Questions tagged [black-scholes]
Black-Scholes is a mathematical model used for pricing options.
3
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2answers
34 views
What are the underlying events that the random variables map to the real line in the derivation of the Black-Scholes PDE?
When we first try and set up a model for the evolution of S, the value of the underlying stock, I have seen in a lot of textbooks that they model the evolution by the formula $$\frac{dS_t}{S_t}=\mu dt+...
0
votes
1answer
43 views
Monte Carlo simulated price and Black Scholes Price are giving a huge difference in my Matlab code
I have written a script for showing Monte Carlo Price for a increasing N.
But comparing with BS results , This indicates a huge difference. Where is the error?
Function :
function [cpay,ppay] = ...
4
votes
1answer
68 views
The choice of portfolio in the proof of the Black-Scholes formula
Consider a stock whose price $S$ satisfies $$dS_t=\mu S_tdt+\sigma S_tdW_t$$ for constants $\mu,\sigma$ and where $W$ is a $\mathbb{P}$-Brownian motion. Further assume that the stock pays out ...
1
vote
3answers
71 views
Can you model the LIBOR rate as a geometric Brownian motion?
i.e. The LIBOR rate is driven in the same way as a stock price in the Black Scholes model.
For example let $R_t$ denote the LIBOR rate at time t.
the stochastic differential equation (sde) would take ...
0
votes
0answers
44 views
Euler scheme to simulate trajectories + Python Code
Question: Euler scheme to simulate trajectories of S + Python code to get independent trajectories of S?
For solving a problem I have the following assumptions:
stochastic basis (Ω,FT,(Ft)t≥0,P) ...
1
vote
1answer
58 views
Calibrate a SABR model?
How do you calibrate a SABR model using R/Python/Matlab?
Using the data example from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2725485
1) How does one calibrate the SABR model?
2) How ...
2
votes
1answer
43 views
Continuous Time Asset Model in Higham
I read Higham's derivation of the Black-Scholes equation in "An Introduction to Financial Option Valuation". The issue I am having is that it relies on some assumptions related to a continuous time ...
4
votes
1answer
64 views
Uniqueness of Risk-neutral measure: Probabilistic view
Suppose we are working on the Black and Scholes Framework. There are only two assets, the risk-less bank account and a stock. The discounted process is a GBM under the physical measure with drift term ...
4
votes
1answer
48 views
Discounted asset price is martingale in BS model
I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that
$$
\mathrm{d}\mathrm{e}^{-r(T-t)}...
2
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0answers
54 views
Black Scholes to Heat Equation - Substitution
Sorry as really basic question. Chapter 8 of Wilmott introduces Q Finance the BS equation is transformed into the heat equation. Firstly by using
$
V(S,t) \rightarrow \mathrm{e}^{-r(T - t)}U(S,t)
$
...
0
votes
1answer
81 views
Why can derivatives be viewed as a portfolio of the underlying and the riskless asset?
I am struggling with the statement:
"Every derivative of the underlying can be viewed as a portfolio of the underlying asset and the riskless asset."
Is this based on the put-call parity?
Also I ...
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vote
0answers
39 views
Black-Scholes to Diffusion Initial Condition
I'm having troubles with the transformation from the Black-Scholes PDE and transforming it to the diffusion equation. I read this other stackexchange post (Here) and I understand most of the process, ...
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vote
0answers
51 views
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
votes
0answers
102 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 ...
1
vote
1answer
67 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
votes
0answers
47 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?
1
vote
1answer
44 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 ...
1
vote
1answer
153 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 ...
1
vote
0answers
101 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 ...
2
votes
0answers
117 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 ...
-1
votes
1answer
63 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:
$dX_t= \mu_t dt +\sigma_t dW_t$...
0
votes
2answers
109 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 ...
1
vote
1answer
60 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
votes
1answer
135 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 \,...
1
vote
1answer
71 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 ...
1
vote
0answers
90 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 ...
1
vote
0answers
70 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 ...
1
vote
1answer
59 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
25 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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votes
2answers
52 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 ...
3
votes
0answers
87 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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votes
2answers
194 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 ...
2
votes
1answer
96 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
94 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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vote
0answers
77 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 ...
1
vote
2answers
223 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
221 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 ...
0
votes
0answers
50 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 ...
2
votes
1answer
148 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
25 views
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
201 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 ...
6
votes
2answers
210 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
227 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 ...
0
votes
1answer
75 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(...
1
vote
1answer
217 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 ...
5
votes
3answers
794 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 "...
8
votes
3answers
396 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
20 views
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
74 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 ...
