Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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Which is more Appropriate way to calculate Leverage with Options Contracts? [closed]

the connecting of words and bold colors was because of something here in the posting feature, it would not let me take it away when originally posted. i have re-copied and tried correct those errors. ...
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Some questions about the pricing and the construction of Fixed Coupon Notes (FCN) [closed]

I'm currently studying Fixed Coupon Notes (FCN) out of my own curiosity. I've already read some articles and watched a video about it. One of the articles about FCN: https://cegafi.medium.com/...
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Monte carlo pricing on zero coupon bon under the Vasicek model [closed]

I would like to price an European call on zero coupon bond under the Vasicek model. I am planning to follow the Excercise 33 (hereby) from Lamberton Lapeyre (Introduction au calcul stochastique ...
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Stopping times, question on exercise

I'm completing exercises from Steve Shreve - Stochastic Calculus for Finance I and I'm stuck on one subtask for which I can't find missing element for 11 stopping ...
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How to determine break-even price on a delta hedge?

Suppose there is a portfolio of short $x$ shares and long 1 call option. This call option has a strike and premium. If the stock moves up, you loss money on the short position but gain on the option ...
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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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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 $X=(X^1,\dots,X^n)$. The derivative value $V_t$ is given by the function $v(t,X)$, so that the hedge ratios for the hedging ...
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Why is call option delta increasing in below setup? [duplicate]

I have a question. I really appreciate if someone can reply. Enter the same strike and stock price to an options calculator. Set the expiration days to say 20 and calculate delta. It comes out around ...
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9 votes
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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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Does Put-Call parity have influence over American Option pricing in practice?

I am learning my options and from what I read it seems that put-call parity is regarded as only being applicable to European options because the time to exercise is known. American options, on the ...
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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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Show that $S_0$ is the smallest value of a super hedging strategy for a Call option in a arbitrage free market

I attempt a proof : We want to show $S_0=\inf\{V_0 : \exists\theta\;s.t\;V_T(\theta)\geq H\}$ Suppose this is not the case. There exists $V_0^{*}$ such that $V_0^{*}< S_0$ and $V_T^{*}(\theta)\geq ...
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The case of the complete Trinomial model

In my journey to hope for a better understanding of incomplete markets I have decided to focus on the Trinomial model (maybe some of you have seen my previous questions). I have decided to consider ...
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Black Scholes and out of the money Index Options

I understand that the Black-Scholes model is not very effective when modeling call options that are deep out of the money. I found a paper on the web by Song-Ping Zhu and Xin-Jiang He related to this ...
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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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Trinomial model

What is the aim of having the price of a self financing portfolio in the trinomial model if we know that the option we are considering is not duplicable ? Do we have to assume that the payoff of the ...
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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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4 votes
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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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Risk-neutral option pricing on a quasi-reverting underlying asset

Consider $(S_t)_{t\geq0}$, on the probability space $(\Omega,\mathcal{F},\mathbb{Q})$, which evolves according to $$\begin{equation} \frac{\mathop{dS_t}}{S_t}=\mu\mathop{dt}+\sigma_{t,S_t}\mathop{dW_t}...
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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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1 vote
1 answer
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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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2 votes
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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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4 votes
1 answer
328 views

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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2 votes
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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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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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1 vote
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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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1 vote
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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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