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Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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How to price equity options using a Black76 implied volatility surface?

I would like to calculate the fair value of american and european options on various equities and indices using QuantLib C++. Since I do have discrete dividends available for most underlyings, I use <...
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Interpretation of Market Price of Volatility Risk

In option pricing with market model equipped with stochastic volatility, there are numerous times mentioning "market price of volatility risk" without even define or give any explanation regarding the ...
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What is the use of undiscounted Futures/Option Prices

Reading the great book of Gatheral on Vol Surfaces (link) I can't help but notice that throughout he uses undiscounted option prices (though he obviously never assumed rates to be zero). See e.g. ...
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79 views

Heston model computations

In the Heston model the dynamics of a single-asset $S$ are given by: $dS_t = rS_tdt+S_t \sqrt{V_t}dW^S$ where $W^s$ is a brownian-motion $W^S$ and the square root variance process $V$ is given by ...
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Asian basket option variance reduction control variates monte carlo

I have priced an Asian put option with three underlying correlated stocks. Now I want to try to reduce the variance using control variates. I have found great ideas when there is one underlying (thus ...
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How does LR binomial Tree Model handle input values which would cause NA result?

I am using C++ to implement a LR binomial Tree algorithm to price American options, but I find it would constantly generate invalid output, which is "nan" value in C++, although the input value seems ...
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Probability of option expiring in money for trinomial model

Are there any formulae for calculating the probability of an option expiring in the money, when using the trinomial pricing model? I have found one for the binomial model, but I am unable to derive it ...
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1answer
61 views

Black Scholes on Eurodollar Options

I am trying to replicate the Black Scholes results of CME option calculator for options on Eurodollar Options. (link) I am trying to replicate the implied volatility result by unaltering the spot and ...
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1answer
32 views

Characteristic function of CGMY model

I have a basic question about the CGMY model which has characteristic function $$ \Gamma(-Y_p)\left((M-iu)^{Y_p}-M^{Y_p}\right)+\frac{C_n}{C_p}\Gamma(-Y_n)\left((G+iu)^{Y_n}-G^{Y_n}\right) $$ whith $...
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1answer
60 views

For equity options, does the implied vol change if the price of the underlying does?

For example, consider S&P options. My reasoning is rooted in the fact that VIX returns and S&P returns have a negative relationship, since VIX is a measure of S&P options' implied vol. ...
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42 views

Pricing a Double Knock In Option

I have been looking at pricing a barrier option that has payoff of your usual European Call option, $\max(S_T - K, 0)$ if the stock price exceeds a horizon $A$ and then afterwards drop under some ...
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Simulating assets of different currencies

I have a situation as follows: One year call option on a Euro stock with a Euro denominated strike. Knock in feature as follows - The option can only pay out if the growth in the Euro stock over ...
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1answer
143 views

Alternative derivation of the Black Scholes formula

I encountered the following derivation of the Black Scholes formula for call price. It may very well be an established method but I had never seen it before so I called it an alternative derivation. ...
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Regression techniques for bermudan Monte-Carlo

One knows that the price of a bermudan claim exercisable at times $T_1, T_2,\ldots, T_N$ is $$V_0 = \sup_{\tau\in\Gamma} \mathbf{E} \left[ e^{\int_0^{\tau} r_s ds} \varphi_{\tau}\left( x_{\tau} \...
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Pricing LIBOR options

I am trying to price interest rate options with the underlying as LIBOR. What rate do I use for the risk-free rate? Should the risk free rate be the LIBOR rate itself?
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Black and Normal Model for Caplet using Python

I am able to Price Caplet using Black 76 model in Python. However, I am unable to price the same with Normal Model. Can anyone suggest what is missing ? I am valuing caplet that caps interest rate on ...
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FX Hybrid Model with Smile

I’ve read the fantastic paper by Piterbarg on long dated FX options here: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=685084. One of the limitations of the paper is that the model mainly ...
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Straightforward question about paper on Arbitrage-Free Smoothing of the Implied Volatility Surface

