Questions tagged [black-scholes]
Black-Scholes is a mathematical model used for pricing options.
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Decomposing option payoffs [closed]
Suppose an option payoff function $$max(min(S-1, 2-S), 0)$$ To value such an option, one would decompose this function, for example, as follows: $$max(S-1, 0) - max(2S-3, 0) + max(S-2, 0)$$ Now, it ...
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Practical use of Dual Delta?
I am wondering what the practical use of the Black-Scholes Dual-Delta is?
I know it is the first derivative wrt the strike price:
$$
\frac{\partial V}{\partial K} = -\omega e^{-r T} \Phi(\omega d_2)
$$...
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Increasing or decreasing BS-formula respect their parameters [closed]
The Black-Scholes formula depends of many parameters, is easy to note that it is increasing respect to the parameter $S_t$, $\sigma$ and $r$, it means to fix a all parameters and vary only one. Is ...
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Commodity forward curve Monte-Carlo
I need to value an Asian commodity option using Monte Carlo and a log-normal model. The inputs are the commodity forward curve and the volatility surface for futures/options expiry. Unfortunately, all ...
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Clarification on a Claim in Black-Scholes-Merton Derivation
In these notes: https://johnthickstun.com/docs/bscrr.pdf, towards the end of the proof of Proposition 5.2 on page 6, the author claims:
$$
\log
\lim_{n \to \infty} \Bbb{E}_\pi \left[\frac {S^*_n} S \...
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Options market making process (step-by-step)
What are the steps involved in options market making?
Does it roughly follow this procedure:
Choose a pricing model, e.g. Black-Scholes.
Calibrate the model, e.g. Volatility.
Quote a bid-ask spread ...
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Is there a Black Scholes PDE for a GBM with path-dependent volatility?
Question: Is there a known path-dependent Black-Scholes PDE?
To be a little more precise, let $S$ be a stock price under a risk-neutral measure such that $S$ satisfies the SDE with path-dependent ...
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Vanna Volga Price of an Up and In Put
In the Vanna-Volga approach to pricing first generation exotics, such as single barriers, as I understand it the pricing is as follows:
Let $K,S_t < B$. I'll choose the ATM IV $I_{ATM}$ as the ...
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Calendar spreads under black scholes world
If IV skew is flat (all strikes with the same IV ss ATM) as in the black-scholes world for all maturities, would calendar spreads be considered as pure arbitrage?
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Forward Black Implied Volatility For Within Risk Neutral European Option Pricing
Going to preface this question with an acknowledgement with how silly the ask is, but alas that is the working world; if anyone can share any ideas I'm all ears.
We're pricing an exotic option in risk ...
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Delta on dividend paying equity index
we calculate the delta as change in NPV for 1% change in spot * 100. Would bumping up the forward price by same 1% produce the same results? I'm assuming F = S *exp(... ) or it's too simplifying ...
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Financial software: academia vs. real world [closed]
I am looking for resources (if they exist) that explain the differences between quant finance software in academia and the real world, or explain how quant software is implemented in practice.
For ...
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Uncertain Volatility Model - Option Pricing R code help
I am trying to price the following call option using the UVM method in R.
The code I wrote below keeps producing the same price for the min and max volatilities, which is wrong, however, I can't seem ...
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Volatility Time and Interest Rate Time
In Sheldon Natenberg's book "Option Volatiliy & Pricing (2nd)", he mentioned that (on page 65), only trading days (roughly 252 in a year) are counted when computing vol time and all ...
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Bisection method for implied volatility not working for European Put Options
I am trying to implement a Bisection method for implied volatility calculation. I use an algorithm from Haug (page 455).
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Cash-Or-Nothing Option pricing formula when volatility and interest rate go zero
How does Cash-Or-Nothing Call Option (Fix payoff if S > K) pricing formula behaves when volatility and interest rate both go zero? Say for the scenario of deep out of money (eg. S = 0.5 x K), or ...
