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.

5
votes
1answer
295 views

At-the-money Call Spread approximation

In a trading manual I got during a course, the value of the ATM Call-Spread is approximated by $CS_{ATM}=\frac{1}{2}StrD+(F-m)\times\Delta CS$ The lecturer skipped the part where he derived this ...
5
votes
1answer
865 views

Better understanding of the Datar Mathews Method - Real Option Pricing

in their paper "European Real Options: An intuitive algorithm for the Black and Scholes Formula" Datar and Mathews provide a proof in the appendix on page 50, which is not really clear to me. It's ...
5
votes
1answer
151 views

Change of numéraire for non-Normal distributions

I'm looking for a resource, a book or an article, that describes the framework of change of numéraire in a broader context than just Brownian motions or Normal distributions. I'm only really ...
5
votes
2answers
1k views

Uniqueness of equivalent martingale measure in Black Scholes-Model

Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
5
votes
1answer
495 views

what's the relationship between forecasted stock volatility and implied volatility?(option)

what's the relationship between forecasted stock volatility and implied volatility? I know that implied volatility is the volatility calculated by BS formula, is there any relationship between implied ...
5
votes
2answers
2k views

How should option prices differ when using the Heston versus the Black-Scholes model?

I am running Monte Carlo simulations for a European Call using Heston Model and I am trying to compare them with prices calculated using Black-Scholes formula. I am not quite sure if the prices I get ...
5
votes
2answers
351 views

Why is the black-scholes model arbitrage free when σ>0?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
5
votes
4answers
302 views

risk-neutral valuation implies no arbitrage?

It is known that in an arbitrage-free continuous time market, the price of every asset is evaluated as the corresponding price in the replicating strategy using risk-neutral valuation. I want to ...
5
votes
2answers
366 views

Can we explain physical similarities between Black Scholes PDE and the Mass Balance PDE (e.g. Advection-Diffusion equation)?

Both the Black-Scholes PDE and the Mass/Material Balance PDE have similar mathematical form of the PDE which is evident from the fact that on change of variables from Black-Scholes PDE we derive the ...
5
votes
0answers
101 views

Pricing and hedging of vanilla options based on non-tradable underlying

Consider a non-tradable stock index $S$ which satisfies: $dS_t=\mu S_tdt+\sigma S_tdW_t$ and a risk-free asset $B$. I want to price an European Call option with the payoff $C_T=max(S_T-K,0)$. The ...
5
votes
0answers
227 views

pricing option with two stocks

Let $\left(S_t^{(1)}\right)_{t\ge0}$ and $\left(S_t^{(2)}\right)_{t\ge0}$ be the price processes of two stocks with dynamics $$ \begin{align} & dS_t^{(1)}=\sigma_{11}S_t^{(1)}dW_t^{(1)} \...
4
votes
1answer
607 views

Black Scholes differential

I'm studying a BS derivation and I don't understand one part .We have a portfolio consisting of $\Delta(t)S(t)+B(t)$ where the first term is risky and the second is a riskless bond. The part i don't ...
4
votes
3answers
806 views

Arbitrage bounds for Black-Scholes

In some implied volatility code I came across, there is a check to ensure there is no violation of the arbitrage bounds based on the inputs to the method. For the call option, if $$P < 0.99 * (S-...
4
votes
1answer
364 views

Can option prices be characterised by an ODE?

If a stock price, $S(t)$, is governed by a geometric brownian motion. Is it possible to characterise the value of an option $V(S,t)$ as an ODE rather than a PDE (given $S$ is itself a function of $t$)?...
4
votes
4answers
11k views

Early execise of American Call on Non-Dividend paying stock.