In the paper Arbitrage-Free Smoothing of the Implied Volatility Surface by Matthias R. Fengler (2005) (https://core.ac.uk/download/pdf/6978470.pdf), if $\lambda =0$ on equation (1) then the solution ...
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Arbitrage free smoothing of volatility smile - cubic spline - implementation procedure

I am studying the paper Arbitrage-Free Smoothing of the Implied Volatility Surface, from Matthias R. Fengler (https://core.ac.uk/download/pdf/6978470.pdf). The problem I want to solve is much simpler ...
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114 views

Expectation of option value

Say we are in a BS world where the (conditional on t) price of a call is given by the usual $$V(S_t)=V(S_t;K,r,\sigma,T|F_t) = \Phi(d_1)S_t - \Phi(d_2)Ke^{-r(T-t)}$$ Now, what about the ...
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Suggestions of papers for computing market implied probability distribution function

I need suggestions of papers that propose simple and fast methods (not heavily dependent on simulations, nut can depend on simulation) to derive the market implicit probability distribution function ...
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SDE for option value

Given an SDE for an underlying: $$dS(t) = \mu(S,t)dt+\sigma(S,t)dW(t)$$ the SDE for the value of the option $V=V(S,t)$ is given via Ito's lemma as: $$dV = V_tdt+V_S\mu(S,t)dt+\frac{1}{2}V_{SS}\...
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1answer
112 views

Python package for option pricing models?

Is there a good python package for various option pricing models, e.g., Heston, SABR, etc? I found that it's even hard to find a good python implementation of Black-Scholes model (i.e., price + IV + ...
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1answer
115 views

condition of risk neutral pricing

The theorem says if $U$ is a numeraire and let $\mathbb{Q}^U$ be the corresponding measure. Then for every tradable asset $S$, the relative price $S_t/U_t$ is a martingale under $\mathbb{Q}^U$. But I ...
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Volatility Index for Industries

The VIX index is based on the S&P, I am wondering the feasibility of creating a VIX index for each of the S&P 500 Industries? Would this even be possible?
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176 views

Hedging with machine learning

I’ve been thinking about an interesting problem lately: Suppose I have a position in an exotic derivative. How can I automate the hedging process? Traditionally, one build a pricing model and ...
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1answer
121 views

Why my implementation of CRR model does not converge?

Recall that CRR (Cox-Ross-Rubinstein) model for option pricing is the usual binomial tree model with $u$ (up-factor) and $p$ (one of the risk-neutral probabilities) defined as follows: $$u = e^{\sigma\...
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Cash-or-nothing and Asset-or-nothing price derivation

I was wondering how to derive the price of a cash-or-nothing and asset-or-nothing option by trying to work out the expectation under the risk-neutral measure, while assuming that the underlying ...
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1answer
54 views

Value simple chooser option as a sum of call and put options

There is a well known formula for valuating the chooser's option price: $H_{chooser}=max\{C(S_t, K, T-t), P(S_t, K, T-t)\}=max\{C(S_t, K, T-t), C(S_t, K, T-t)+Ke^{−r(T-t)}−S_t\}=C(S_t, K, T-t) + max\{...
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1answer
102 views

Pricing a call option with pay-off function max{$S_T - S_{T/2}, 0$}

Pricing a call option with payoff function $C=\max\{S_T - S_{T/2}, 0\}$, where $S_T$ is geometric brownian motion. I appreciate any help! Please close this question if this is a duplicated question. ...
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1answer
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Approximate Hagan formula for SABR model with negative beta

While looking into fixing the $\beta$ parameter (based the following regression: $\text{ln } \sigma^{ATM}_t = \text{ln } \alpha - (1-\beta)\text{ln }F_t$, as explained in West (2004), page 6) before ...
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2answers
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Why do we need approximation in option pricing?

