Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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15 views

Risk-neutral price of $H=e^{X_T^1+X_T^3}$

Let $B=(B_t^1,B_t^2,B_t^3)$ is a $\mathbb R^3$-valued Brownian motion. Let $r_t$ (risk free rate) be bounded and deterministic. Let consider the DISCOUNTED market $$dX_t=\frac52dt+2dB_t^1-dB_t^2-dB_t^...
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85 views

Intuition for consistent Derivative Prices under different Numeraires and Measures

This is essentially the Fundamental Theorem, however I am not asking for a thorough proof, I am more interested in the general intuition. In words, it makes sense that whatever your unit of account (...
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77 views

To use daily volatility or annual volatility

From Joshi's Quant Interviews books: The statistics department from our tell you that the stock price has followed a mean reversion process for the last 10 years, with annual volatility 10% and ...
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78 views

Simulation Heston Model, markovianity

I am trying to simulate the instanteneous volatility of a Heston process. My equations are the following : wealth process: $$dX_t = r_t X_t + \theta \sqrt {V_t} u_t dt + u_t dW_{1t}$$ Volatility: $$...
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34 views

Binomial Model Is Leading to More Expensive Options as Number of Periods Grow

I've coded up a binomial model. It spits out the right numbers that the book I'm currently reading is using for the given inputs. For example, Stock Price 100 Strike Price 100 Number Of Periods 3 ...
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1answer
56 views

Making portfolio Delta and Gamma neutral using 2 derivatives

We have an option portfolio with delta =2 and gamma 3 and we want to making this portfolio delta and gamma neutral using two derivatives D1 and D2: ...
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1answer
38 views

Historical volatility calculation to price options with the Black-Scholes formula

I'm looking for a reference algorithm for calculating historical volatility to price options. I know there are several volatility calculation models that use the time series of the underlying's ...
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2answers
69 views

Are there volatility models dependent on returns?

When I look at the relationship between volatility and price, I see a clear negative correlation as shown in this figure (SPY and VIX prices today looking back 1 year). The common volatility models (...
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30 views

Do daily returns from a distribution with skew and/or kurtosis lead to options implied volatility skew?

I've been trying to price a call option using a Monte Carlo approach with the specific goal of showing implied volatility skew. I'm using the sinh-arcsinh transformation to make the random numbers I ...
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118 views

What's the point of having an accurate option pricing model?

Just curious what's the actual reason of having an accurate option pricing model? For e.g. an option pricing model fits the volatility surface incredibly well, then what? Do practitioners actually use ...
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51 views

Formula for the discounted payoff of a digital option

In "Heard on the Street" it states that the expected discounted payoff of a digital option is $$H\exp^{-r(T-t)}N(d_2)$$ where $H$ is the payoff of the option, the exponential is the discounting. ...
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44 views

Differences in bull put spread option strategy

I am supposed to construct a profit and loss diagram for a bullish spread strategy: −1put($X_{1}$) + 1put($X_{2}$) and compare it to the profit and loss diagram for the strategy: −10put ($X_{1}$)+ ...
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34 views

Where could I get European non-dividend option data

I am pretty new to option pricing. I got a task asking me to price a stock option, which should be an European non-dividend option, and compare my price to its quote. I used to use TSLA data ...
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95 views

Nonlinear Black-Scholes model Vs linear Black-Scholes

I am working on a project related to Nonlinear BS partial differential equation, with terms for transaction costs and/or discrete hedging. I have two questions: Is there any exact solution to the ...
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140 views

Mark Joshi uses forward price to price an option that pays $S_t^2-K$ if $S_t^2>K $ and zero otherwise? Why can we do that?

The following question is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $6.6$ Suppose a stock follows geometric Brownian motion in a Black-Scholes ...
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137 views

In Carr-Madans option pricing method, why do they use FFT?

In the famous fourier option pricing method by Carr-Madan, (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.348.4044&rep=rep1&type=pdf), the crucial formula is They evaluate this by ...
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61 views

Show that $Ae^{rt}$ is a solution of the Black-Scholes equation. Why should this be so?

The following is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, exercise $5.6$. Question: Show that $Ae^{rt}$ is a solution of the Black-Scholes equation. ...
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161 views

Periodic functions when determining No Arbitrage price

Is it possible to value a T-claim which has a periodic component? For example a claim such as $X = cos(S(T))$. We assume here that $S(T)$ is the stock price derived from the dynamics $dS(t)=rS(t)dt+\...
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37 views

Confusion about bid- ask- and last-prices from option prices data

I’m struggling with the interpretation of quoted option prices I obtained from Bloomberg. The call options prices are available for a daily time series with different strikes at a given day. I ...
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2answers
74 views

Price American call equal to price European call (non-dividend-paying stock)

Let $\tilde{C}_K(t,T)$ be the value (price) of an American call option at strike $K$ and maturity $T$, and $C_K(t,T)$ the value (price) of a European call option at same parameters. For a non-...
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59 views

Formula for quantiles of swaprates in the 1-factor Hull-White model

Is there a closed formula to approximate the quantiles of swaprates in the 1-factor Hull White model? Background The Hull-White is a Gaussian model for the short rate. Its mean and covariance ...
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143 views

QuantLib Python: caplet/swaption pricing under dual curve

Is there a way to price caplets/swaptions in QuantLib python (v 1.6.2) under dual curve i.e. pass projection curve for forwards and discounting curve for discounting the cash flows? Goutham has an ...
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181 views

Monte Carlo option pricing with R

I am trying to implement a vanilla European option pricer with Monte Carlo using R. In the following there is my code for pricing an European plain vanilla call option on non dividend paying stock, ...
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1answer
55 views

Arbitrage opportunity between two call options with strike price \$40, \$30 and cost \$4, \$3 respectively?

