Questions tagged [black-scholes]
Black-Scholes is a mathematical model used for pricing options.
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Random walk with continuous deterministic dividend [on hold]
I am studying the basics of quant finance, and stumbled upon the SDE of a stock price with continuous dividend yield (Wilmott, Equation 3.19):
$dS=\sigma S dX+(\mu-q)S dt$
Wilmott says that the ...
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2answers
98 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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0answers
49 views
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
42 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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35 views
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
69 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
76 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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0answers
53 views
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
100 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
72 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
70 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 \...
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1answer
55 views
How to calculate the price of an asset using Black-Scholes equation?
I'm trying to solve this problem given:
The dividend yield for asset 1 (asset 2) is 0.05 (0.03), and it is
also given the time zero stock prices, and both assets' Black-Scholes
equation.
I ...
2
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1answer
128 views
How to compute the dynamic of stock using Geometric Brownian Motion?
I have been given the following question:
Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$
For the first ...
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2answers
87 views
Is American option price lower than European option price?
I used to think under the same condition, the American option is always more expensive than the European option, because American option can be exercised at any time (has more rights than European ...
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1answer
108 views
Arbitrage when risk-free portfolio earns less than riskless portfolio
I'm currently reading Paul Wilmott's excellent book on option pricing.
Near the beginning, he constructs a risk-free portfolio using an option, and a short on the underlying to hedge the risk. I'm ...
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1answer
88 views
Binomial Tree Option Pricing Model. Lets talk dividends and futures
I am writing an option pricing model for production use. Its not for arb or anything so it doesn't need to be 100% as accurate as possible. Just good enough for "what happens to my book if we jump 10 ...
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2answers
96 views
Conceptual explanation of the relationship between gamma and vega plotted against delta for a European call option
I recently plotted Gamma and Vega against Delta for a European call option and found that the graphs look very similar. This makes sense to me mathematically since the two formulas are pretty much the ...
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1answer
107 views
Do *all* non-dividend paying assets have the risk-free instantaneous return rate under the risk-neutral measure?
For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). ...
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1answer
50 views
Black Scholes modified boundary conditions
Compute the price of the payoff $(2\log(S(T))-K)^+$. Before I do any algebra, I want to make sure I understand. To solve this problem, I need to solve the Black Scholes PDE with boundary condition $C(...
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Volatiliy in a at-the-time call option [duplicate]
I understand that the vega of the Black-Scholes equation is a positive function, which means the value of the option is an INCREASING function of the volatility, since vega is the derivative of the ...
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0answers
42 views
If the value of a call option is not dependent on the drift of the stock, why does a higher stock price mean a higher call option price [duplicate]
I have read that the price of an option is not affected by the drift of the stock since the drift term doesn't appear in the Black Scholes PDE. I become confused because to me, this implies that the ...
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1answer
60 views
Why does a higher stock value imply a higher call option value [closed]
This may seem like a very dumb question, but if the underlying stock price is greater, then why should a call option be worth more.
My reasoning is that, if the option price is not affected by the ...
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0answers
56 views
Black Scholes Replicating Portfolio Riskfree Asset
Im having a question about this standard derivation of the Black-Scholes formula:
http://www.soarcorp.com/research/BS_hedging_portfolio.pdf
The paper states
$$C=\Delta S+B$$
and finally $\Delta = ...
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1answer
49 views
Is it possible to calculate the equity required (or expected) return using Black-Scholes option pricing model?
I know the method of calculating the equity value as a European call option (using Black-scholes formula). My question is: Is it possible to calculate the expected (or required) return of equity when ...
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2answers
207 views
What is the Brownian motion in the model for the return of a stock price trying to capture?
I have read that in the derivation of the Black-Scholes PDE, we assume that the return of a stock $S$ is given by
$$\frac{dS}{S}=\mu dt+\sigma dB$$
where $\mu$ is the average growth of $S$, $\sigma$ ...
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1answer
49 views
Proper maturity in the Merton's model
I am working on a credit rating project using Merton's model. Basically it adopts Black-Scholes that equity value can be viewed as a call option with a strike price of face value of debts. Since the ...
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2answers
44 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 $$\frac{dS_t}{S_t}=\mu dt+...
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1answer
56 views
Monte Carlo simulated price and Black Scholes Price are giving a huge difference in my Matlab code
I have written a script for showing Monte Carlo Price for a increasing N.
But comparing with BS results , This indicates a huge difference. Where is the error?
Function :
function [cpay,ppay] = ...
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1answer
89 views
The choice of portfolio in the proof of the Black-Scholes formula
Consider a stock whose price $S$ satisfies $$dS_t=\mu S_tdt+\sigma S_tdW_t$$ for constants $\mu,\sigma$ and where $W$ is a $\mathbb{P}$-Brownian motion. Further assume that the stock pays out ...
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3answers
95 views
Can you model the LIBOR rate as a geometric Brownian motion?
i.e. The LIBOR rate is driven in the same way as a stock price in the Black Scholes model.
For example let $R_t$ denote the LIBOR rate at time t.
the stochastic differential equation (sde) would take ...
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1answer
136 views
Calibrate a SABR model?
How do you calibrate a SABR model using R/Python/Matlab?
Using the data example from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2725485
1) How does one calibrate the SABR model?
2) How ...
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1answer
51 views
Continuous Time Asset Model in Higham
I read Higham's derivation of the Black-Scholes equation in "An Introduction to Financial Option Valuation". The issue I am having is that it relies on some assumptions related to a continuous time ...
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1answer
75 views
Uniqueness of Risk-neutral measure: Probabilistic view
Suppose we are working on the Black and Scholes Framework. There are only two assets, the risk-less bank account and a stock. The discounted process is a GBM under the physical measure with drift term ...
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1answer
58 views
Discounted asset price is martingale in BS model
I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that
$$
\mathrm{d}\mathrm{e}^{-r(T-t)}...
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0answers
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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)
$
...
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1answer
83 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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0answers
55 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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0answers
71 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 <...
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0answers
106 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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1answer
82 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 ...
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0answers
53 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
60 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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1answer
379 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
120 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
141 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
66 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
146 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
61 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 ...
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1answer
143 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 \,...
