Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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15 views

Books on options trading with a practical bent?

just curious to see if anyone here has come across a book or books on options trading with the practitioner in mind? My lecture slides for instance, go through black scholes and the ins and outs of ...
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Problem with the maximum likelihood for a GARCH-type of model

I'm currently working with the following GARCH process from Heston and Nandi (2000): \begin{align*} r_{t+1} - r_f &= \lambda h_{t+1} - \frac{h_{t+1}}{2} + \sqrt{h_{t+1}}z_{t+1} \\ h_{t+1} ...
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Does the Black-Scholes formula work when unit of time is in hours?

In the Black-Scholes formula, the unit of time is usually in years from what I understand. An online calculator I found allows the users to input the time in days and years. Would the formula still ...
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Interpolation implied volatility put pricing

My goal is to price a put (for exemple with maturity = 0,5 and K(strike price)=250 S(asset price)=247,74 and r=1,11%) through B&S formula. I know that I have to choose a value for implied ...
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19 views

Option Delta Calculation - Local Vol Model vs Black-Scholes Model

I am looking to get the greeks for option chain. Which model does work better for greeks calculation especially the delta. I am having issue with the Black-Scholes Model Delta since it always ...
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24 views

Fourier transform Carr-Madan method on an arbitrar initial $S_0$ values

As mentioned in Carr-Madan's paper, here, the European call option is: $$ C_T(k)=\frac{e^{\alpha k}}{\pi}\int_0^\infty\mathcal{Re}\left(e^{-iuk}\psi(u)\right)du $$ where $$ \psi(u)=e^{-rT}\frac{\phi_T(...
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380 views

Suppose that we are wrong about the relevant class of distributions for financial economics and econometrics. Now what?

I read a very interesting paper by Harris (2017) where he points out some interesting link between market microstructure and the distribution of returns on equity. You can make a good case that the ...
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21 views

Dupire Vomma and Stochastic volatility

Suppose that you are short an option on asset $X_t$ following a pure diffusion. Suppose you are hedging your position using (Dupire) Local volatility model. Suppose that the option is concave with ...
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19 views

Replication Strategy of European Call Option? [closed]

I am having trouble with the following question. Your input is greatly appreciated. Thanks.
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37 views

Implied volatility and greeks of options

When we are calculating deltas or vegas for different strikes should we use the underlying asset's volatility or should we use the implied volatility for the specific strikes at a fixed maturity? ...
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59 views

Black-Scholes and solving for both $r$ and $\sigma$ ; Do I have a unique solution?

Below is a problem that I am working on. I believe that my incomplete solution is correct as far as it goes. I would like to know if my solution is incorrect. I plan to solve the system of two ...
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Connecting the dots: Black Scholes, Volatility and Implied Volatility

I am a first year Management & Finance undergrad preparing for my second year Finance courses, given that term 3 and exams have pretty much been cancelled for all British first years. During that ...
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What is the delta of an at-the-money European call option with respect to volatility?

Question: What is the delta of an at-the-money European call option with respect to volatility? Note that $$\frac{\partial\Delta}{\partial\sigma} = N'(d_1) \frac{\partial d_1}{\partial\sigma} = N'(...
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67 views

Why is there a theoretical lower bound on the price of call options?

From my textbook, I see that the theoretical lower bound for the price of a European call option on a non-dividend-paying stock is: $S_0 - \mathrm{Ke}^\mathrm{-rT}$, where $S_0$ is the current stock ...
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131 views

What stochastic volatility models are industry standard for option pricing and how do they work?

I've started reading up on stochastic volatility models and it seems very difficult to discern which ones are used in practice and which have been mostly left alone in theory. What are the popular ...
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Pricing options using the IG component GARCH model of BCHJ(2018)

Babaoglu, Christoffersen, Heston and Jacobs (2018) introduced a component GARCH model with inverse Gaussian innovations and an exponentially quadratic pricing kernel back in 2018. The article shouldn'...
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Two- (multi) dimensional geometric Brownian Motion

I am trying to calculate the value of a Basket Option with two stocks and the following information: S1 = 100, S2 = 120, r = 0.06 L = Volatilitymatrix = ((0.3, 0.1), (0.0, 0.2)), weight of Stock 1 = 1/...
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105 views

Why multiply stock returns with $\sqrt{252}$?

