Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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How to price a risk reversal for common dice gain with chance to re-roll

I was just thinking about an extension to the common dice throwing interview expected value question: Question: Imagine a game where you throw a die and get a payoff equal to the number shown by the ...
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3 votes
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Independent variable in pricing of strongly path dependent options

I am reading Paul Wilmott on quantatative finance where he discuss the pricing of strongly path dependent options.The payoff at expiry T depends on the path taken by the asset in the sense that it ...
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Valuation of non-deliverable option

What is the difference between valuation of deliverable and non-deliverable European options? I am not asking settlement-wise, but daily valuation. Will Black-Scholes be used for both?
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Derivation of static replication formula

I know that a way of computing the price of a derivative paying $S^2$ at time $T$ is by making use of the following strategy: $V=\int_{0}^{\infty} s^2 \frac{\partial^2 C}{\partial K^2}(K=s)ds$ Where $\...
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How is delta defined as a unit?

This is going to be a embarrassingly basic question. But the answer seems to be hard to find. What does, say, selling, $d$ delta of calls mean? How is the "delta" defined? I am not asking ...
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3 votes
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Reduced volatility in local stochastic volatility model

in Local Stochastic Volatility models I always read or hear "first the stochastic volatility model is calibrated to reduced vols and then the local volatility model corrects it" also I head ...
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Dice question - expected winnings of rolling dice $2$ times

Typical trading interviews consider gambling problems such as rolling a dice and winning its face value. The expected winnings are $\\\$3.5$, $\\\$4.25$, $\\\$\frac{14}{3}$ for one throw, two throws, ...
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Question about pricing forward start option with Heston Monte Carlo

I'm trying to price a forward start option with payoff $\Big(\dfrac{S_{T_2}}{S_{T_1}}-1\Big)^+$ with Heston Monte Carlo. Heston Model: $$ dS_t = rS_tdt + \sqrt{v_t}S_tdW_t^1$$ $$ dv_t = \kappa(m-v_t) +...
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FX option premium conversion from one currency to another

Suppose we have an FX option with the underlying FX rate $X^{FOR/DOM}$ and suppose we used a Black like formula to get the price, which is given in domestic currency. How can one simply convert the ...
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Montecarlo pricing

I have some problems with the Montecarlo simulation to price a generic Call option. I want to explain something regarding MC simulation with a simple cases, and after that I am going to talk about my ...
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No free Lunch and weak-star topology

The no free lunch is stated as follows What is the significance of the weak-star topology here .Also as far as I understand the weak-star topology is defined on the dual of a Banach space.So what is ...
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No free lunch with bounded and vanishing risk

I am reading a book which states 'No free lunch with bounded risk as follows where $\tilde{V}_t$ is the discounted value of the portfolio.Then it states the following theorem EMM is the equivalent ...
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Path-dependent options valuation

Assume that we have an arbitrage-free and complete market. The well known formula for the arbitrage-free price of an attainable derivative $X$ at time $0 \leq t \leq T$ is given by: \begin{align*} V(t)...
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How to identify between Analytical, Numerical and ML Model based option pricing? [closed]

I am new to Quantitiative Finance. Coming from Computer Science domain, I wanted to clear the key distinguishing factor between analytical, numerical and ML based models for option pricing. As far as ...
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Method to retrieve implied density for a mixture of local volatility model

Given a mixture model of two local volatility models, the price for an option is given by: $$V(K,T) = p V_{loc1}(K,T) + (1-p) V_{loc2}(K,T)$$ where $V_{loc}(K,T)$ is the price of the option given a ...
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How to get Risk-Neutral Drift for Trading Volume from Time Series

I am trying to price an option with Monte-Carlo simulation, where the payoff depends on some constants and a time-series (trading volume) which I model to follow a GBM. Now if I understood it ...
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Why Vasicek model on a tree is a bad choice for pricing American option on credit prepayment?

