# Questions tagged [black-scholes]

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb{Q}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $$C(S,T)=\max(S-K,0)$$...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 $x= S_0$, ...
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### Black-76 Model for Swaption Price and Greeks

I'm in the early stages of developing a swaption pricing model. Suppose $t_1$ is the tenor of the swap rate in years, $F$ is the forward rate of the underlying swap, $X$ is the strke rate of the ...
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### 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 ...
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### 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 \...
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### 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 ...
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### 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 ...
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### Exercise Probabilities Vanilla Cap/Foor

When looking at the discounted pay-off formulas of a vanilla caplet and a vanilla floorlet $\frac{\Delta\tau}{1+r_k\Delta\tau}\max(r_k-r_{cap},0)$ \$\frac{\Delta\tau}{1+r_k\Delta\tau}\max(r_{floor}-...
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### 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 (...
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### 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 ...
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### 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 ...
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### 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? ...