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Questions tagged [black-scholes]

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

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Is there a method to interpolate the volatility smile?

I have a small question of interest. During my classes at the university I have learned about the Nelson-Siegel method to fit interest rate curves. With this method you are able to determine interest ...
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61 views

Derivative of the Black and Scholes equation [on hold]

What is the financial interpretation that the derivative of the Black and Scholes equation is equal to 0? St n(d1) - Xe^-rt n(d2) = 0
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Fast implied volatility for american options

Peter Jäckel has developped a method to compute implied volatilites from option prices, called "by implication", see the papers : By Implication Let's be Rational on its website -- as well as a ...
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What are the main problems for calculating the implied volatility of in the money American put options?

As stated in the question I have a problem with calculating the implied volatility for in the money put options I have a data set of 2.6 million American style plain-vanilla call and put options. For ...
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Why do I get a curved line when I plot “implied interest rate” on the strike price?

Currently, I am working on my thesis (MSc. Finance) and I run into an interesting “phenomenon”. I have option data for a non-dividend paying stock. In class I have learned, how to calculate the ...
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1answer
179 views

Options Pricing and Mean Reversion

I'm confused about the impact that a mean reverting stock price process has on the value of an option on it. Several sources say that there is indeed an impact on the price of an option: Option ...
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81 views

Swaption : Bloomberg Black implied volatility quotes and pricing in the Black model

I used a lot Bloomberg's VCUB for data, but never used its integrated swaption pricer "Quick Pricer for Swaptions", nor Bloomberg's "full" swaption pricer from "SWPM -OV". I am retrospectively quite ...
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55 views

Correlation between Two Factor Gaussian Shortrate Model and Black Scholes Model

I want to implement a two factor Gaussian Shortrate Model \begin{align} r(t) & = x(t) + y(t) + \phi(t), \\ dx(t) & = -ax(t)dt + \sigma dB_1 (t), \\ dy(t) & = -by(t)dt + \eta dB_2(t), \end{...
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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
169 views

Expectation of $\frac {S_{T_2}} {S_{T_1}}$ at $T_0$

Is my below computation correct (assuming flat volatlity Black Scholes model, flat interest rate curve): $\mathbb{E}(\frac {S_{T_2}} {S_{T_1}}| \mathcal{F}_{T_0})$ $ = \mathbb{E}{\frac{S_{T_0}e^{(r-\...
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How can I graph futures options profit/loss when the options have different underlyings?

Consider a portfolio of vanilla SPX monthly options that consists of two components, a SEP 2019 3000 Call and a DEC 2019 3000 Call. It's easy to graph these as they both share the same independent ...
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1answer
34 views

Stochastic solution (mean, variance) to lognormal drift and normal volatility

I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model I ...
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1answer
153 views

Pricing under risk-neutral probabilities for weird derivatives?

I would really appreciate some help to value a weird derivative that I've found in an assignment: $$ X=(S_{T_1}-k)^{+} = \max(S_{T_{1}}-k;0) $$ which expires at time $T_{2}$ and uses the price at ...
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1answer
83 views

Hedging with different volatility (Ahmad and Wilmott paper)

In their paper they show that: - if you hedge with the realised volatility, the present value of the total p&l is the difference between the option value based on the realised volatility and the ...
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40 views

Calculation of Conditional Expected Value and Pay-Off Diagram

I have a stock with mu 6% and sigma 20% following a random walk and I would like to to calculate the Conditional expected Value of the stock in 10 states with equal probability (10%). Meaning, I would ...
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Numerical Solution to 3 Dimensional Backward BS PDE

I have a three dimensional backward BS PDE. $$ \frac{\partial V}{\partial t} + a(t) S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma(t, S)^2 \frac{\partial^2 V}{\partial S^2} + b(t, M) \frac{\...
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Pricing a transfer option for oil

Need some input in how to attack this problem. Given are 8 timeseries: UK Oil price, Delivery Quarter 1 2020 UK Oil price, Delivery Quarter 2 2020 UK Oil price, Delivery Quarter 3 2020 UK Oil price, ...
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3answers
105 views

Delta hedging pnl to recover option price

In Black Scholes framework, assuming zero interest rates and realized volatility to be same as implied volatility, gamma pnl is exactly same and opposite of theta pnl. So if I buy an option and delta ...
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1answer
84 views

How does the Black Scholes Model Incorporate Log Prices Into Model?

I am still not understanding the link between log prices and how that is incorporated into the BS model. I understand why log(S) is assumed because it makes math easier and it prevents ending prices ...
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0answers
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Implied volatility as break-even delta hedge volatility

There have been some posts on this topic, but not what I am looking for, so a new post on an old topic.. I think some/most of us here are familiar with the following formula expressing implied ...
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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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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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1answer
70 views

How would you do valuation of a bear put spread?

