Questions tagged [black-scholes]

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

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59 views

Implied Volatility of a call plus its delta

I would like to understand if exists a smart way to imply the volatility from a quote that is the sum of a call and its delta: is there any method other than simple iterative minimization?
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75 views

Different scaling conventions for greeks

I have been following this tutorial (http://gouthamanbalaraman.com/blog/value-options-commodity-futures-black-formula-quantlib-python.html). It says in the conclusion and I quote:It is worth pointing ...
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49 views

Pricing call option on S&P 500 [duplicate]

How to price December 2020 maturity European call option of S&P 500 (Strike of 3000). What should be the risk free rate ?
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133 views

Interpretation of drift parameter $\mu$ in GBM

Currently studying Ito's calculus. Looking on the GBM model: $ \frac{d S_t}{S_t} = μ dt + \sigma d B_t$ we end up on the expected stock price at time t: $E[S_t]=s_0 e^{\mu t}$.What does actually $\mu$ ...
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1answer
118 views

Quoting options with reference price and delta

I always thought equity options where quoted with implied volatility, the price being given by the Black-Scholes price of the option with volatility equal to the implied volatlity. But apparently ...
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51 views

Is this the right formula to use implied volatility to gauge probability of a stock being within a certain range? [duplicate]

I read online somewhere, and I can't find it now, that to find the probability of a stock hitting a certain price within a certain time frame, we can use Implied Volatility: ...
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738 views

What is Dual Delta?

I understand that it is the partial derivative of option price with respect to strike. What is it used for though? What does your dual delta signify?
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124 views

Isn't this modified stop-loss strategy an arbitrage?

In John Hull's The Book, section 18.3 he briefly discussed a stop-loss strategy for writing a call option: buy one share of stock whenever $S_t>K$ and sell it otherwise (except at time $0$: if $S_0\...
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1answer
3k views

Monte Carlo European Option Pricing

I've written code below that simulates GBM paths for determining the price of a given European call option and put option. The stock is priced at 150 USD, strike price at 155 USD, risk-free rate was ...
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132 views

Black's formula for a call option on a non-tradable underlying

I am looking for an explanation of the following fact, which seems to be rather simple yet I am missing something. Say that $S_t$ is a stock following GBM $$ dS_t = r S_td_t + \sigma S_t dW_t,$$ and ...
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264 views

Are there any papers measure the accuracy of various option pricing models against real market price?

There are many stochastic volatility option models not only require significant more computation/simulation comparing to the standard BSM model but also introdue large source of possible problems at ...
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1answer
107 views

To Collar or not to Collar

I have a conundrum. I have a stock that has had considerable price appreciation over the past year. Well over 100%. I no longer see any factor (or fundamentals) supporting it's current price (in the ...
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1answer
233 views

Derivation of Black Scholes using expected payoff [closed]

The payoff function of a call is $f(S_T, K) = (S_T - K)^+$, so the expected payoff should allow me to value the price of this call. $$ \mathbb{E}[f(S_T, K)] = \mathbb{E}[(S_T - K)^+] = \mathbb{E}[(...
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381 views

What is the intuition behind the equivalent martingale measure result?

"Suppose that f and g are the prices of traded securities dependent on a single source of uncertainty and define phi = f/g. The equivalent martingale measure shows that, when there are no arbitrage ...
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119 views

zero curvature boundary condition

Assume I am solving numerically Black Scholes PDE $$u_t+0.5\sigma^2s^2u_{ss}+rsu_s-ru=0$$ and I decided to have boundary condition on the right boundary as $u_{ss}=0$. One way is to write the discrete ...
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1answer
65 views

Impact of the interest rate volatility in the valuation of a bond

I am currently valuating a bond whose cupons have the following structure: $\left\{ \begin{array}{rcl} H_j-2\% & \mbox{if} & R_j<H_j-2\% \\ R_j & \mbox{if} & H_j-2\%\leq R_j\leq ...
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1answer
144 views

Physical Option Implied Distribuition

So I got risk neutral probabilities from stock option prices. How can I then map them to a physical measure?
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137 views

Black (1976) model: boundary conditions with non-convergence of spot and forward prices

Let's suppose we have a futures contract F in a market where the relation $$F(t,T)=S(t)e^{r(T−t)}$$ doesn't hold. What are the the boundary conditions for the derivation of the Black (1976) formula?...
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1answer
105 views

Black Scholes: two assets, same $W$-process

Consider a Black Scholes model with two risky assets that are driven by the same $W$-process, and then 1 risk-free asset. When is this model arbitrage-free and complete? We have only 1 driving ...
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3answers
182 views

Does the Ito correction term in GBM result in 'real money', or is it illusory?

