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Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

231 questions with no upvoted or accepted answers
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35
votes
0answers
966 views

How to show that this weak scheme is a cubature scheme?

Weak schemes, such as Ninomiya-Victoir or Ninomiya-Ninomiya, are typically used for discretization of stochastic volatility models such as the Heston Model. Can anyone familiar with Cubature on ...
19
votes
0answers
218 views

Is there a relationship between Risk Neutral Pricing framework and Nash Equilibria?

Based on the Fundamental Theorem of Asset Pricing, the risk neutral price of a contingent claim on an asset in a liquid, arbitrage free market can be determined by switching to an equivalent $Q-$ ...
12
votes
1answer
470 views

Transformation of Volatility - BS

I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev \begin{equation} \sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}} \end{...
9
votes
0answers
293 views

Jim Gatheral's ansatz

In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes $$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$ where $\hat w_t:=\int_0^t \hat v_s ds$ is the expected total ...
8
votes
0answers
210 views

Basket option density in BS model

Let X and Y be two GBM’s, they have each a univariate log-normal distribution for some time t, that is $X_t\sim{LnN(µ_x, σ^2_x)}$, $Y_t\sim{LnN(µ_y, σ^2_y})$ and $Z_t=[X_t,Y_t]\sim{ MvLnN(μ, Σ)}$ ...
7
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0answers
259 views

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 ...
7
votes
0answers
122 views

recent developments in American options?

I have read the paper written by Egloff (2005) using machine learning techniques to solve the optimal stopping problem. Is there any development in pricing American options during 2005-2016? (based ...
6
votes
0answers
250 views

implied volatility and strike price

Assume for simplicity that the expiration time of an option is $1$ the initial stock price is $1$ and there is no dividend yield and the risk free return is $0$. How is it possible to show that the ...
6
votes
0answers
318 views

How are quants able to verify whether their calculated prices are any good

This question is related to the discussion on Model Validation Criteria However it appeard to be very high level to me and I would like to go more into detail. Not working at a pricing desk the ...
5
votes
0answers
98 views

Implied Funding/Borrow Costs in Short-Dated ETF Option Prices

I'm struggling with some anomalous behavior in an analysis I'm running and was hoping for some advice/insights. I'm attempting to extract the implied funding/borrow costs from ETF option prices (say ...
5
votes
0answers
115 views

Pricing and hedging of vanilla options based on non-tradable underlying

Consider a non-tradable stock index $S$ which satisfies: $dS_t=\mu S_tdt+\sigma S_tdW_t$ and a risk-free asset $B$. I want to price an European Call option with the payoff $C_T=max(S_T-K,0)$. The ...
5
votes
0answers
100 views

Complete Financial Market: Integrability condition for Contingent Claims

Consider an arbitrage-free and complete financial market with underlying filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\,\in\,[0,T]},\mathbb{Q})$, where $T\in(0,\infty)$ is ...
5
votes
0answers
235 views

pricing option with two stocks

Let $\left(S_t^{(1)}\right)_{t\ge0}$ and $\left(S_t^{(2)}\right)_{t\ge0}$ be the price processes of two stocks with dynamics $$ \begin{align} & dS_t^{(1)}=\sigma_{11}S_t^{(1)}dW_t^{(1)} \...
5
votes
0answers
192 views

Pricing interest rate options in emerging markets

I've been thinking how to price the early payment of mortgages in banks from emerging markets, where swaptions/caps/floors aren't available, and how to hedge this kind of options. At first I thought ...
5
votes
0answers
204 views

Quantitative approaches to measuring the effectiveness of a Stock Option Pricing Model?

