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Questions tagged [black-scholes]

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

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9
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222 views

No arbitrage conditions for normal implied volatility

usually the term implied volatility refers to Black-Scholes implied volatility (also Log-Normal volatility): it is defined as a quantity which when plugged in the Black-Scholes formula returns the ...
6
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250 views

implied volatility and strike price

Assume for simplicity that the expiration time of an option is $1$ the initial stock price is $1$ and there is no dividend yield and the risk free return is $0$. How is it possible to show that the ...
5
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0answers
62 views

How to explain the asymmetry of vanilla Volga?

I've plotted the charts of Volga of Vanilla Call/Put using finite difference method, and found they are the same, and an asymmetrical shape of observed for both. Any intuitive way to explain the ...
5
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0answers
117 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
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236 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
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0answers
75 views

Is there an arbitrage free option model that treats volatility as a deterministic function of strike?

I am trying to get a good understanding of the different models out there, and thus be able to study hedging errors, and strengths and weaknesses. My understanding of the Local Volatility model in ...
4
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0answers
264 views

Finding the dynamics of a dividend paying asset under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
4
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0answers
351 views

Black-Scholes PDE to heat equation, nonconstant coefficients

Can someone provide me with details or a reference on how to transform the Black-Scholes PDE with nonconstant coefficients (i.e. $r=r\left(S,t\right)$, $\sigma=\sigma\left(S,t\right)$) to the heat ...
3
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82 views

Deriving the black-scholes formula for the European asset-or-nothing call option

I would like to find out what boundary/final conditions i should be using to find the formula for a European asset-or-nothing call option, as i feel that is where I'm making my mistake. I've read ...
3
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62 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 ...
3
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0answers
55 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 ...
3
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110 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 ...
3
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90 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 ...
3
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0answers
341 views

Why is Bachelier implied volatility more skewed than the Black-Scholes implied volatility?

I found the following explanation in a paper by Grunspan (see attached paper page 6) but have trouble understanding it: By differentiating Formula (3) with respect to m, it turns out that the ...
3
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556 views

Black-Scholes PDE - Change of Variables

In the derivation below, I cannot figure out how to solve for "Step 3". Can anyone help me walk through the steps in detail? Derivation:
3
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0answers
109 views

Pricing Options on Fixed Income ETFs

The market for trading options on fixed income ETFs like HYG has become increasingly prominent in the past couple years, but I've been unable to find any discussion related to the pricing methodology ...
3
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0answers
593 views

Black-Scholes explicit Euler implementation python

I've written some code for the explicit finite difference method to solve the BS equation. For certain sets of parameters (time-steps and asset-steps) I get a stable but wrong solution. For others, ...
3
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0answers
73 views

Range options in BS

I know how barrier options are priced in Black-Scholes scheme. I'm wondering if an analytical formula exists also for range (corridor) digital options i.e. options paying only if the price remains ...
3
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0answers
271 views

PDE and Black Scholes problem

Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
3
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3k views

Black Scholes diffusion well coded in Python

I have some trouble with the following code. Some jump and a decentered path are present but it's not the case, normally for Black Scholes diffusion! Is anyone see a problem in my code ? ...
3
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0answers
500 views

Pricing a Power Contract derivative security

I'm trying to price a "power contract" and would appreciate guidance on the next step. The payoff at time $T$ is $(S(T)/K)^\alpha$, where $K > 0$, $\alpha \in \mathbb{N}$, $T > 0$. $S$ is ...
2
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55 views

Alternative derivation of Black Scholes by Merton

I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I ...
2
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0answers
49 views

Develop an option pricing equation by Ornstein Uhlenbeck process

I know that Black-Scholes equation is based that the Equity price has a Geometrical Brownian movement. Can I develop from the same principles( now with transaction cost) that Black Scholes is ...
2
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0answers
72 views

Does an option need to be tradable for Black Scholes pricing formula to hold?

Given the classic Black-Scholes model, e.g. $dS(t)/S(t)=rdt+\sigma dW^{\mathbb{Q}}(t)$ with $S(0)=S_0$ and $dB(t)=rB(t)dt$ with $B(0)=1$, whereby $r$ and $\sigma$ are constants and $\mathbb{Q}$ ...
2
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0answers
30 views

Dimension reduction for worst of basket on $min(S_1, S_2)$

Suppose we want to price an exotic equity which is a function of $min(S_1, S_2)$. To do this, I'm trying to compute an implied volatility surface for $min(S_1, S_2)$ and then price the option using ...
2
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0answers
77 views

Fast implied volatility for american options

Peter Jäckel has developped a method to compute implied volatilites from option prices, called "by implication", see the papers : By Implication Let's be Rational on its website -- as well as a ...
2
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0answers
82 views

What are the main problems for calculating the implied volatility of in the money American put options?

