Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [black-scholes]

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

0
votes
1answer
27 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....
0
votes
0answers
24 views

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. ...
1
vote
0answers
29 views

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] \...
1
vote
0answers
58 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 ...
3
votes
2answers
148 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 ...
1
vote
1answer
70 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,...
1
vote
0answers
49 views

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 ...
2
votes
0answers
98 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: $$C(S,T)=\max(S-K,0)$$...
2
votes
2answers
77 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 ...
0
votes
2answers
101 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?
3
votes
0answers
51 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 ...
0
votes
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:...
1
vote
0answers
39 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{\...
2
votes
1answer
70 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 ...
1
vote
1answer
109 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 <...
2
votes
0answers
57 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:             &...
3
votes
0answers
47 views

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 ...
2
votes
2answers
103 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 + ...
1
vote
2answers
75 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 ...
2
votes
1answer
78 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 \...
1
vote
1answer
56 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
votes
1answer
143 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 ...
0
votes
2answers
96 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 ...
2
votes
1answer
111 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 ...
2
votes
1answer
101 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 ...
3
votes
2answers
109 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 ...
3
votes
1answer
120 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). ...
0
votes
1answer
51 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(...
0
votes
0answers
32 views

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 ...
0
votes
0answers
43 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 ...
0
votes
1answer
61 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 ...
1
vote
0answers
66 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 = ...
2
votes
1answer
51 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 ...
3
votes
2answers
212 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 $S$, $\sigma$ ...
1
vote
1answer
53 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 ...
3
votes
2answers
46 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
59 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
91 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
101 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 ...
2
votes
1answer
218 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
53 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
96 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
69 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
votes
0answers
122 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
86 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 ...
1
vote
1answer
94 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, ...
3
votes
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 ...
1
vote
1answer
92 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
55 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
82 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 ...