When converting daily volatilities to annual volatilities one need to multiply with $\sqrt{252}$. But I found this piece of code this piece of code who calculate log-returns in the following way: In ...
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50 views

Parameter Inference Stochastic Volatility

When calibrating the Heston model for instance; \begin{align} d S_{t}=\mu S_{t} d t+\sqrt{\nu_{t}} S_{t} d W_{t}^{S} \\ d \nu_{t}=\kappa\left(\theta-\nu_{t}\right) d t+\xi \sqrt{\nu_{t}} d W_{t}^{\nu}...
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251 views

Pricing call option using risk-neutral martingale approach with squared stock price boundary?

I have to use the risk-neutral martingale 5 step approach under BS pricing framework to price the following call option at time 0: $$X = \begin{cases}1, &{if} &S_T^2\geq K,\\0, & {...
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28 views

How to replicate an exchange option?

The exchange option has payoff $\text{max}\Big(S_{1}(T)-S_{2}(T),0\Big)$. Dynamic quantities of what instruments should I have to replicate it?
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Finite difference valuation with early exercise

I am implementing a finite difference pricer for American options using local volatility models. The pricing PDE is given by $$ \frac{\partial V(t,S)}{\partial t}+\left(r(t)-q(t)\right)S\frac{\partial ...
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Oughtn't option premiums increase by the same amount as strike prices? [closed]

Pls see this question's title. In the screenshot below, as the strike prices below increase by +1, oughtn't the option premiums increase by +1 too? Why buy the \$104 put for \$13.71? The \$105 put ...
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39 views

Why'd put options with lower strike prices cost more?

I can't fathom the option premiums for the put options offered below : can someone please ELI5? Don't strike prices vary directly with option premiums? Liquidity doesn't appear the hitch that would ...
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45 views

Black-Scholes Theory vs Actual Market Price

I have a question of which I am uncertain on how to answer, that is: Assume the Black and Scholes differential equation for option pricing with constant risk free rate, $ r $ and constant volatility $...
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68 views

Option Volatility Smile vs Delta

I am new to options trading and have been trying to better understand the relationship between implied volatility, delta, and moneyness. I was wondering how a call option's implied volatility can go ...
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Relation between volatility and exercise timing of American Options

Hopefully someone can help me with intuition. Suppose that we have a stock whose value evolves per the geometric brownian motion $dX_t=X_t\mu dt+X_t\sigma dW_t$, for $\sigma>0$, $\mu\in\mathbb{R}$ ...
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55 views

For any twice differential continuous function C(T, K), does there exist a sigma(t, S) that can reproduce C(T, K)?

In the Dupire's paper, he assumes that there exits a function $\sigma(t,S)$ that can reproduce $C(T, K)$. My question is that: is the assumption true for any twice differential continuous function $C(...
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88 views

Is Dupire's local volatility model path independent to recover historical option price?

Generally when we implement Dupire's local volatility model, we follow the steps below: Calculate implied volatility from given historical data Fit the implied volatility skew. So we also know the ...
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59 views

Some basics of option pricing

I am a mathematician trying to learn finance on my own. Try to avoid financial lingo in your answer when not necessary. So I am trying to understand (European) option pricing under the no free lunch ...
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42 views

Holiday handling in Monte Carlo Simulation

For Monte Carlo, usually people will take the days per year as 252 or 250 or 260 to avoid the holiday issue. However, for some stupid reasons, I need to handle the holiday issue with time step as ...
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91 views

Black-Scholes formula and implied vol

Is the Black-Scholes formula the only way "implied volatility" is calculated/defined in markets?
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Are there trades that long gamma (convexity) and short volatility at the same time?

