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Questions tagged [black-scholes]

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

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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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37 views

Why Can I not estimate a CVAR from Heston Model

I fit the parameters of Heston model, using option data for SPX. Now I have the process S and P 500 is expected to follow. I make 100,000 simulations of this process and then calculate the expected ...
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How accurate are Black-Scholes estimates of Vega, Volga, Vanna

Wikipedia provides analytical formulas for calculating Greeks. I can get Delta, Gamma, Theta all from Bloomberg. I need Vega, Volga, Vanna for my research. Should I use these analytical formulas for ...
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Cash-or-nothing and Asset-or-nothing price derivation

I was wondering how to derive the price of a cash-or-nothing and asset-or-nothing option by trying to work out the expectation under the risk-neutral measure, while assuming that the underlying ...
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Value simple chooser option as a sum of call and put options

There is a well known formula for valuating the chooser's option price: $H_{chooser}=max\{C(S_t, K, T-t), P(S_t, K, T-t)\}=max\{C(S_t, K, T-t), C(S_t, K, T-t)+Ke^{−r(T-t)}−S_t\}=C(S_t, K, T-t) + max\{...
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Remaining variance and historical variance in Black-Scholes with term structure

When pricing an European vanilla option in a Black-Scholes world with deterministic volatility term structure, what matters is the remaining variance between today $t$ and maturity $T$, i.e. the ...
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48 views

Why can a deterministic portfolio only grow at risk free rate

In black scholes derivation we assume that portfolio grows at risk free rate because the process is deterministic, my question is why is it riskfree rate? If i have information about some event in 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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Calculating historical Volatility for the Black Scholes Model [closed]

Below is a problem from the book "Options, Futures, and other Derivatives" by John C. Hull. I did the problem but I am fairly sure that my answer is wrong. I am hoping that somebody can tell me where ...
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1answer
56 views

Expected value of delta-hedged portfolio

Consider portfolio in black-scholes world $\Pi = \Delta S - V$, where $S$ is the stock price and V is the price of the option. I have read that if we set $\Delta = \frac{\partial V}{\partial S} $ ...
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Oscillating errors in finite difference Black Scholes

I am writing an implementation of the explicit finite difference method to price a standard european call option, and comparing the results to the corresponding analytical value to gauge the error ...
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86 views

P&L Calculation of Option Strategy

I have designed a call writing option strategy, where I am rolling the options upon expiry, i.e., my portfolio consists of one short call position at any given time. I have a time series of the value ...
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basic difference between interest rate models

I am reading up on interest rate models, but currently confused about difference in the two types of models: no arb models like ho-lee, vasicek etc. others like nelson siegel, pca models etc. While ...
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32 views

Distribution of maximum of stock price for lookback option

I'm trying to solve the following question: I've solved part (a) and got: $$\gamma = N(-d_-)e^{r(T-T_0)} - N(-d_+)$$ $$-d_± = \frac{(r±0.5\sigma^2)(T-T_0)}{\sqrt{\sigma^2(T-T_0)}}$$ I'm stuck on ...
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Interest rates forward implied volatility models

I'm trying to find out which model to use to price a pur forward volatility product named VolBond marketed by structuring desks currently. Let me introduce the products first: Example 1: You pay 100 ...
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153 views

Does Black Scholes need to assume no arbitrage?

Since Girsanov's theorem guarantees a risk neutral measure for Geometric Brownian motion, by the fundamental theorem of asset pricing there can be no arbitrage. So, why does the model assume no ...
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37 views

Unique risk neutral measure in Black Scholes vs Merton Model

I was going through a question on the unique risk neutral measure in the Black Scholes model : Unique risk neutral measure for Brownian Motion One of the answers said it is essentially because there ...
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113 views

Black-Scholes: Delta/probability of exercise increases with volatility

The delta for an ITM call option with increasing volatility initially decreases, reaches a global minimum, and then increases. If we consider delta as a representation of risk-neutral probability of ...
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Pricing options with 0 or negative underlying values

I am trying to calculate the value of an option whose underlying is the calendar spread between two months for a commodity (front month Brent vs 2nd month), usually known as a calendar spread option. ...
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1answer
109 views

Option greeks as dollar P&L

If I write the value of an option as O(S, K, T, V), where S is the underlying price, K is the strike, T is the time to expiry and V the implied volatility, how can I compute the dollar amount that I ...
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Vega of exotic options

I'am wondering if there is a standard definition to the Vega of an exotic product when the underlying model is not Black-Scholes. Let me give some examples : What is the Vega if the price is ...
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1answer
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Black-Scholes pricing of binary options

I'm trying understand something basic about Black-Scholes pricing of binary options. In my example above, the current price is over the strike price. The volatility is extreme but I'm still having ...
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Question about calculating asset volatility using Black-Scholes and the Merton Model (Differentiation Question)

I have a problem where I need to relever equity volatility and take into account debt. I'm trying to solve a system of nonlinear equations for $\sigma_v,V$ using $f(\sigma_v,V) = VN(d_1)-De^{-R_ft}N(...
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81 views

Black-Scholes for Binary Option

Something is wrong with this python code designed to apply Black Scholes to the price of a binary option (all or nothing, 0 or 100 payout). The results I get here is 0.4512780109614. Which I know ...
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Forward implied volatility

Can one price accurately by only using vanilla options a derivative that is exposed/sensitive mainly to the forward volatility ? Here are some examples : a) In equity markets : ...
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1answer
149 views