Let us consider an American call option with strike price K and the time to maturity be T. Assume that the underlying stock does not pay any dividend. Let the price of this call option is C$^a$ today ...
4
votes
3answers
2k views

Deriving the Black-Scholes formula as the expected value on the payout of an option

My question concerns the Black-Scholes formula for the value of a European option, namely \begin{align} C(S_t, t) &= N(d_1)S_t - N(d_2) Ke^{-r(T - t)} \\ d_1 &= \frac{1}{\sigma\sqrt{T -...
4
votes
4answers
6k views

Which risk-free interest rate to use in Black-Scholes equation

Sorry but i'm new in quantitative finance. According to BS derivation the risk-free interest rate is the rate to wich the rate of a particular investment tends when the risk tends to zero. Suppose i ...
4
votes
1answer
348 views

Equivalent form of Black-Scholes Equation (to transform to heat equation)

I am trying to understand the transformation of the Black-Scholes equation to the one-dimensional heat equation from Joshi, M. (2011). The Concepts and practice of mathematical finance. 2nd ed. ...
4
votes
2answers
210 views

Option Valuation

Can Black-Scholes option values be derived via the Capital Asset Pricing Model, without resort to the use of a risk-free portfolio being created from the option and a Delta determined quantity of the ...
4
votes
1answer
320 views

Risk-neutral expectation equation with collateral and funding costs

I am looking at a paper by V. Piterbarg, Funding beyond discounting: collateral agreements and derivatives pricing, that you can download on the following link, in which the author adapts the Black-...
4
votes
5answers
820 views

Black-Scholes formula proof, without stochastic integration

I've looked into many books at my academic library, and very often it goes like this: Brownian motion Then, stochastic integration (Itô's formula etc.) Application: Black-Scholes formula for price of ...
4
votes
2answers
791 views

Expected value of Black-Scholes

(Apologies for any formatting mistakes) Within the Black Scholes model, given that you are estimating the volatility from historical data - and all other parameters assumed exact - one usually ...
4
votes
2answers
6k views

price of a “Cash-or-nothing binary call option”

I'm stuck with one homework problem here: Assume there is a geometric Brownian motion \begin{equation} dS_t=\mu S_t dt + \sigma S_t dW_t \end{equation} Assume the stock pays dividend, with the ...
4
votes
1answer
190 views

Options Pricing and Mean Reversion

I'm confused about the impact that a mean reverting stock price process has on the value of an option on it. Several sources say that there is indeed an impact on the price of an option: Option ...
4
votes
2answers
1k views

Black Scholes in Practice: Delta Hedging

From the Wikipedia page, we know call option as an example is price through delta hedging. $$\Pi=-V+V_SS$$ and over $[t,t+\triangle t]$ $$\triangle\Pi=-\triangle V+V_S\triangle S$$ My questions ...
4
votes
2answers
200 views

Why is the rate of change of a stock price proportional to the stock price?

When deriving the Black Scholes equation, it is usually stated "we assume the change in the stock price is": $dS=\mu S(t) dt + $random term My question is why is the change in the stock price always ...
4
votes
2answers
252 views

Time-zero price of two specific contingent claims

I am unsure how to start with the following problem. I have two contingent claims where contingent claim (1) pays $\int_0^T S_u du$ and contingent claim (2) pays $(\log S_T)^2$ at time $T$ Now I ...
4
votes
1answer
93 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 ...
4
votes
1answer
149 views

Notion of risk-less portfolio in derivation of Black-Scholes

EDIT: As pointed out by Gordon in the comments, the portfolio I considered in my original post is neither self-financing nor (locally) risk-free. Though the central question is still open. Suppose ...
4
votes
2answers
1k views

How to use the Black-Scholes formula with LIBOR rates?

I want to price an FX option using the Black-Scholes model, but I don't know the risk free rate, nor the volatility. I only know the LIBOR rates, the strike, and that the expiration day is 87 days ...
4
votes
1answer
5k views

Find call and put volatilities using ATM, Risk reversal and Butterflies volatilities

I have to plot the implied volatility surface for EUR/USD. So, my goal is to produce something like that, from put delta 10 to call delta 10: Searching for informations, I found that I could find ...
4
votes
2answers
1k views

Black-Scholes PDE: what is the form of the boundary conditions

I'm working on the Black-Scholes equation, but I'm pretty new to financial modeling. Right now, I am trying to understand the Black-Scholes PDE. I understand that the Black-Scholes equation is given ...
4
votes
3answers
562 views

Can I get Black-Scholes option price from greeks?