I have an American option on a credit prepayment, i.e. the holder of the option can prepay the remaining credit if the interest rate falls below the initial strike. The pricing of this option was done ...
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What do Future price and Forward price represent

In Shreve's Finance and Stochastic calculus, definitions are: Forward Price: The $T$-forward price $For_S(t,T)$ of this asset at time $t$, where $0\leq t\leq T$, is the value of $K$ that makes the ...
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QuantLib Inaccurate - American Put Option with Discrete Dividends

I'm trying to use the QuantLib library to price American options that pay discrete dividends. The call options are priced with good accuracy (generally <0.1% error), however the same inputs for a ...
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QuantLib Two Asset Barrier Option

I am trying to price a two asset barrier option where each asset has its own barrier and both barriers have to be met for the payoff. The experimental TwoAssetBarrierOption class seems to accept only ...
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1 answer
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Importance sampling for Monte Carlo with local volatility in practice

I am given a diffusion with a local volatility to price barrier options: $$dX(t)=X(t)\mu dt+X(t)\sigma(t,X)dW_t$$ I want to use Importance Sampling to price barrier options "far" out of the ...
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option pricing: monte carlo simulations that include expected return [closed]

In the book Derivatives Markets (McDonald, 3rd edition), there's a chapter on Monte Carlo valuation of option prices. It starts with simulating stock prices (p578) with the following equation: ...
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Why calibrate volatility Models to volatility surfaces rather than underlying's historical price data?

I'm trying to grasp the rationale for calibrating stochastic volatility models (i.e. Heston model) to empirical IV data from market prices. Doesn't this assume that the options are fairly priced and ...
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142 views

Early expiry option implied volatility

I’m new to this forum so first of all I wanna welcome everyone here. I am a commodity trader, mostly covering option books (vanilla and structured one) and I would ask more expert people how they can ...
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8 votes
1 answer
270 views

Realised variance under simple rough volatility model

Using the Mandelbrot-Vann Ness representation of fractional Brownian motion in terms of Wiener integrals, increments of the logarithm of realized variance $v = \sigma^{2}$, under the physical measure $...
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Compound Option Monte Carlo Methods Reference Request

I am interested in valuing option where the underlying security is similar (not exact) to an option. The underlying security might be a Preferred Share (option to convert into common share plus ...
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1 vote
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How to extract volatility smile implied by a mixture model?

If one had to extract the implied volatility smile from a local volatility model, one can simply use the relationship: $\sigma^2_{imp}(t, x)T = \int_t^T \sigma^2_{loc}(s, x)ds$ with $\sigma_{loc}$ the ...
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2 votes
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Relationship between VIX and Vega

Assuming that all other factors (such as underlying price, strike price, etc.) remain unchanged, I want to see how a spike in VIX would affect the price of the average call option? Assume Vega is ...
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Question on model recalibration upon a spot shift scenario analysis

I am given a plot of the fair value of a complex derivative against a scenario spot shift for a range odd possible shifts (-40% to 40%). Let us say the pricing model is a local vol model. I am unable ...
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2 votes
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Delayed Settlement Option- how will values in Black Scholes change

If there is an option that expires a year from now, but is settled after 2 years, how would the Black Scholes formulation for such a situation look like? Will the risk free rate now be for 2 years or ...
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change in implied volatility with respect to change in spot

It's clear that IV increases as spot decreases, and vice-versa. In pricing an option, is there any model that is useful in estimating the change in IV with change in spot price? For example, if the ...
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Monte Carlo Simulation of GBM Process has a Very High Variance - Explanation Needed as to why?

I use Geometric Brownian Motion (GMB) to simulate a share price from March 24, 2020 to March 24 as follow: \begin{equation} S_t=S_{t-1}exp((rf-0.6\sigma^2)*(2)+\sigma*sqrt(2)*\mathcal{N}(0,1)) \end{...
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Derivation of Bergomi model

In Stochastic Volatility Modeling, L. Bergomi introduces in Chapter 7 the pricing equation (7.4) : $$ \frac{dP}{dt}+(r-q)S\frac{dP}{dS}+\frac{\xi^t}{2}S^2\frac{d^2P}{dS^2}+\frac{1}{2}\int_t^Tdu\int_t^...
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Exotics - Combination of different payoffs using Black-Scholes

I'm currently struggling with the derivation of a formula to price the following exotic option with Black-Scholes. The option has the maximum payoff of $(S_T-z)$ and $(y - S_T)$, where $S_T$ is the ...
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Bergomi's model normalisation

