Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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ATM Implied Volatility and Expected Variance

This answer claims that $$\sigma^2_{ATM}\approx E^Q\left(\frac{1}{T}\int_0^T\sigma^2_t dt\right)$$ ie implied ATM vol = risk-neutral expectation of integrated variance. Is there some proof available? ...
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When is the gamma of an iron butterfly spread positive? (Assuming stock price at t=0 is equal to the highest strike price)

I know the Gamma of a butterfly using calls is $$\Gamma_{butterfly} = \Gamma_{C_{K_3}}-2\Gamma_{C_{K_3}}+\Gamma_{C_{K_3}}$$ Where K3-K2 are the same as K2-K1 and S=K1, But under what condition is the ...
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101 views

Payoff of a Butterfly spread under risk neutral measure is always positive for any t<T

In a situation where $$K_3-K_2=K_2-K_1=h>0$$ and $$K_1\le S_t\le K_3$$ where $$S_T=S_t.e^{[(r-\sigma^2/2)(T-t)+\sigma(W_T-W_t)]}$$ (i.e. Stock process follows GBM under the risk neutral measure). I ...
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How to compute the price range for an American call and put option?

A non dividend paying stock has the following details for its European option: Time to expiry – 1 year, Risk free interest (Continuous)- 5%, Exercise price = 42, Current Stock Price = 40, Call option=...
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What is the optimal time for exercising American call and put option?

A 9 month American option (underlying) is known to pay dividend of USD 1 and USD 0.75 at the end of the ...
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implied vol smile relative to atm vols

Am I correct in saying that most stochastic vol models are meant to behave in a way that as atm vol goes up the smile comes down and risk reversals become "less stretched?" - by that i mean ...
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Deriving strike from Delta

According to the following thread: How can I calculate the strike price or implied volatility from a given delta? To back out some strike given some Delta, you simply use realized vol (plus a few ...
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Calibration and pricing with the Stochastic Local Volatility model

I'm reading the stochastic local volatility model literature, e.g., the Heston Stochastic Local Volatility model (https://ir.cwi.nl/pub/22747/22747D.pdf); but I'm a bit unsure about its calibration ...
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Some thoughts over risk-neutral pricing vs real world expectations

I am trying to connect risk-neutral and physical measure expectations, to understand the difference between a no-arbitrage price and an expected terminal value. Imagine I have a European derivative ...
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How does the issuer of a Barrier Reverse Convertible determine the coupon?

I am looking into BRC's, and I keep reading about their relatively high coupon rates which are pre-determined by the issuer. However, I can't seem to find any good resources on HOW they pre-determine ...
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Has anyone ever derived an analytical basket option which gives terminal asset prices individually, by asset?

Random thought I had around what would be an ideal analytical basket formula. If the formula gave terminal prices of each asset instead of a single basket price, you could price any number of exotic ...
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Monte Carlo Pricing of Barrier Options - can't figure out where I'm wrong

I'm trying to price a simple Up-and-out Barrier option using Monte Carlo; haven't even implemented the variance reduction but it's already glitching. The code seems right, but I'm not sure where it's ...
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1 vote
1 answer
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Computing Delta-Hedged Option Returns

I was reading some papers on delta-hedged option returns and came across an intriguing paper that I found quite interesting. However, I was a bit confused on the authors' methodology of computing ...
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Volatility Options

I'm working on a scenario needs to price an option on volatility. After googling, there is no clear relevants in this field. VIX options pricing is one way, while seems it takes the VIX (a volatility) ...
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Why are options on Leveraged ETFs cheaper than ETFs — on the same underlying index and expiration?

I had always reckoned that IVs on Leveraged ETFs (LETF) are "increased by the same amount as the leverage i.e. if it is 2x then IV will be 2x. This will essentially double your cost (in my ...
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Showing that pnl from gamma and theta cancel

I've seen a few questions state without proof that $0.5 \Gamma S^2 \sigma^2 = \Theta$. That is, the gamma and theta pnls cancel out. For example: Relationship between time decay and gamma My question ...
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Extend basket analytic solution (equal weighted) to a various weight basket, also put formula

So I coded up the solution from here: Do basket options have a closed form valuation formula? Which provides a good solution for equally-weighted underlyings under a Black model. The simplified ...
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Is the market price of an asset always lower than the expected discounted value under the REAL WORLD measure?