Question: Given two call options $c_1$ and $c_2$ with strike price $30$ and $40$ respectively. If $c_1$ costs \$3 and $c_2$ costs \$4, is there an arbitrage opportunity? My attempt: Short $c_2$ and ...
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41 views

Graph of European call option value versus future price

Given a standard European call option on a non-dividend-paying stock. Draw the graph of call price at time $t$ versus the future price $F(t,T)$. The future price $F(t,T)$ is observed at time $t$, ...
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66 views

Delta of an option which is approaching expiration when stock price decreases

The following is an interview question. It is 10 months since you sold a one-year European call option to a customer. You have been delta-hedging your exposure to the written call since it was sold....
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78 views

Why are put and call options worth the same despite that put has no upside whereas call has unlimited upsides?

The following is an interview question. All Black-Scholes assumptions hold. Assume no dividends. Consider a standard European call and a standard European put on the same stock. Assume that each ...
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74 views

How do you price an option on multiple things>

Suppose, for simplicity, I want to cover the U.S. stock market by buying ETFs for the Russell 1000 and Russell 2000. But I want to overweight small cap, so the Russell 3000 won't do. Also, let's ...
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81 views

Martingale representation of European option

Let stock price $S$ satisfy $$S(t)=S(0)e^{(\int_0^t\sigma(s)dB_s-\frac{1}{2}\int_0^t\sigma(s)^2ds)}$$ I want to calculate the Martingale representation $V(t)=E(F|F_t)$ of European option with strike ...
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106 views

Pricing European call with Feynman-Kac

I am trying to calculate the solution to the Black-Scholes (BS) equation using the Feynman-Kac (FK) formula for a simple European call. According to FK, the solution to BS is the discounted average of ...
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2answers
482 views

Risk Neutral and Real World Valuations using Monte Carlo

Assume I'm an investor that wants to sell exotic put options. No one else is selling my kind of put option, so I need to determine my own "Market Price" through Monte Carlo simulation. I know that by ...
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208 views

Is Local Stochastic Vol needed in order to price barrier options?

I'm trying to understand when it is appropriate to use stochastic local volatility models rather than local volatility ones. More precisely, for which products is it appropriate to introduce a ...
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161 views

Cos Method in Finance / Practice

A lot of my professors advised me on doing an undergrad thesis that has something to do with the "relatively new" cosine method (~10 years). What applications are there in Finance of the FFT/Cosine ...
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76 views

Black Scholes theta as function of time to maturity

I would like to understand why the Black and Scholes greek letter theta for european call option behave in the following way: as time to maturity is far away (right part of the x-axis in the the ...
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30 views

Fourier transform method: the reason why it's beneficial to put points of interest on the middle of the “time-domain”?

I was trying to solve European option pricing problem using Conv method (introduced by Lord in 2008 https://pdfs.semanticscholar.org/0632/460bd50b2151f74ac40028df4cc60e73a884.pdf). The final step of ...
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1answer
60 views

Variance of a spread for options on spreads

I was reading the paper: https://people.umass.edu/nkapadia/docs/Negative_Vega.pdf In the equation $(5)$, he is defining the variance of the spread as: $$\sigma_1^2S_1^2 + \sigma_2^2S_2^2 - 2\...
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56 views

Solving for unknowns in Black-Scholes equation using Python

I have defined the Black-Scholes equation in Python as follows: ...
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48 views

Call price in case of AOA

I have this exercice, and for the last question, i tried to say that with lower bound, $C > S_0 - Ke^{-rT}$ which is $-8$ something but it doesn't make sense so i don't know what to do. Could we ...
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154 views

Floating Strike Lookback Call Option

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate $r$, stock drift $\mu$ and volatility $\sigma$). If $r=\...
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26 views

Option Bounds in a risk-averse incomplete market

I was reading the article "On option pricing bounds" by Ritchken(1985). It uses linear programming to determine options upper and lower bounds. Given a single period model, the stock price will have ...
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50 views

Why does the price of a butterfly spread increase are rate exponential [closed]

I know that stock prices are assumed to be Stochastic processes that follow Geometric brownian motion. The expectation of stock prices at time T given stock price at time 0 is: $e^{-rT}S_0$. However, ...
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74 views

Replicating portfolio with stock, bond and call option

I am trying to interpret: I am having trouble interpreting the replicating strategy: Context: $\phi$ is a generic payoff function, 0 < S < $\infty$, assumed throughout to be twice ...
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81 views

Alternative derivation of Black Scholes by Merton

I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I ...
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29 views

Reading this ichimoku cloud how do you read this wdfc chart?

Having trouble reading the charts as the breakouts aren’t clear What do you see in this chart?
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45 views

Quesion about the VBA function of continuous cap look up [closed]

Here is the VBA function to calculate the cap price Can anyone tell me what is N and t0? From the textbook, N = the number of reset (or payment) dates and t0 = time until the first reset date. But ...
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Pricing Equity Swaptions

Consider a swaption to enter into a standard equity swap as a fixed-rate payer, equity receiver, in which the notional principal is fixed. If the strike is K. the underlying swap starts at time 0=n ...
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101 views

A relatively useless but fun question for quants and physicists

Suppose I am a trader travelling in a very fast spaceship. What kind of SDE and corresponding PDE should I jot down and work with in order to ensure that there is no arbitrage (or if there is ...
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1answer
60 views

European Call option replication

An asset $S_t$ is evolving according to the Black-Scholes model. We want to replicate a call option on this asset by holding Delta units of the asset at every time. I use a Monte Carlo algorithm to ...
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44 views

Strictly increasing asset price under a risk-neutral probability measure?

I am reading a paper on option pricing under jump processes in continuous time. There is a section labeled examples where the authors work under a risk neutral probability measure and derive option ...
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69 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 ...