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Difference Between Option market price and Theoretical price? [closed]
So I am working on strategies that depends on the difference between Actual market price of option and price derived using black and scholes model.
For eg: Spot 19000 , strike 19200 . It is OTM call ...
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Modelling the Implied volatility of an asset
I want to estimate the implied volatility of an asset which has not historical implied volatility data. I do have the historical realized volatility ( I have the historical prices). What would be some ...
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Option Pricing for Illiquid case
I am currently studying crypto options trading and have observed that there is often a lack of liquidity for options (such as BTC Options) on various exchanges, including Binance. In many cases, there ...
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asian geometric option valuation-- unable to get monte carlo simulation to converge to analytic value
I'm trying to price asian put options in which the averaging window begins immediately (T=0). currently, I'm trying to match up geometric averaging between my Monte Carlo simulations and my attempt at ...
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Closed-form equation for geometric asian call option
I'm looking to use the geometric asian option as a control variable for a monte carlo simulation. However, I have an issue with the closed-form equation to get the geometric price.
I'm using the ...
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Analytical formula for discounted exposure of a European Put on a stock in Real-World measure
Is there an analytical formula to approximate the discounted exposure for a European Put on a Stock in the Real-World measure? This is just an initial phase to be able to assess the accuracy of using ...
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Convergence in the CRR model
Under certain conditions, the option price of the CRR (Cox-Ross-Rubinstein) Binomial model converges to the Black-Scholes price as the maximal step size of the partition converges to zero (i.e. a ...
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How to apply put-call parity in volatility surface construction?
How to make the volatility surface free of put-call parity arbitrage? If I bootstrapped the implied vol from a call price and plugged it into the BS model to have a put price, what if it violates the ...
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Why is the stochastic process of the volatility of a stock price square integrable?
I am taking a course in financial mathematics(Ito-Integrals, Black-Scholes,...) and there is something that is not immediately clear to me. When constructing our stock price model, the integral $\...
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m-of-n Soft Call Trigger
I'm trying to do some convertible bond pricing. Typically, there's a soft call provision which is triggered if the value of the underlying equity is above x% of par for m of the last n days. (Usually ...
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Black-Scholes implied volatility using a GARCH model
Why I'm not getting the same Black-Scholes implied volatility values as the ones given in the paper "Asset pricing with second-order Esscher transforms" (2012) by Monfort and Pegoraro?
The ...
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Valuation via decomposition or via simulation of the underlying?
My question might be very straight forward but I have seen both approaches being followed in practice so I am curious to see if there are arguments in favor or against each one. I am explaining my ...
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Is my solution on Black Scholes correct? [closed]
Question: Consider in the context of the Black Scholes model with stock price dynamics $d S_{t}=\mu S_{t} d t+\sigma S_{t} d W_{t}$, initial stock price $S_{0}>0$ and with money market account $B_{...
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Can the PDE of Black and Scholes really be derived from the CAPM?
Black and Scholes (1973) argue that their option pricing formula can directly be derived from the CAPM. Apparently, this was the original approach through which Fischer Black derived the PDE, although ...
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Delta-hedge experiment of American Put option
I am trying to run a delta-hedge experiment for an American Put option but there's a (systematic) hedge error which I cannot seem to understand or fix.
My implementation is found in the bottom of this ...
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Foreign equity call struck in domestic currency
I'm trying to get a solution for the foreign equity call struck in domestic currency, where the foreign equity in domestic currency is defined as $S=S^fX^\phi$ with $0<\phi<1$, instead of the ...
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Derivatives without analytic expressions? [closed]
I was wondering if there exist options or other derivatives that do not have a known closed-form analytic expression (i.e., some sort of Black-Scholes PDE) and are usually priced using Monte Carlo ...
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Homogeneity of BS Formula
I'm reading M. S. Joshi's paper "Log-Type Models, Homogeneity of Option Prices and Convexity", and I'm having problem understanding equation 1:
$$
C(S_0, K) = \int (S - K)_+ \Phi\left(\frac{...