I am unpleased with current Interactive Brokers risk graph for option strategies, so I'm planning on writing an application myself to plot it. My initial idea is to get the option greek values from ...
4
votes
2answers
2k views

Why gamma and theta have opposite signs?

I saw some textbooks use B-S equation to explain why gamma and theta have opposite signs in most of the cases. For example, John Hull's classic book. The explanation is, first write B-S equation in ...
4
votes
1answer
154 views

Black-Scholes: If exercise probability is 0.5, should $D_2$=0?

Let's say we have option strike price equal to current stock price. And we have zero risk-free rate. In this case I assume that probability of exercise is 0.5 because chances that price will go up or ...
4
votes
2answers
905 views

Interpertation of delta hedge error in Black Scholes

I have spent some time to prove the delta hedge error as described in this paper paper page 16-17 by Davis. The proof is discussed here Deriving Delta Hedge error in the B-S setup (part 2) (a post by ...
4
votes
1answer
377 views

Why must the risk free rate be free from risk in risk neutral valuation?

I am reading through documentation related to Funding Valuation Adjustments (FVA) which discuss risk free rate and funding matters and the following question came to my mind: in risk neutral valuation ...
4
votes
2answers
276 views

Smoothing of the payoff function as a terminal condition for numerical option pricing

I am interested in using a 4th order finite difference method in (underlying asset) space to price a European call basket option. I have developed the solver and everything works as expected, except ...
4
votes
2answers
167 views

What's the explanation for the formula for the volatility of a stock / volatility of the continuously compounded return of a stock?

I am self-studying for an actuarial exam, Models for Financial Economics. It's stated as a given in my manual that $\sigma$ is the volatility of the stock, $\sqrt{\text{Var}(\ln(S_t/S_0))}$ and that ...
4
votes
5answers
5k views

Option pricing ? Where to get the dividend yield from?

I'm trying to apply Black & Scholes formula for a real example to price a vanilla equity option but I'm strugling a little bit whith the dividend yield. Let's assume I have a stock that trades at ...
4
votes
1answer
4k views

What exactly is the OIS Black VOL?

While poking around in Bloomberg I stumbled upon the following data set: EUR SWPT BVOL OIS for various maturities. Obviously OIS must suggest OIS-discounting but how is it related to the Black-...
4
votes
2answers
2k views

Drawbacks of Black-Scholes option pricing model

Will highly appreciate if anybody can provide logical financial proof why the Black-Scholes option pricing model overestimates the value for long-term options as described in this paper "Warren ...
4
votes
2answers
4k views

Trading days or calendar days for Black-Scholes parameters?

Black-Scholes requires volatility estimated in trading days. How does this affect other parameters? Specifically, should the time-to-expiration also be in trading days? And how does this affect the ...
4
votes
1answer
4k views

volatility input for black scholes formula

I am not a mathematician but want to try and understand the BS model for option pricing. I get the intuitive sense of it but am unable to figure out calculation of volatility (as an input). Some ...
4
votes
1answer
295 views

Black-Scholes Model for portfolios

Given Black and Scholes model, consider the portfolio $a_t$ = 1/2, $b_t$ = $1/2$$S_t$ $exp(-rt)$. Show that this portfolio replicates one share of stock. Show if it is self-financing. Find another ...
4
votes
1answer
299 views

Solve Black scholes PDE without using any transformation

I know that one of the methods of solving the black scholes PDE given by : $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV ...
4
votes
2answers
499 views

Solution for american perpetual put

I have been attempting an exercise in which I have to determine the value of an american perpetual put, $P$ in terms of the asset value $S$. The solution to the exercise says: When $S>S_f$ (the ...
4
votes
1answer
169 views

Expectation of $\frac {S_{T_2}} {S_{T_1}}$ at $T_0$

Is my below computation correct (assuming flat volatlity Black Scholes model, flat interest rate curve): $\mathbb{E}(\frac {S_{T_2}} {S_{T_1}}| \mathcal{F}_{T_0})$ $ = \mathbb{E}{\frac{S_{T_0}e^{(r-\...
4
votes
1answer
138 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
82 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)}...