Questions tagged [black-scholes]
Black-Scholes is a mathematical model used for pricing options.
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How to price equity options using a Black76 implied volatility surface?
I would like to calculate the fair value of american and european options on various equities and indices using QuantLib C++. Since I do have discrete dividends available for most underlyings, I use <...
3
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0answers
81 views
How to interpret CDF($d_1$)/PDF($d_1$) from BS model ?
In my research on put options, I come across the ratio:
$\frac{(1-\mathcal{N}(d_1))}{\mathcal{N'}(d_1)}$
where $d_1=\frac{\log(S/X)+(r+\sigma^2/2)t}{\sigma \sqrt{t}}$ and $\mathcal{N}(.)$ is the ...
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0answers
25 views
Simating a Cppi strategy with stochastic interest rates
I Hope someone can help me despite my english Is not that good.
In trying to set up a similation code for a costant proportion portfolio insurance. My portfolio Is composed by risky assets and free ...
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1answer
61 views
Black Scholes on Eurodollar Options
I am trying to replicate the Black Scholes results of CME option calculator for options on Eurodollar Options. (link)
I am trying to replicate the implied volatility result by unaltering the spot and ...
2
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0answers
42 views
Effect of mean reverting Volatality in Black and Scholes? [closed]
Can someone please elaborate what would be the effect of a mean reverting volatility (instead of a constant volatility) in pricing options using BS ?
Also how would the greeks vary?
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1answer
35 views
Relationship between asset volatility and debt and equity value
So how I understand it, higher asset volatility implies a higher call option price. The Merton Model holds that the value of equity is a call option. This therefore implies that the equity value must ...
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0answers
50 views
Pricing collateralised derivatives with stochastic interest rates
I was reading the article by V. Piterbarg, Funding beyond discounting: collateral agreements and derivatives pricing, that you can download on the following link.
He states that when the derivative is ...
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1answer
108 views
Positive theta on a long put?
I am trying to hand-price options under the Black-Scholes model.
Given the following parameters:
Stock price: $12.53$
Strike price: $14.00$
Risk-free rate: $0.03$
Annualized Volatility: $0.10$
Time ...
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0answers
94 views
Arbitrage from ATM option trading?
So I was testing out a collar options strategy (long put, short call, and long shares of the underlying stock) in a backtest for a school finance project, and the profits & losses are given by the ...
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0answers
114 views
Expectation of option value
Say we are in a BS world where the (conditional on t) price of a call is given by the usual
$$V(S_t)=V(S_t;K,r,\sigma,T|F_t) = \Phi(d_1)S_t - \Phi(d_2)Ke^{-r(T-t)}$$
Now, what about the ...
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1answer
61 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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2answers
95 views
Proof Black Scholes Theta
I saw the following proof of theta in a paper I read, and I thought it looked pretty neat. Unfortunately I don't understand the step that they do.
This is what they do:
Now, I don't get how they go ...
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1answer
59 views
Cash deposit in replicating portfolio for BS equation unnecessary?
The book on Option Valuation Methods that I currently study (Higham 2013) constructs a replicating portfolio $\Pi = A(S,t)S + D(S,t)$ for deriving the BS PDE, where $D$ is a cash deposit. $D$ does not ...
3
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1answer
130 views
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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1answer
66 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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0answers
85 views
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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0answers
48 views
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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1answer
54 views
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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0answers
21 views
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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2answers
50 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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0answers
85 views
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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2answers
158 views
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
86 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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0answers
26 views
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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2answers
93 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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0answers
74 views
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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vote
2answers
207 views
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 ...
5
votes
2answers
208 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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0answers
41 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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1answer
130 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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0answers
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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
170 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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2answers
196 views
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 ...
3
votes
1answer
197 views
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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1answer
68 views
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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1answer
167 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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3answers
690 views
Forward implied volatility
Can one price accurately by only using vanilla options a derivative that is exposed/sensitive mainly to the forward volatility ?
If it is impossible in the real world, what do we mean by saying "...
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4answers
349 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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0answers
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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 ...
0
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1answer
70 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 ...
3
votes
2answers
258 views
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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0answers
58 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
251 views
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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0answers
68 views
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),$$
where $d_1=\frac{1}{\sigma\sqrt{T-t}}\big[\ln(\...
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0answers
85 views
Beginner question on Black Scholes
Would you please confirm whether my understanding is correct please? (Sorry a lot of questions...)
1) BS is derived based on the assumption that during an infinitesimal time, we can replicate the ...
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2answers
156 views
Heging against stochastic interest rate
I am working on an Index and I am trying to price Call options on it. I work with the 3 Months LIBOR as Cash.
I use the following Black-Scholes formula
$$C_{t} = S_{t}e^{-q_{t}(T-t)}\mbox{N}[d_{1}(t)]...
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1answer
88 views
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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0answers
42 views
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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0answers
82 views
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 ...
3
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1answer
218 views
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{...