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Three mathematical mistakes in Black-Scholes-Merton option pricing?
In this preprint on arXiv (a revised version of the one discussed in a post here) we show that there are three mathematical mistakes in the option pricing framework of Black, Scholes and Merton. As a ...
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BS price as the first term of option price expansion
I recently saw someone write, on a generally non-technical platform, that the Black-Merton-Scholes vanilla option price is the first term of an expansion of the price of a vanilla option.
I get that ...
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Black-Scholes formula is a (probabilistic) convex combination
[![enter image description here][1]][1]A call price is bounded when $\sigma\sqrt{T}$ goes to $0$ and $\infty $ by:
$$C_{inf} = e^{-rT}[F-K] \leq C \leq C_{sup}=S $$
Now a simple rearrangement of Black-...
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Option time value is Nd1-Nd2
I can't find the below statement anywhere (rearrangement of Black-Scholes formula) :
$C(0, S) = e^{-rT}N_2[F-K] + [N_1-N_2]S$
$F$ being the forward, it reads as a straightforward decomposition to ...
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Implied Volatility Surface Interpolation for fixed moneyness and maturity on each day of the calendar
I'm new to quantitative finance and interested in performing a PCA on the implied volatility surface. However, my dataset displays certain point changes over time. As a result, I need to interpolate ...
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Effect of number of monitoring points on Asian Option Price
I want to understand conceptually the expected effect of the number of monitoring points used during the average calculation on Asian options pricing and the reason of such effect.
Asian Options ...
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How to use GARCH/ARCH/EGARCH volatility forecasts to compare the Black Scholes constant volatility assumption with GARCH/ARCH/EGARCH volatility
I should preface this by saying I am an undergraduate physics student, this is more of a side interest to me, so I apologise if I am missing something obvious. I am not following a formal class or ...
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Problem matching prices of Black-Scholes vs. GARCH(1,1) in Duan (1995)
In the paper of Duan (1995) the author compare European call option prices using Black-Scholes model vs. GARCH(1,1)-M model (GARCH-in-mean). To be brief, the author fits the following GARCH(1,1)-M ...
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Is there anyway to compute the CEV-implied volatility from option prices?
Under Black-Scholes, there exists a solution for the option price for a volatility. The volatility can then be backed out from the option price using numeric methods.
For the constant-elasticity of ...
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Exact delta-hedging for endogenous payoffs
I would like to derive the exact delta-hedging strategy in the Black-Scholes market to replicate the following non-standard endogenous payoff. The particularity is that the payoff does not only depend ...
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Binomial tree convergence tree towards BS equation - Struggle with a limit
I am trying to prove that the Binomial tree pricing method converges towards the Black and Scholes model, but I am struggling on a specific step.
I don't understand how the limit of p*(1-p) is ...
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Solve for spot price given delta [closed]
I can use Black Scholes or Bjerksund Stensland to solve for delta given spot price, strike, expiration, vol, interest rate, etc. But is there a direct solution to solve for spot price given delta, ...
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Does skew flatten with a decline in volatility?
In Trading Volatility by Bennett, he says:
If there is a sudden decline in equity markets, it is reasonable to
assume realised volatility will jump to a level in line with the peak
of realised ...
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Get strikes from delta works with put but no with call function
From call-put parity one can derive that $\Delta_C - \Delta_P = 1$. We also know that $\Delta_C = e^{-qt}N(d_1)$. If $e^{-qt} = 0.85$ then there is no value for $d_1$ for a $\Delta_C = 0.9$ as $ 0 <...
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How does Bloomberg calculate Interest Rate Caps/Floors with Black Scholes Merton Model and Volatility set as "Normal"?
While valuing Interest Rate Caps/Floors in Bloomberg, I saw that we have an option for selecting both Model and Volatility. So, my question is how exactly does Bloomberg value the cap/floor, when we ...
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