There are two ways to think about investment returns and randomness. First is sort of like 'bank interest', with randomness. Suppose we invest 100 units of currency. Suppose each year there is a ...
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1answer
685 views

Price of Geometric basket call option

I wonder if someone can explain how this should be solved: Compute the arbitrage free price at t=0 of the Geometric basket call option (My remark: the payoff function is $\max\left(\left( \prod_{...
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1answer
387 views

Option on Futures - Black Equation Derivation

How to derive Generalised Black equation for Option on Future using generalised Black Scholes Equation $$F=exp(r(T−t))S$$ $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2F^2\frac{\partial^2 V}{...
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1answer
33 views

Nature of the sample [closed]

When we calculate a difference between the market price and modelled price. so to test the significance whether we apply dependent test or independent test.
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1answer
67 views

Differential product Correlated processes

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...
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1answer
146 views

SPY American option Greeks and Premium

I am trying to replicate Ivolatility.com's option calculator for a client. Here's the example Using standard Black Scholes model, I can replicate the exact calculations if there is no dividend. With ...
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1answer
238 views

How to derive the Black Scholes partial differential equation from a stock log-normal distribution?

Is there a way to go from this $$\ln S_t=\ln S_0+(\mu-\sigma^2/2)t+\sigma W_t $$ $$\ln S_t\sim N[\ln S_0+(\mu-\sigma^2/2)t, (\sigma^2)t]$$ To the Black-Scholes partial differential equation?
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1answer
160 views

Calculating the volatility for Black Scholes

The following problem is from the book by Hull. I did it but I am not sure it is right. I am hoping that somebody here can tell me if I did it right and if not where I went wrong. Thanks Bob Problem:...
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1answer
397 views

How would you price an option with payout ln(St) where St is the stock price at time t

I know it has to be done through martingales, but I am not fully sure how to do this BSM pricing.
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1answer
564 views

What is the unit of $T$ in the Black-Scholes formula?

When the Black scholes formula is derived, $T$ is just some time in the future. We don't specify what it is. So why is it that if you go to an option pricing calculator, it asks specifically for days ...
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1answer
102 views

straddle return [closed]

I have the following options data. This is just a snippet but this data is available every day Period 30 to 720 and Out of the Money : 0 to 60 in increments of 5 I would like to compute short ...
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1answer
70 views

How to compute Pr(S>100) when S follows Geometric Brownian Motion?

I have been trying to resolve this problem, under (b), but I cannot find the correct answer. For i=1, my ultimate answer (P=1) deviates from the correct answer (P=0.7580). Please let me know whether ...
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1answer
236 views

Increasing the correlation of two asset reduce the value of spread option.

We know the payment function of Spread option is $$\max\{X_T - Y_T-K,0\}$$ here $$d X_t = (\mu_x - D_x)X_t dt + \sigma_xX_td W^x_t$$ $$d Y_t = (\mu_y - D_y)Y_t dt + \sigma_yY_td W^y_t$$ $$d W^x_td W^...
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1answer
413 views

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
97 views

What is the value this “special” forward contract at maturity?

Background Information: I am not sure this is relevant: Terminal value pricing: If the derivative $X$ equals $f(S_T)$, for some $f$ then in the value of the derivative at time $t$ is equal to $V_t(...
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1answer
48 views

Which expression of $S_t$ to use under the Black-Scholes model?

I am currently looking at example exam questions relating to the evolution of a stock price under the Black-Scholes model. However, I am confused by seemingly inconsistent expressions used for the ...