On his book https://www.amazon.fr/dp/B019FNKQS8/ref=dp_kinw_strp_1 Bergomi derives a multifactor mean reversible volatility of the volatility such that : \begin{equation*} d \xi_{t}^{T}=\omega(\tau) \...
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4 votes
1 answer
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Understanding Monte Carlo to solve option price with local volatility

I have read this question pricing using dupire local volatility model which seems to have an answer from here https://www.csie.ntu.edu.tw/~d00922011/python/cases/LocalVol/DUPIRE_FORMULA.PDF Both of ...
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4 votes
1 answer
214 views

Why does volatility increase the expense of delta-hedging?

Consider someone that writes a call, and wishes to delta-hedge against it to remain delta neutral. For this to be profitable, the price they sell this option for should be greater than or equal to the ...
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Black-Scholes option pricing [duplicate]

Consider Black-Scholes (B, S) market model. Let $r = 0$ (hence, $B_t ≡ 1$), $S_0 = 0 $. Stock price is described by $dS_t = σS_tdW_t$. Find the price of the option that pays $(S_T^3 - S_T^2 )_+ = max(...
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1 vote
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Fourier transform of a European put

In book The concepts and practice of mathematical finance, in the context of illustrating the stochastic volatility model, the Fourier transform $\hat{P}(\xi, V, T)$ of a European put $P(x, V, T)$ is ...
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1 answer
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SDE into ODE problem

Let S be the solution of the SDE: $dS_t = g(S_t)dt + \sqrt S_tdW_t, \; S_0 ∈ (1, 2)$, where $g(·)$ is a bounded function. Let $τ$ be the exit time $τ = min(t ≥ 0 : S_t ≥ 2 \; or \; S_t ≤ 1)$. Obtain ...
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2 votes
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Finding Option Probability Density Using Local Volatility from Dupire Model

This question is different than pricing using dupire local volatility model and Is Dupire's local volatility model path independent to recover historical option price? I also asked this on Math ...
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4 votes
1 answer
246 views

Deriving the Heston-Hull-White PDE

I'm trying to derive the Heston-Hull-White PDE. The correct backwards PDE is equation (1.3) of this paper on page (2). I will begin deriving the forward PDE, but switching between the two is trivial. ...
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In what cases characteristic function of (log-)price process is known?

Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant ...
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6 votes
1 answer
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How to simulate Levy processes

Hey how to simulate Levy processes? I have no problem with Wiener process and compound Poisson process, I also know how to simulate Variance Gamma process but I have no idea how to simulate for ...
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how can properties of transition matrix be applied in the transcation cost of option

I am currently reading the PP BOYLE's article ' Option Replication in Discrete Time with Transaction Costs' written in 1992. Here is one place i couldn't figure out: Where does that $\widehat{p}$ ...
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2 votes
1 answer
114 views

FX Asian Option Moment-matching in Harmonic case

I need to price a "foreign-paying" fixed-strike Asian (i.e., average) option. Thus, the payoff is: $$\left(\frac{A_T - K}{A_T}\right)^{+} = \left(1 - \frac{K}{A_T}\right)^{+} = K \left(\frac{...
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-2 votes
1 answer
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Why does bull call spread shows higher payoff than bull put spread?

I am trying to compare bull call spread and bull put spread for equity index option. For the options where the put call parity holds, I am getting a different payoff for bull call spread and bull put ...
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3 votes
3 answers
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How to derive a pricing PDE for an asset that follows a mean-reverting process?

I want to derive a Black-Scholes type partial differential equation to price options on an asset that follows a mean-reverting process (Schwartz model). My attempt follows the methodology of deriving ...
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-1 votes
1 answer
170 views

Barriers on structured notes

I asked a question here: Structuring and Customization Thanks to all the contributors. However, I now have a follow-up question. I would like to buy barrier options and I was informed from that post ...
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3 votes
2 answers
365 views

Structuring and Customization

It seems complex derivatives in particular exotic options are not available at any retail broker. Can a regular retail trader get access to these instruments? Maybe through prop firms or banks? ...
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