Questions tagged [option-pricing]
Questions about models for the valuation of option contracts.
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What is the minimum price of an option, given no information about Greeks? [closed]
I was asked this interview questions for an analyst level structuring role and it has been bothering me since I can't figure it out:
Assuming the price of an equity is 100, what is the minimum price ...
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Given $\mathbb{E}[X]$, $\mathbb{E}[\max(0,X)]$, and $\mathbb{E}[\min(0,X)]$, what is $\mathbb{E}[f(X)]$
Let $X$ be any random variable with any distribution. Given that we know $\mathbb{E}[X]$, $\mathbb{E}[\max(0,X)]$, and $\mathbb{E}[\min(0,X)]$, can you write a formula for $\mathbb{E}[f(X)]$ where $f$ ...
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I can’t understand why the premium of two butterflies with same strike but different broadness are approximately the same
Consider the following premiums of calls option with different strikes.
C90 = 57.35
C95 = 52.55
C100 = 47.3
C105 = 42.9
C110 = 38.25
In this case, the butterfly 90-100-110 cost 1 and the 95-100-105 ...
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How do you explain consistently making money with discrete hedging a call option?
In a backtest I did, I'm selling a call option and buying a delta amount of the underlying (calculated using implied vol). Now I know in the limit case of continuous hedging I end up paying a PnL ...
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Is Local Volatility a function of the Strike or the Underlying price?
Long story cut short: I am asking why the Local Volatility function can be thought of as a function of the underlying, when in fact it appears to be a function of the strike.
Additionally, I wonder ...
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Pricing options in underlying problem
Let us look at options, which are cash settled, but instead of receiving cash, you receive the proportion
from underlying asset with the same value as cash. Moreover, you can pay for these options in ...
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How do you handle non integer time intervals in Quantlib for options pricing (ie intraday pricing)
I'm using QuantLib (python version) for options pricing, and am trying to figure out how to handle non integer dates. If you try and price an option at 9:30 AM and at 4 PM, you should get different ...
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Exotic options with lookback features [closed]
I am trying to value an american call option with a lookback feature. So the holder can choose to exercise either based on a fixed strike (K) or a floating strike equal to 10-day moving average (MA). ...
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Mid-curve swaption pricing - how to get the spread vol?
I believe I understand the following (from the accepted answer to the Quantitative Finance question called "volatility of a mid curve option"):
A swaption in which the underlying swap ...
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Implied volatility skew decay over expiry
I seem to remember the implied volatility skew of European options decreases as the expiry increases. It is true for the Heston model under some approximation. What are the good references that prove ...
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Probability density from COS method too sensitive to truncation range
I have a long-standing confusion around the truncation range of the COS method proposed by Fang and Oosterlee because I find that the results are highly volatile given the different truncation ranges.
...
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VIX options: vertical mark price vs. term structure
Does anyone have any hypothesis why, for options on a future series, vertical spreads priced at half the strike difference should exhibit the same strike across all expirations, regardless of term ...
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How do I proceed with pricing after calculating local vol volatilities?
What are the next steps for pricing under the local volatility formula? It feels like I'm missing the trick (as I only see the numerical methods used to obtain prices after calculating local vol).
I.e....
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FX Option Vol Quotes (Days or Days+Time to Expiry)
I understand FX Options are often quoted via ATM, RR, BF for 10/25 deltas. There are many resources that outline how to convert those quotes back into absolute strike space (using spot delta or ...
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Very close local volatility and implied volatility using Dupire's equation
I used Dupire's equation to calculate the local volatility as in https://www.frouah.com/finance%20notes/Dupire%20Local%20Volatility.pdf and Numerical example of how to calculate local vol surface from ...
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Black and scholes option pricing
I have to solve the following problem in the Black and scholes model: find the price at anty $t\in[0,T)$ for an option whose payoff at the maturity is:
\begin{equation}
0 \ \ \ \text{if} \ S_T<K_1\\...
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In the derivation of the Black-Scholes PDE, using delta hedging, how is this linked to the risk neutral valuation? [closed]
I was reading this paper:
http://www.columbia.edu/~mh2078/FoundationsFE/BlackScholes.pdf
I don't understand the paragraph here:
"The most interesting feature of the Black-Scholes PDE (8) is that ...
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Why should delta-neutral backspread always result in credit?
Natenberg mentions in chapter titled "Volatility Spreads" :
under the assumptions of a traditional theoretical pricing model, a delta-neutral ratio spread
where more options are purchased ...
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Any innovations in mathematical processes behind option pricing models?
I am working on my thesis about option pricing models beyond classical Black-Scholes Model by looking for some recent innovations on mathematical processes behind the pricing structures. By that I ...
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How to calculate the local volatility from implied volatility in practice
The local volatility can be derived from the implied volatility. But in practice how we deal with the first-order and second-order derivatives?
I have seen this formula
$$
\sigma_{\mathrm{Dup}}(T, K)^{...
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Converting implied volatilities into digital option prices
I have Black and Scholes (1973) implied volatilities computed and I would like to convert these IVs to digital option prices using a Black and Scholes type of formula, I can't find a formula to do ...
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Option pricing Greeks in Python - incorrect Gamma with MC option pricing (Black) using AAD autograd / JAX libraries - but works with closed form?
I am attempting to use AAD (Adjoint Algorithmic Differentiation) with a simple Black MC pricer, and found that the Gamma is incorrect. The output was compared to Black analytical Greeks, as well as ...