We know that we can get a closed form for European option price. And we can calculate directly the normal distribution accumulation. But I saw that people use many approximation methods such as ...
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Why do we require a continuous volatility calibration while pricing Options [closed]

On pricing Options the volatility surface is represented by a mathematical model (with parameters). What does it mean to calibrate the volatility surface How often has the volatility surface to be ...
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1answer
67 views

Basic Replication of European Call Option

I am looking at the very basics of replicating an option with a portfolio of risky and risk free assets. As such we can define a portfolio of $x$ no. of shares, $y$ bonds & $z$ options at time $(T)...
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Usages of variance swap

I’m interested in variance swap. Considered from its feature, variance swap is used for betting the (historical) volatility of underlying asset. If we use it for hedge tool of Vega or Volga, does it ...
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Why futures pricing not calculated like options?

I have read about futures and options ( from online resources ). I only have the basic understanding,not math heavy ( for eg. for Black Scholes I know only the intuitive idea from the khan academy ...
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Calculating historical Volatility for the Black Scholes Model [closed]

Below is a problem from the book "Options, Futures, and other Derivatives" by John C. Hull. I did the problem but I am fairly sure that my answer is wrong. I am hoping that somebody can tell me where ...
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1answer
59 views

Option price of a future

This must be a dumb question. Consider a European option $V$ on a (stock) futures $F$. The hedging condition seems to be the same as that for a stock $$d\Big(V-\frac{\partial V}{\partial F}F\Big)=r\...
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Oscillating errors in finite difference Black Scholes

I am writing an implementation of the explicit finite difference method to price a standard european call option, and comparing the results to the corresponding analytical value to gauge the error ...
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71 views

Basic question about Structured Product Greeks

Say I'm dealing with a structured product (SP): a short put is financing a coupon which is dependent on the paths of the underlying. So typically pricing of an autocall. What's the delta of this ...
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1answer
77 views

Asset price simulation under Monte Carlo for option pricing using market data

I am trying to use Monte Carlo to price some exotic options. I have in mind to simulate asset prices under GBM (say S&P prices) using Monte Carlo and price the option accordingly from the payoffs ...
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Checking arbitrage for the SABR model - analytical vs numerical approach

I wish to check if the fitted volatility smile/surface from the SABR model for a fixed time period is arbitrage free. Through my research, I've learnt the following need to be checked: The RND (risk ...
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1answer
80 views

HJM or Short rates model?

When market practitioners do prefer HJM models to short rates models when it comes to pricing derivatives (other than swaptions and caps, let say light exotics to exotics) ? To be more specific, ...
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2answers
207 views

Interest rates forward implied volatility models

I'm trying to find out which model to use to price a pur forward volatility product named VolBond marketed by structuring desks currently. Let me introduce the products first: Example 1: You pay 100 ...
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208 views

Does Black Scholes need to assume no arbitrage?

Since Girsanov's theorem guarantees a risk neutral measure for Geometric Brownian motion, by the fundamental theorem of asset pricing there can be no arbitrage. So, why does the model assume no ...
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Pricing caplet with Bachelier (normal dynamic) using forward measure

I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution. Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing : $C_t(T,T+d)$ ...
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1answer
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Using Geometric Brownian Motion for Index Options

As far as I understand, in most of the cases we derive the option valuation assuming that the log-return of the asset is partly driven by its own Brownian motion, and we use Geometric Brownian motion (...
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62 views

option price change

I am trying to match change in European Call option price to greeks using the calculator here e.g. for S=95, K=100, r=0, V=25, t=5 and dividend=0, I get ...
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Pricing European Call on Coupon Bond in Lattice

What's the best approach to pricing a par call option on a coupon paying bond? Is it to discount the greater of the price and strike through the lattice? And for this, is the price used the dirty or ...
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1answer
68 views

Perpetual Put vs European Put

I am looking at a perpetual put option where the strike price is initially the stock price $K(0)=S(0)$ (i.e. at the money), but the strike price grows at the constant risk-free rate $r$ [i.e. $K(t)=S(...