The risk neutral measure is often said to reflect the risk aversion of investors. So intuitively, I would think that an asset's expected discounted value should be lower under the risk neutral measure ...
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How to apply the Spanning Formula (Carr-Madan) on European Call-option?

In the paper Optimal positioning in derivative securities (Carr & Madan, 2000) the so-called "Spanning Formula" for replicating payoffs is presented in section 2.1 as equation (1). It ...
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Equivalent martingale measure and derivative pricing [duplicate]

So I just recently saw in class that to price a derivative you use what is called an equivalent martingale measure which allows you to compute the price of the contract which then will be the expected ...
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GARCH option pricing

I have been trying to implement GARCH(1,1) model for pricing call options. Suppose I have calibrated Garch(1,1) model for modelling the conditional volatility using the historical data of an equity ...
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2 answers
101 views

What is Phi in Cox-Ross-Rubinstein Binomial Model?

I have a question regarding the Cox-Ross-Rubenstein (CRR) model (Cox et al.,1979). While I do understand how the model is constructed, I am now trying to put it into code and am not sure how to ...
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2 votes
1 answer
218 views

Quantlib: day-by-day evaluation of option value

I'm using Quantlib in Python to price an FX option. I'm comparing the result to Bloomberg, to make sure the code is working correct. I want to calculate the P&L of a certain option trading ...
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3 votes
1 answer
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Quantlib: Greeks of FX option in Python

I'm using Quantlib in Python to price an FX option. I'm comparing the result to Bloomberg, to make sure the code is working correct. I also want to calculate all the Greeks, and eventually use those ...
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What is the name and payoff of this exotic option (where the holder can lock in a price)?

An exotic option is described as follows: Let $S_t$ be the underlying at $t$. The holder has the option to lock in the current price during the lifetime of the option, which he does for $S_{t}=50$. ...
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2 votes
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Calculating model-free implied volatility [closed]

I am trying to come up with model-free implied volatility as in Britten-Jones, M. and Neuberger, A. (2000) Option Prices, Implied Price Processes, and Stochastic Volatility, Journal of Finance, 55, ...
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Option Chain Simulator Using Historical Index Future data, VIX, Implied volatility for Calculation ( Pls Review the Idea & give your suggestions )

Recently I started trading in options, for Learning purpose I am Planning to Create old European Option chain like previous week or last year particular entire week Weekly expire option chain with the ...
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3 votes
1 answer
181 views

Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$

Assume $(X_t)_{t\geq 0}$ follows an SDE of the form: $$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$ where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\...
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What does a rainfall index-based insurance contract looks like?

I would like to know what a rainfall index-based crop insurance contract looks like. My understanding is like There is pre-agreed index say $I_0$ at time $0$ between an insurance company and a holder,...
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Expected Exposure for OTC and ETD trades

I found the implementation of Expected Exposure measure for ETD trades. In the code the Exposure at time $t$ was equal to $$Ex(t) = V(t) - V(0) - VM(t) - IM(t).$$ I don't understand why we are ...
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1 vote
1 answer
106 views

American option under Ornstein-Uhlenbeck stock price

I came across with the following problem: For the Ornstein-Uhlenbeck process $(X_t, 0\leq t\leq T)$ with initial condition $X_0 = x$, find the stopping time $\tau$ that maximizes $\mathbb{E}[e^{-r\...
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1 answer
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Black-Scholes differential equation rewritten [closed]

I have seen that the Black-Scholes equation $$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+ rS\frac{\partial V}{\partial S}-rV=0$$ can also be written in the ...
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2 votes
1 answer
134 views

Analytical evaluation of the following caplet-type product under lognormal assumptions

Let $n \geq 2$, and consider a tenor discretization: $0 = T_{0} < T_{1} < ... < T_{n}$ and associated forward rates evaluated at time $t$, as $L_{i}(t):=L(T_{i},T_{i+1};t)$ for any $i = 0,...,...
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146 views

