is risk neutral) related to ...
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Arbitrage bounds for Black-Scholes

In some implied volatility code I came across, there is a check to ensure there is no violation of the arbitrage bounds based on the inputs to the method. For the call option, if $$P < 0.99 * (S-...
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Price of an American call option [closed]

I'm working through revision questions at the moment and we are asked to compute the price of an American call option. Suppose that $dS_t = \sigma S_t dW^*_t, S_0 >0$ Let $0<U<T$ be fixed ...
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150 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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489 views

Black-Scholes under stochastic interest rates

I'm trying to implement the Black-Scholes formula to price a call option under stochastic interest rates. Following the book of McLeish (2005), the formula is given by (assuming interest rates are ...
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60 views

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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1answer
96 views

What is the distribution assumption of the black scholes model

As per wikipedia the Black Scholes assumption is: (...
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306 views

Why is $C(t,S_t)/B_t$ a martingale?

In the derivation of the Black-Scholes formula given by Joshi (extract below), he says $C(t,S_t)/B_t$ is a martingale. Why? I understand this can be deduced from the Black-Scholes PDE since the drift ...
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88 views

Black Scholes Model and Dividends

My question can be summarised as such: Consider a portfolio. Say it has a price $\Pi = x$. Portfolio consists of a stock and a sequence of call options underlying on the stock. It has been announced ...
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1answer
246 views

Black Scholes Formula, drift term

In the formula, the stock return is modelled as a brownian motion that is a drift + a stochastic term, ok I get that. But the drift term is then modelled as r - volatility ^ 2 / 2. I am not sure how ...
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How can I make this portfolio self-financing?

$a_t S_t$ = number of shares ($S_t$ is stock price at $t$), $S_0 = 1$ $b_t \beta _t$ = saving account value , $d \beta_t = r \beta_t dt$, $r=$ interest rate So the value of the portfolio: $$V_t = ...
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Why is the black-scholes model arbitrage free when σ>0?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
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Show that the equation solves the Black-Scholes PDE

I have the solution as given Based on this, I have to show that this solves the Black-Scholes formula It means that I should take the partial derivatives of the solution above and then receive the ...
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1answer
122 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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114 views

Pricing call option

Question: The price of a stock is 100. With equal probabilities, it either goes up to 130 or down to 70. What is the price of a 1 year call option with exercise price 100. Risk free rate is 5%. ...
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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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1answer
248 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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1answer
125 views

Gamma derivation from the expectation

I am trying to derive Gamma from the expectation principle (differentiating under expectation sign). I understand these steps $\frac{d^2 C}{d x^2} = e^{-r\tau} \mathbb{E} [ \frac{\partial}{\partial x}...
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Delta derivation from the expectation

I'm trying to understand the following transformation leading to Delta $\frac{dC}{dx} = e^{-r\tau} \mathbb{E}[ \frac{\partial}{\partial x}\text{max}(xY-K,0)] = e^{-r\tau} \mathbb{E}[Y \textbf{1}(xY&...
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Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
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2answers
134 views

Using Black-Scholes to price a geometric average price call

Sorry if this is the wrong exchange for this question. It seems to be the most relevant, anyway. I'm trying to learn and understand the Black-Scholes framework, with a focus on the stochastic ...
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1answer
353 views

Method for finding a arbitrage opportunity when market price of call is incorrect

The solution of the Black-scholes equation is the price of a European call. And the option price assumes the underlying stock is a geometric Brownian motion with volatility $\sigma_{1}>0$. Suppose,...
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Time value of option not always leading to an increased option value

My understanding was that as you increase the time to expiry of an option, the value of the option increases. However, I have run a bunch of scenarios and have realized that if you assume a dividend ...
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2answers
311 views

Implied Volatility Calculation

I want to calculate the implied volatility from the option data that I took from Bloomberg (call Option written on S&P500 index with the maturity of 19-Dec-2009 and strike of 1300), but volatility ...
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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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1answer
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Effect of massive volatility on BS formula

I am experimenting with very high volatility on the standard Black-Scholes formula. I set risk free to zero, time to expiry to 1, volatility to 1 (=100%), and underlying to 1. Then I simulate the ...
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Why the Black-Scholes formula can be used in the real world?

The BS formula is deduced using the risk neutral measure. Why can it be used in the real world?
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269 views

Does a forward price have a drift component in any measure?

Going by intuition, a forward price should already take into account the drift in the underlying price process. Further, assuming interest rates are deterministic, the stochasticity in the forward ...
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1answer
105 views

The role of Gamma in replicating a put

I am analyzing portfolio protection by replication of a put. Having my portfolio with value $V$ I could buy put giving me the payoff $P$ resulting in a call like pay-off scenario $C=V+P$. Say, I don'...