My question contains many parts, but I will try to keep it somewhat focused. I am primarily looking for a framework to evaluate the accuracy of a stock-focused Options Pricing Model. One of the ...
5
votes
0answers
80 views

How to price lookback american option when its payment is distributed during its life

I would like to price a floating strike american lookback with a particular feature: I don't want to charge upfront the client, rather I would like to insert a "running fee", some sort of a dividend. ...
5
votes
0answers
332 views

Algorithmic Trading Model Calculation and Stale Data

I'd like ask everyone a more concurrency programming but definitely quant-finance related question. How do you deal with staleness of data in market hours as quote ticks are streaming and your model ...
5
votes
0answers
536 views

Pricing with collateral

I have been confused about many things concerning the princing of securities with collateral. We can prove that today's price of a security( fully collateralized and within the same currency) is the ...
5
votes
0answers
212 views

Use of Local Times in Option Pricing

I know two applications of local time in option pricing theory. First, it allows a derivation of Dupire's formula on local volatility in a neat way (i.e. without resorting to differential operator ...
4
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0answers
127 views

Model-Free Option Pricing

From Breeden and Litzenberger (1978) and subsequent work, we may find the risk-neutral density $q_{S_T}$ of $S_T$ from European option prices - assuming there are enough traded options (e.g. SPX) via ...
4
votes
0answers
146 views

How to price the american options using local volatility

I have given with a surface of american option prices $C_{am}(T, K)$. From these american option prices the implied volatility surface is deduced. Now I want to find the local volatility $\sigma(s,t)$...
4
votes
0answers
75 views

Is there an arbitrage free option model that treats volatility as a deterministic function of strike?

I am trying to get a good understanding of the different models out there, and thus be able to study hedging errors, and strengths and weaknesses. My understanding of the Local Volatility model in ...
4
votes
1answer
194 views

Numerical simulation of Heston model

I am trying to simulate on Python random paths for a general asset price as described by the Heston model: \begin{equation} \begin{aligned} dS_t &= \mu S_t dt + \sqrt{\nu_t} S_t dW^S_t \\ d\nu_t &...
4
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0answers
65 views

Stochastic Long-Run Mean Instantaneous Variance in Heston Model (and extensions)?

I'm working on my dissertation in Financial Economics, focusing on the topic of Stochastic Volatility Jump Diffusion models; and I'm playing around with some ideas for model extensions. In particular, ...
4
votes
0answers
58 views

Why does risk-neutral price processes do not, in general, compose all arbitrage-free price processes?

I was reading reviewing my mathematical finance notes and I came across a remark I cant understand fully Remark :Contrary to discrete time models, the risk-neutral price processes do not, in general, ...
4
votes
1answer
240 views

How to determine the risk-neutral measure in a Heston model?

To clarify, I'm quite familiar with the risk-neutral pricing framework, and I know one can efficiently Monte-Carlo a Heston model via the non-central $\chi^2$ distribution approach. But so far we're ...
4
votes
0answers
76 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 ...
4
votes
0answers
79 views

Structured Energy Option Pricing

Let's say I have an option with the following terms. This is for an energy product (ie natural gas) The contract will last for 6 months The payoff is the difference between the first of month index ...
4
votes
0answers
75 views

Spread in Option Quotes

Let's take a look at market-maker's option quote in vol terms: 8.5 / 9.5. In that example bid-ask spread equals 1.0 point of vol. Can anybody clarifying how market-maker choose amount of spread in ...
4
votes
0answers
120 views

How can a beginner trader make use of 'volatility of volatility'

For a beginner option trader in equity options, how can he use this metric that is provided by his broker/data vendor? How can he use this metric to gain an added understanding of the option pricing/...
4
votes
0answers
250 views

negative transition probabilities in the heston model

I've been trying to implement a bivariate tree for pricing american options with the heston model in R using the paper of Beliaeva and Nawalkha (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=...
4
votes
0answers
642 views

ATM volatility versus OTM volatility and directional standard deviation

The forward instrument vol curve is skewed to the downside (50 delta risk reversal, 25 put, 25 call) were trading several ticks to the put). Is there a smaller standard deviation (in price terms) to ...
3
votes
0answers
67 views

working with eurodollar options with strikes > 100?

I am trying to extract the implied volatility from options on Eurodollar futures. My understanding is that I should be converting the Underlying Price and Strikes to rates (S = 100 - FuturesPrice, K* =...
3
votes
0answers
41 views

Rainbow option pricing formula under *Bachelier* model

Let's consider a call on min option on two underlying arithmetic Browniation motions $V_t$ and $H_t$ (no drift). Let $P_t$ denotes the price process of the option, $r$ the riskfree rate, $\tau$ the ...
3
votes
0answers
45 views

Why not discount the dividend in the european put lower bound condition?