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Special Exotic Option Pricing Approach [closed]
I am currently stuck with the following problem:
You need to price the following exotic option, where the share price of Stock ABC is the underlying:
• Time to maturity: 2 years
• Right to exercise: ...
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Cash balance sign in hedging portfolio
Consider a derivative which depends on $n$ assets with price vector $X=(S^1,\dots,S^n)$. The derivative value $V_t$ is given by the function $v(t,S)$, so that the hedge ratios for the hedging ...
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What is the market standard for IR option pricing when moving to SOFR
From books it looks like market standards to price IR options, like swaptions, are SABR, LMM or mix of the two (SABR-LMM).
But LMM models the forward LIBOR rate. What will happen to it once LIBOR ...
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How to price american barrier with Local-Stochastic Volatility
I have attended a conference where one speaker mentioned that the market standard to price FX and Equity derivatives is now the Local-Stochastic volatility model.
I understand this class of model is a ...
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What are some good books to get started with option theory? [duplicate]
Recently graduated in econometrics but starting to realize my knowledge is limited. Any and all tips are welcome!
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Pricing FX options on pegged currencies
I'm wondering what's the standard (if any) for practitioners to trade volatility on pegged currencies. Is there any specific convention? I'm thinking situations like EURCHF before the unpeg, how were ...
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Hedging possibility in a market with more state of the world than asset (discrete time)
For a European Call option, by proposing the initial price of the underlying asset I am sure to be able to meet my commitments, however this result is not true for a Put option. However, by proposing ...
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Difference between closed form binomial option value and monte carlo simulation
I am trying to calculate the price of a European call option using both the the closed form expression and a monte carlo simulation. But the value's I get from both these methods are not the same:
...
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Trinomial model option pricing
If I have well understood, in the trinomial model we have a kind of risk neutral pricing formula that depends on a parameter. This means thaht as in the binomial model, we could use directly this ...
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Superhedging in Cox-Ross-Rubinstein model revisited
I am doing the following exercise from a math finance textbook but I got stuck at the end of the part 2. I found nothing on the internet concerning solutions of exercises from this textbook (called ...
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Can the Feynman-Kac formula be used for asset classes that don’t have options?
So rather than a call option C(S_t,t) we have some type of asset with asset price is given by S(x,t) where x is any type of variable that the asset price depends on. I.e Price of wooden desks, W(x,t) ...
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Pricing & hedging vanilla interest rate options with SABR LMM
Are there any advantages of pricing and hedging plain vanilla interest rate options with more complex SABR LMM instead of simpler SABR model? Should one always go with the SABR LMM as a universal ...
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Perpetual Option Paying Chooser Option
A perpetual option solves the ODE
$$rSV_S+\frac{1}{2}\sigma^2S^2V_{SS}-rV=0$$
The general solution is $$V(S)=aS+bS^{\gamma}$$ where $\gamma=-\frac{2r}{\sigma^2}<0$.
For an American put option with ...
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Extension of CRR model
I'm considering an extension of the binomial model where the risky asset can take three values at each node, that is $
S_{t+1}=\left\{
\begin{array}{ll}
S_t\cdot u\\\nonumber
...
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SABR LMM vs no-arbitrage term structure of SABR parameters
There exists a LIBOR Market Model with stochastic volatility for pricing and hedging exotic (e.g. path-dependent) interest rate options with smile. However let us consider the following approach:
...
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Why do we worry about the bid/ask spread when pricing option in incomplete market?
Several resources I saw introduce the notion of bid/ask spread when trying to price options in incomplete market, I don't understand why the notion is introduced since we are interested on the price ...
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Equivalent BS volatility formula under the Heston model?
Is there an equivalent BS volatility formula for the Heston model, something like Hagan's formula for the SABR model? Of course, such a formula will be an approximation as in Hagan's formula.
Under ...
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Find the lower bound of a contingent claim in incomplete market
I'm trying to justify the lower bound for the price of a contingent claim (a European one) which is not marketable in an arbitrage free market. I would like to have your advice on my way to do it:
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Are European call and put option useful ? [Cox-Ross-Rubinstein model]
I'm new to the world of option market, but after having studied CRR model I'm wondering if European call and put option are very useful since a talk with my professor that piqued ma curiosity. In the ...
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Confusion about "cost" in option pricing paper by Cox-Ross-Rubinstein paper
I am trying to understand the paper "Option Pricing: A Simplified Approach" by Cox-Ross-Rubinstein (available online here).
To my frustration, I already don't understand the paper starting ...
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European option with payoff $X_T^2$ [closed]
I have been ask to price a European option with payoff $H(X_T,T) = X_T^2$ using the equivalent martingale measure (EMM).
For this I used the process:
\begin{equation}
dX_t = r X_t dt + \sigma X_t d\...
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Use of markov process in option pricing
In several books on asset pricing and more particularly when it concerns option pricing, I see the use of Markov process, they argue the computation is made easier with such process. Is this ...
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Option pricing in incomplete CRR model
I'm studying the way option can be priced in an incomplete market and I have found an example talking about the Cox-Ross-Rubinstein model with three path possible instead of 2, making the model ...
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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)$$
ie implied ATM vol = risk-neutral expectation of integrated variance.
Is there some proof available? ...
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Payoff of a Butterfly spread under risk neutral measure is always positive for any t<T
In a situation where $$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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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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