Bloomberg OVML| FX option pricing | Python

Wanted to check if any API for python is available to replicate Bloomberg's OVML. The objective is to perform FX options pricing for multiple positions, and we are getting stuck in calculating ...
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1 vote
1 answer
390 views

Pricing binary options under volatility smile

I was asked to show that the price of a digital/binary option $D$ while a volatility smile $\sigma(K)$ is present is given by $$D= \exp(-rT)( \Phi(d_2) - K \sqrt{T} \phi(d_2) \sigma ' (K))$$ Where $\...
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69 views

Interest Rate Options - OTC vs Exchange, vol difference

I understand that Exchange Traded Interest Options (USD Libor 3m or Euribor 3m) trade with a lower volatility than the respective Cap or Floor for an equivalent structure. Can anyone give any colour ...
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Buying a double no touch

Why does buying a double no touch get me long butterfly exposure? I understand short volatility exposure but can’t see why I would get long butterfly as I would’ve thought I would want wing volatility ...
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1 answer
163 views

Implied volatility plotted against the strike price in Heston model

How can I reproduce the implied volatility curve (plotted against the strike price) in the Heston model (i.e. the blue line in the graph below)? What's the equation that I have to set up and solve? ...
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4 votes
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Comparison of Option-Pricing Models (volatility models) vs Product-Mapping

I scoured this forum, looking for some indicative (updated as of year 2021) comparison of volatility/option-pricing models. There were some, but they seem dispersed and lacking in general details... ...
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1 vote
1 answer
156 views

How do trading platforms estimate options pricing P&L graphs?

Using the profit/loss calculator for equity option strategies of a trading platform, it displays estimated P&L curves for some date in the future and across the prices of the underlying with a ...
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  • 113
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1 answer
141 views

Calibrating Heston model using implied volatilities

I'm trying to understand how the authers of a paper calibrated their model. We got data on European type options on the S&P500-index period from early 2005 to mid-2009. We have daily data on ...
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1 vote
1 answer
135 views

FX Euro-American Knockout Option Pricing

this is my first time asking questions here. I want to look for some calculation method to price a very exotic option. The FX Euro-American Knockout Option (EAKO) is an option that has an American ...
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1 answer
88 views

How to find state prices?

I am trying to find out how to solve state prices, but I do not know what I am supposed to do, my professor has given a solution to this problem as being (0.060 0.417 0.476), but I can't figure out ...
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5 votes
1 answer
585 views

Where does 1/2 in Fourier Transform method of pricing options come from?

I am reading Jianwe Zhu's Applications of Fourier Transform to Smile Modeling. On page 26, the author is describing how to use the Fourier tranform to price vanilla European call options. If $f_j$ is ...
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1 vote
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SciPy Calibrating Heston call option

I have been attempting to calibrate my Heston model, but I am running into issues with scipy.optimize module. I have tried various scipy optimizers, but they all return the error "TypeError: can ...
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0 votes
1 answer
244 views

Quant Interview - Options pricing [closed]

I am fairly new to all this, merely read the first few chapters of "The Concepts and Practice of Mathematical Finance". I recently had a job interview and was asked a simple question about ...
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1 answer
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Bachelier Spread formula - input spread volatility question

I don't have much experience using Bachelier's single factor spread option formula, but I know it takes a dollar volatility of the spread as an input. What I don't know, is that just the standard ...
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1 vote
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108 views

How does $(d_2/\sigma) = (1-d_1)$ while deriving the Vanna Formula from BSM? [closed]

Just realized there was a quant finance board, so I figured I'd post it here instead. I'm trying to derive Vanna from the Black-Scholes Model (BSM) equation, but had a hook up on one of the ...
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1 vote
0 answers
43 views

Assymetric Rate Distribution

The pandemic has disavowed any notion of nominal rate distributions to being truncated at 0%. However, if Central Banks at Debtor nations are conflicted in that they are incented to suppress interest ...
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Noob Question - Monte Carlo vs BAPM European Option Pricing Discrepancy

It's winter break (happy new year!), and I'm trying implement a few options pricing models (bapm, tapm, monte carlo, Fast Fourier etc.) for practice. The issue: My BAPM CRR model converges to 8....
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