Likewise, are there trades that short gamma and long volatility at the same time? Under fixed income context, are there trades that short convexity and long volatility at the same time?
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108 views

Throwing a dice and risk neutral probability

Consider the game of throwing a "fair" dice. Not sure if the answer is obvious but is there any proof (e.g. replication argument) that under the risk neutral measure the probability of any outcome is ...
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Intuition behind the Carr and Wu (2014) static hedging for ordinary options

Let $(S_t)_{t \geq 0}$ be the price of an underlying asset, $r$ be the risk-free rate of return, $q$ the dividend yield, $C_t(K,T)$ is the price of a call option written on $S_t$ at time $t$ with ...
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Vanilla option pricing at different points in time

Let $C(t) = C(t; S,K,T)$ the price at time $t$ of a plain vanilla call option with maturity $T$ and strike $K$ on an underlying $S$; if for $t_1<t_2$ we have $C(t_1) > C(t_2)$, it could not be ...
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46 views

MonteCarlo option pricing error estimate

Consider the problem of pricing an option via MonteCarlo with 10000 simulations. If the variance of the simulation is 100, which is the MC estimate of the error on the price?
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87 views

Why does a Bermudan option have a higher implied volatility than its European counterpart?

I get that the premium for an earlier exercise should be higher to compensate the seller but intuitively you would think that the spot has "less room to run" in a potentially shorter period of time (...
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108 views

Going from $\mathcal{P}$ to $\mathcal{Q}$

Under $\mathcal{P}$, we have the Heston Model given by: $$ d S_{t}=\mu S_{t} d t+\sqrt{\nu_{t}} S_{t} d W_{t}^{S},\\ d \nu_{t}=\kappa\left(\theta-\nu_{t}\right) d t+\xi \sqrt{\nu_{t}} d W_{t}^{\nu}. $...
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80 views

Which method is used to price highly exotic options in exotic models?

What is the go-to method to price exotic options in exotic models? If we are in Black Scholes, then this is hard to answer, since we can both do various sorts of Monte Carlo or solve various sorts of ...
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How to price a put option on a multi-asset fund? Confused by risk-neutral pricing implicaton on real world

The fund has super track record with stable vol. The chance for this Put to pay out is very low in real world, but a B/S risk-neutral pricing would give a very high cost. I am struggling with the ...
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Delta hedging an option with earlier expiry

The answer here states: For instance a volatility product that would expire at 10:42 am on a random day would be off term. One that expires at the same time than a major listed contract would ...
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73 views

How to calculate the prices of option instruments for a new underlying

Can someone with practical experience with implementing and verifying please point me in the right direction. Let's say I have 3 months of data for an underlying. I want to generate theoretical ...
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1answer
73 views

Monte carlo delta calculation for Worst/Best Of Option

I try to calculate the Delta for WO by finite difference. For example, $K = 1.$ $$ S_t = S_0 e^{(r - d_1 - \frac{\sigma_1^2}{2})t + \sigma_1 W_t^1} $$ $$ F_t = F_0 e^{(r - d_2 - \frac{\sigma_2^2}{...
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Weighting function for parametric estimation of the Risk-neutral density function

I would like to estimate the Risk-neutral density function (RND) implicit in financial Call option prices by a parametric approach where the parameters of the RND (for instance mean and variance for a ...
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60 views

Lower bound for Bermudan Option Price

i have the following question. The price of an Bermudan option is given by \begin{align*} V_{0} = \sup_{\tau \in \mathcal{T}(0,\dots, T)} \mathbb{E}[f_{\tau}(X_{\tau})]. \end{align*} It is ...
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66 views

Option on a dice game with three dices and min. value

We have a call option on 3 dices with strike 3. What's the fair value of the call when it pays the min value of the 3 dices? E.g if we throw and have 426, the min is 2 here and so call is OTM (S < ...
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How can the claims of this paper be true (on speed of Carr-Madan method for option pricing)?

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2815371 This paper states that a strike-optimized version of the Carr-Madan method for option pricing is faster than the original equivalent that ...
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59 views

Risk Neutral Density Curve for SPY Options looks very weird

I have created a risk neutral density curve using SPY weekly options and the RND package in R. I calculated the risk neutral density for the Feb07 options. The curve looks very weird when I look at ...
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105 views

Why sub-replication is not studied in literature

There are numerous paper about super-hedging and super-replication in an incomplete market where the risk neutral measures are not unique. The most fundamental result is that the super-replication ...

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