I have a CVR (Contingent Value Right) that behaves as a European long put and short put, with strike prices of 175 and 150. It is possible to value this instrument by Black and Scholes?
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2answers
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Expectation of Gamma times S$^2$ in Black-Scholes model

Can somebody prove that: $$E[S_t^2 \times \Gamma(t,S_t)] = S_0^2 \times \Gamma(0,S_0)$$ where $S_t$ follows a lognormal process as in the Black-Scholes model, and Gamma is the second derivative $\...
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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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1answer
27 views

Security value based on futures contracts of a traded and non-traded assets

S1 - index with dividend a, S2 - non-traded asset. A security pays off $S_{1T}S_{2T}$ upon its maturity S1 and S2 are uncorrelated and follow geometric brownian motion. What is the value of ...
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0answers
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Pricing a power barrier option

I wish to price an option with payoff $S_T^2{1_{\left\{ {\mathop {\max }\limits_{0 \le t \le T} {S_t} \ge B} \right\}}}$ in the usual Black Scholes setup with zero interest rate. Now the pricing isn't ...
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Is there an arbitrage free option model that treats volatility as a deterministic function of strike?

I am trying to get a good understanding of the different models out there, and thus be able to study hedging errors, and strengths and weaknesses. My understanding of the Local Volatility model in ...
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2answers
185 views

When should we delta hedge?

Let's say I'm the seller of a European call option on a non-dividend paying stock. I pocket the premium $c_0$ of the call at $t=0$. If I start to delta-hedge right away, this is equivalent to ...
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Free Call Option [duplicate]

Suppose we follow the assumptions of the Black-Scholes Model, including unlimited borrowing, continuous prices, and frictionless markets. For simplicity assume the risk-free rate is 0. In this world, ...
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1answer
60 views

Black-Scholes volatility implied by stock prices only

I was solving Problem 2.47 from T.F. Crack's "Heard on the Street". I think that the answer given in the book is not correct and I would be thankful if you tell me, where I am mistaken. Question 2....
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How is a LIBOR Market Model volatility skew determined?

LIBOR based interest rates are derived from the prices (supply / demand) of swaptions, caps and floors. These prices are generally quoted in yield vols. Their prices are given by the Black formula. ...
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Poisson parameter in Merton's Jump-Diffusion Model to price call option

I've been taught the following European call valuation formula under jump-diffusion model: \begin{equation} price = E[e^{-rT}max(S_T-K,0)] =\sum_{j = 0}^\infty e^{-rT}P_j(\lambda)E[max(S_T-K,0)|J=j] \...
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66 views

Can implied volatility be 0?

I am calculating IV for intraday options and sometimes I am getting the value as "0"? Is that possible? For example: Strike = 26700 PE Fut = 26962.55 Spot = 26902.55, TimeToExpiry = 797340sec. Price ...
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2answers
167 views

How to calculate implied correlation via observed market price (Margrabe option)

I can't seem to figure out how to do the following: compute the implied correlation $ρ_{imp}$ by using the observed market price $M_{quote}$ of a Margrabe option, and solving the non-linear equation ...
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1answer
84 views

Black Scholes- Options and OIS

I have 2 questions. In the Black Scholes formula for currency options, where does forward premium come in? Volatility will be a historic parameter, so which component considers fwd premia. Typically,...
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Pricing Knock Out Barrier Options by solving Black Scholes PDE (MATLAB)

This question is based on MATLAB functions. Suppose there is a stock S following the process $dS_t=(r-q)S_tdt+\sigma(S_t,t)dW_t$ r - risk-free rate, q - dividend yield, W - Weiner process The ...
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187 views

Black-Scholes equation to Heat equation .(Boundary conditions)

I have been given a problem to code the heat equation which is transformed from B-S equation (European call option) . Now the boundary conditions are for European call option: $$C(S,T)=\max(S-K,0)$$...
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2answers
87 views

Calculation Error or High Vega? How to interpret?

I am trying to calculate/interpret Vega. For the example below I get a Vega of ~36.36. I have checked my math multiple times, but would appreciate anyone pointing out any error that I have made. If ...
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2answers
114 views

What is the price of the European option with the payoff of $\max(S^a-K,0)$?

I interpret such an option as a power option but I do not find any literatures or existing methods to price it. Can it be priced with Black-Scholes with simple changes?
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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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1answer
43 views

Simulating stock prices with and without intermediate paths

So I am simulating stock prices with what I believe to be geometric Brownian motion using parameters from the usual Black-Scholes framework (Please correct me if I am wrong) with the following formula:...
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How to solve for K when setting the differential of a vega option with respect to K equal to 0?

The question is as follows: Let $v = S_0 \phi(d_1)\sqrt{T}$. Solve the following equation for $K$. $$ \frac{\partial v}{\partial K} = 0 $$ By finding $\frac{\partial v}{\partial d_1}$ and $\frac{\...
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1answer
75 views

Pricing an fx option in the same currency

Let imagine we have an option from EUR to USD priced in EUR, therefore the payoff for a call is: $$\frac{(S - K)^{+}}{S} = K (1/K - 1/S)^{+}$$ This is basically the payoff of a price of a put on 1/S ...
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1answer
199 views

python scipy optimize minimize arguments for Implied Volatility

I am having some trouble getting the 'correct' solution to a function where I am trying to utilize scipy.optimize.minimize. In the code below, I create a function <...
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SDE of futures price under non-constant interest rate and volatility process

I'm trying to figure out the form of the SDE of futures price under the risk neutral measure, when stock price follows GBM:             &...
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0answers
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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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2answers
116 views

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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2answers
79 views

Option value with different spot prices [closed]

I found this post online which is plotting different results for option value and greeks depending on spot price. Why would someone want to do calculate the value of the option with different spot ...
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1answer
120 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 \...