Measure theory in quantitative finance

When I read up on stochastic modeling, the use of "measure" comes up a lot. So far I just read the word "measure" as "probabilities" or "distribution" and was able to get away with it when trying to ...
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Black-Scholes Greeks Calculation with Daily Rates [duplicate]

Background: To simulate a set of closing prices (about a months worth) for a stock, I've been using the average, standard deviation and variance of the daily returns of the stock( LN(S1/S0)) over the ...
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1answer
60 views

Dynamic hedging pnl when pinning

Dynamic hedging, if successfully implemented, should ensure the dynamic hedge earns the exact opposite of the corresponding option position. However, if we buy an otm option, and the stock goes in ...
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2answers
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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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56 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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1answer
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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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Best Way of Interpreting Black-Scholes Formula [duplicate]

I'm curious to know the best interpretation of the Black-Scholes formula for a European equity call option: $$C(S,t)=S_tN(d_1)-Ke^{-r(T-t)}N(d_2),$$ where $d_1=\frac{1}{\sigma\sqrt{T-t}}\big[\ln(\...
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Beginner question on Black Scholes

Would you please confirm whether my understanding is correct please? (Sorry a lot of questions...) 1) BS is derived based on the assumption that during an infinitesimal time, we can replicate the ...
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2answers
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Heging against stochastic interest rate

I am working on an Index and I am trying to price Call options on it. I work with the 3 Months LIBOR as Cash. I use the following Black-Scholes formula $$C_{t} = S_{t}e^{-q_{t}(T-t)}\mbox{N}[d_{1}(t)]...
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Pricing an option with sparse data, high underlying volatility and returns

I'm currently pricing American and European options on an underlying with sparse data (interpolated), high annual volatility and returns over the last year around 300%. The product isn't similar to ...
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Return / yield of an ATM-Call Option on a zero coupon bond

The zero-coupon bond with unit face value and maturity S for a call option with maturity T and strike K is given by: The bond prices $P(t,T)$ and $P(t,S)$ $$\begin{aligned} ZBC(t,T,S,K) = & P(t,S) ...
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Longstaff Schwartz Algrorithm in R

I recently discovered the LSMonteCarlo library in R which basically determines the price of American options via Longstaff Schwartz method. I tried the ...
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1answer
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derivation of general black-scholes formula

I would like to find a derivation for the Black-Scholes fomrula in the general case (i.e., where the volatility function $\sigma : [0,T] \to \mathbb{R}^+$ and the investment rate $r: [0,T] \to \mathbb{...
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Trying to understand Strike Adjusted Spread, can someone explain using a simple example?

I should start by saying that I am not a quant, I am someone interested in options but I perhaps lack the mathematics background to always follow along. I recently stumbled upon a terrific article ...
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Option pricing and mean reversion

In different books one can find a formula for option pricing when we assume that $\ln(S)$ follows a mean reversion process $$ dS_t/S_t=\kappa(\theta-\ln(S_t))dt+\sigma dZ$$ If we calculate an ...
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Put Call Symmetry for arbitrary $t\in [0,T]$

I want to assume I am in a general Black Scholes Model with $r=0$ and $\delta=0$ and the typical filtered probability space. I know that $Call^{BS}(0, x, K, T) = Put^{BS}(0, K, x, T)$ with $x= S_0$, ...
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Put Call Symmetry

I want to show the Put Call Symmetry without using the explicite Black Scholes formula. In other words I want to show Call(t, x, K, T) = Pull(t, K, x, T) where $S_t = x $ for $t \in [0, T]$. I ...
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What is the domain of the Black-Scholes operator?

By the Black-Scholes operator I mean the following. $$L_{BS}u(x) = \frac{1}{2}\sigma^2x^2\frac{\partial^2}{\partial x^2}u(x) + rx\frac{\partial}{\partial x}u(x) - ru(x)$$ Obviously, the domain of $...
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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
250 views

Which volatility as input in Black Scholes formula?

I am trying to price an option on an Index using Black Scholes formula. I estimated the daily volatility $\sigma_{day}$. My question is should I use an annual volatility based on the business days of ...
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1answer
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For a market with a bank and risky assets $S_1, S_2$ with different volatility, what should be the short interest rate in this market?

Let there be two assets $S_1$ and $S_2$ s.t.for $\sigma_1 \neq \sigma_2$ $$dS_{1t}=\mu_1 S_{1t}dt+ \sigma_1S_{1t}dB_t \\dS_{2t}=\mu_2 S_{2t}dt+ \sigma_2 S_{2t}dB_t$$ . If there exists a bank, what ...
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How might I answer this past exam question relating to the value of a European option under the BMS market model?

The following question, which is not homework, was taken from a past paper for a module I will soon be sitting: Consider a Black-Merton-Scholes stochastic market with drift $\mu = 1$, volatility $\...
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1answer
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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 ...
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1answer
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How might I answer this past exam question relating to the limiting price of an option?

The following image shows a past exam question that I am attempting to answer (for which I do not have a mark scheme): I believe that under the BMS model, the payoff of a stock at maturity $T$ is ...
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1answer
120 views

How to estimate Black Scholes parameters using Maximum Likelihood estimate method

It might be a naive question but I'm new to finance. I've been trying to get my head around this question from a long time and still totally clueless about this. Suppose that the observed jumps in ...