According to the european put lower bound condition: $ p \geq max(D + K \cdot e^{-r(t_2-t_0)} - S_0, 0)$ where $t_0$ is now and $t_2$ is maturity. Say $t_1$ is the dividend release time where $t_0&...
3
votes
0answers
61 views

Two barrier options puzzle

I come across an interesting question about barrier option as shown below. Two barrier options are given with the same parameters including the barrier level. The first one is knocked out when it ...
3
votes
0answers
123 views

Is there a more efficient data structure to implement binomial trees than 2d array?

I'm just curious what is the "industry standard" for implementing a binomial tree (if "standards" exist in this case). For simplicity, let's just talk about the simplest trees with recombining nodes. ...
3
votes
0answers
120 views

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 ...
3
votes
1answer
254 views

Mixing Black Scholes with SABR

I am new to the whole concept of stochastic volatility so I am experimenting with option pricing. I think the concept is really difficult to understand / grasp. I was wondering if the following ...
3
votes
0answers
161 views

Early Exercise Options and Coin Flipping

This problem was presented in an interview, and I know I got it roughly correct. But I am still not entirely understanding the early exercise component of it: Say I am advertising a game where I ...
3
votes
0answers
109 views

Pricing Options on Fixed Income ETFs

The market for trading options on fixed income ETFs like HYG has become increasingly prominent in the past couple years, but I've been unable to find any discussion related to the pricing methodology ...
3
votes
0answers
170 views

Relationship between Implied Volatility Curve Derivatives and the Underlying's Moments

Very probably this question has been posed before, so if someone can pose the link to the relevant question, it would be appreciated. What is the relationship between the implied volatility skew and ...
3
votes
0answers
74 views

ODE Solution in Carr's Randomized American Put

In Carr's 1998 paper Randomization and the American Put, he sets up the following ODE for the value of an American put with expiration given by the first jump time of a Poisson process with rate $\...
3
votes
0answers
133 views

Example of optimal delta hedging in G. Barles, H.M. Soner option pricing paper

There is a paper Option pricing with transaction costs and a nonlinear black-scholes equation by Guy Barles and Halil Mete Soner. And there is a section about optimal (delta) hedging, which I do not ...
3
votes
0answers
128 views

Approximate asian geometric option with Heston

I am trying to implement Theorem 1 from this Journal in RStudio. The journal says the it is possible to find a approximate price of a geometric asian option in a Heston setup this way: $$X_{1cGAO}=e^{...
3
votes
0answers
498 views

Pricing a Power Contract derivative security

I'm trying to price a "power contract" and would appreciate guidance on the next step. The payoff at time $T$ is $(S(T)/K)^\alpha$, where $K > 0$, $\alpha \in \mathbb{N}$, $T > 0$. $S$ is ...
3
votes
0answers
203 views

How to find the upper bound of a digital option given some market data?

Given the price of a call equals to 5 with Strike 100, please find the upper bound (sup) of the digital option with strike 105. I am not sure about the solution, but I write the condition like this, ...
2
votes
0answers
50 views

Pricing of future options

I have the following question on futures options: There is a Black’s model, which is a variant of the Black-Scholes formula that is used to price stock options. The Black’s model prices future ...
2
votes
0answers
38 views

what is the state of the art method for hedging barrier options?

I want to create my own Barrier options for some security, I want to trade. I did some literature review, and found a static replication method, and many dynamic replication methods. I want to know ...
2
votes
0answers
71 views

Does an option need to be tradable for Black Scholes pricing formula to hold?

Given the classic Black-Scholes model, e.g. $dS(t)/S(t)=rdt+\sigma dW^{\mathbb{Q}}(t)$ with $S(0)=S_0$ and $dB(t)=rB(t)dt$ with $B(0)=1$, whereby $r$ and $\sigma$ are constants and $\mathbb{Q}$ ...