As stated in the question I have a problem with calculating the implied volatility for in the money put options I have a data set of 2.6 million American style plain-vanilla call and put options. For ...
2
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0answers
238 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
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0answers
77 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:             &...
2
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0answers
214 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) $ ...
2
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0answers
157 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
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0answers
183 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 ...
2
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0answers
129 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 ...
2
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0answers
60 views

Put Call Symmetry for arbitrary $t\in [0,T]$

I want to assume I am in a general Black Scholes Model with $r=0$ and $\delta=0$ and the typical filtered probability space. I know that $Call^{BS}(0, x, K, T) = Put^{BS}(0, K, x, T)$ with $x= S_0$, ...
2
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0answers
302 views

Understanding put-call parity

I'm a person with math background trying to break into quantitative finance, and there's something about put-call parity that is not making sense to me. Below I'll detail my understanding of the ...
2
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0answers
37 views

Transforming and minimisation of the BS PDE

I'm trying a novel numerical substitution/fitting method to solve the BS PDE, but the issue is that due to the large range of magnitude of prices $V(s,t)\in[10^{-20},10^1]$, when I try to minimise the ...
2
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0answers
271 views

Pricing of multi strike rainbow options

I am looking at the pricing of a two asset multi strike option in the Black Scholes framework but I am struggling with coming up with a pricing formula. The payoff of the option at maturity is \...
2
votes
0answers
146 views

Option pricing formula for deep in-the/out-of money options?

I am learning option pricing and trying to calculate the call and put price using the Black-Scholes Formula. I have calculated the historical volatility to be 0.232. The formula is gives value close ...
2
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0answers
190 views

Normalized Gains Process is a Q-Martingale - Proof and Intuition

I'm trying to work the proof that the normalized gains process, $G^z_t = \frac{S_t}{B_t}+\int^t_0\frac{1}{B_s}dD_s$ is a Q-martingale under Q (the risk-neutral measure). I'll show what I've worked ...
2
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0answers
737 views

Problem with R code, with option pricing

I have a problem with my R code not producing accurate results. I am trying to implement the Carr-Madan approach to option pricing, using the Black-Scholes model. The formula can be found in equation (...
2
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0answers
106 views

Problems with a Black-Scholes modified equation

I haven't really studied much financial mathematics until about 2 months ago so I'm quite new to this stuff, so I'm sorry if this is a trivial question. At the moment I'm trying to work out what the ...
2
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0answers
78 views

Capital increase: which stock price to use as input to Black-Scholes formula?

For an exercise we have to calculate the theoretical value of a scrip / preferential right on its issue day (23 April) in the context of a capital increase. The scrips are issued on 23 April. The ...
2
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0answers
114 views

Which option pricing models agree best with the market, given the asset price is known?

Assuming you can somewhat forecast the underling asset price movement, and you want to translate this value into the corresponding option price. In practice, which are the better models for this task? ...
2
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0answers
443 views

Black Scholes with Dilution

I've seen two ways to account for dilution when valuing a European option using Black Scholes. I'm not sure which is the correct way and why these methods differ. The two ways I've seen are: 1) ...
2
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0answers
76 views

What is the main reasons to use Miltersen & Schartz (1998) model for commodity futures options

versus a standard Generalised Black and Scholes model (if there are any?) I have read the paper but I am not to sure about its practical implications as would people with more experience using this ...
2
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0answers
83 views

Calculating the error of a Trinomial Model

I've been trying to find a formula to obtain the maximum relative error a trinomial model with n timesteps will incur given all other inputs as compared to the standard BSM model. I'm concerned mostly ...
2
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0answers
86 views

Changes to option valuation for dollar-pegged underlying

In Russia, options on futures on the RTS index are priced in points instead of currency, with points being directly related to the value of the US dollar such that, for example, if the dollar rises, ...
2
votes
1answer
53 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 ...
1
vote
0answers
35 views

Option pricing with definite integral

I would like to consider a slight generalisation of this question, which I recall here: At date of maturity $T_2$ the holder of a financial contract will obtain the amount: $$ \frac{1}{T_2−T_1}\...
1
vote
0answers
111 views

Classic dynamic delta-gamma hedging in Python

I am trying to run a delta-gamma hedge for a Black-Scholes model in Python.The Euler disceretizatioin of the paths is the simplest possible. I wrote the code below but the PnL looks undesirable and ...