Questions tagged [option-pricing]
Questions about models for the valuation of option contracts.
1,776
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Calibration and pricing with the Stochastic Local Volatility model
I'm reading the stochastic local volatility model literature, e.g., the Heston Stochastic Local Volatility model (https://ir.cwi.nl/pub/22747/22747D.pdf); but I'm a bit unsure about its calibration ...
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How does the issuer of a Barrier Reverse Convertible determine the coupon?
I am looking into BRC's, and I keep reading about their relatively high coupon rates which are pre-determined by the issuer. However, I can't seem to find any good resources on HOW they pre-determine ...
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Has anyone ever derived an analytical basket option which gives terminal asset prices individually, by asset?
Random thought I had around what would be an ideal analytical basket formula. If the formula gave terminal prices of each asset instead of a single basket price, you could price any number of exotic ...
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Monte Carlo Pricing of Barrier Options - can't figure out where I'm wrong
I'm trying to price a simple Up-and-out Barrier option using Monte Carlo; haven't even implemented the variance reduction but it's already glitching.
The code seems right, but I'm not sure where it's ...
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Computing Delta-Hedged Option Returns
I was reading some papers on delta-hedged option returns and came across an intriguing paper that I found quite interesting.
However, I was a bit confused on the authors' methodology of computing ...
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Volatility Options
I'm working on a scenario needs to price an option on volatility. After googling, there is no clear relevants in this field. VIX options pricing is one way, while seems it takes the VIX (a volatility) ...
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Why are options on Leveraged ETFs cheaper than ETFs — on the same underlying index and expiration?
I had always reckoned that IVs on Leveraged ETFs (LETF) are "increased by the same amount as the leverage i.e. if it is 2x then IV will be 2x. This will essentially double your cost (in my ...
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Extend basket analytic solution (equal weighted) to a various weight basket, also put formula
So I coded up the solution from here: Do basket options have a closed form valuation formula?
Which provides a good solution for equally-weighted underlyings under a Black model. The simplified ...
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Is the market price of an asset always lower than the expected discounted value under the REAL WORLD measure?
The risk neutral measure is often said to reflect the risk aversion of investors. So intuitively, I would think that an asset's expected discounted value should be lower under the risk neutral measure ...
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GARCH option pricing
I have been trying to implement GARCH(1,1) model for pricing call options. Suppose I have calibrated Garch(1,1) model for modelling the conditional volatility using the historical data of an equity ...
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What is Phi in Cox-Ross-Rubinstein Binomial Model?
I have a question regarding the Cox-Ross-Rubenstein (CRR) model (Cox et al.,1979). While I do understand how the model is constructed, I am now trying to put it into code and am not sure how to ...
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Quantlib: day-by-day evaluation of option value
I'm using Quantlib in Python to price an FX option. I'm comparing the result to Bloomberg, to make sure the code is working correct.
I want to calculate the P&L of a certain option trading ...
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Quantlib: Greeks of FX option in Python
I'm using Quantlib in Python to price an FX option.
I'm comparing the result to Bloomberg, to make sure the code is working correct.
I also want to calculate all the Greeks, and eventually use those ...
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What is the name and payoff of this exotic option (where the holder can lock in a price)?
An exotic option is described as follows:
Let $S_t$ be the underlying at $t$. The holder has the option to lock
in the current price during the lifetime of the option, which he does for $S_{t}=50$. ...
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Calculating model-free implied volatility [closed]
I am trying to come up with model-free implied volatility as in Britten-Jones, M. and Neuberger, A. (2000) Option Prices, Implied Price Processes, and Stochastic Volatility, Journal of Finance, 55, ...
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Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$
Assume $(X_t)_{t\geq 0}$ follows an SDE of the form:
$$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$
where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\...
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American option under Ornstein-Uhlenbeck stock price
I came across with the following problem:
For the Ornstein-Uhlenbeck process $(X_t, 0\leq t\leq T)$ with initial
condition $X_0 = x$, find the stopping time $\tau$ that maximizes
$\mathbb{E}[e^{-r\...
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Black-Scholes differential equation rewritten [closed]
I have seen that the Black-Scholes equation
$$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+
rS\frac{\partial V}{\partial S}-rV=0$$
can also be written in the ...
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Analytical evaluation of the following caplet-type product under lognormal assumptions
Let $n \geq 2$, and consider a tenor discretization: $0 = T_{0} < T_{1} < ... < T_{n}$ and associated forward rates evaluated at time $t$, as $L_{i}(t):=L(T_{i},T_{i+1};t)$ for any $i = 0,...,...
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Bloomberg OVML| FX option pricing | Python
Wanted to check if any API for python is available to replicate Bloomberg's OVML.
The objective is to perform FX options pricing for multiple positions, and we are getting stuck in calculating ...
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Pricing binary options under volatility smile
I was asked to show that the price of a digital/binary option $D$ while a volatility smile $\sigma(K)$ is present is given by
$$D= \exp(-rT)( \Phi(d_2) - K \sqrt{T} \phi(d_2) \sigma ' (K))$$
Where $\...
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Interest Rate Options - OTC vs Exchange, vol difference
I understand that Exchange Traded Interest Options (USD Libor 3m or Euribor 3m) trade with a lower volatility than the respective Cap or Floor for an equivalent structure.
Can anyone give any colour ...
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Implied volatility plotted against the strike price in Heston model
How can I reproduce the implied volatility curve (plotted against the strike price) in the Heston model (i.e. the blue line in the graph below)?
What's the equation that I have to set up and solve?
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Comparison of Option-Pricing Models (volatility models) vs Product-Mapping
I scoured this forum, looking for some indicative (updated as of year 2021) comparison of volatility/option-pricing models. There were some, but they seem dispersed and lacking in general details...
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How do trading platforms estimate options pricing P&L graphs?
Using the profit/loss calculator for equity option strategies of a trading platform, it displays estimated P&L curves for some date in the future and across the prices of the underlying with a ...
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Calibrating Heston model using implied volatilities
I'm trying to understand how the authers of a paper calibrated their model.
We got data on European type options on the
S&P500-index period from early 2005 to mid-2009. We have daily data on ...
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FX Euro-American Knockout Option Pricing
this is my first time asking questions here.
I want to look for some calculation method to price a very exotic option.
The FX Euro-American Knockout Option (EAKO) is an option that has an American ...
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How to find state prices?
I am trying to find out how to solve state prices, but I do not know what I am supposed to do, my professor has given a solution to this problem as being (0.060 0.417 0.476), but I can't figure out ...
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Where does 1/2 in Fourier Transform method of pricing options come from?
I am reading Jianwe Zhu's Applications of Fourier Transform to Smile Modeling. On page 26, the author is describing how to use the Fourier tranform to price vanilla European call options. If $f_j$ is ...
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SciPy Calibrating Heston call option
I have been attempting to calibrate my Heston model, but I am running into issues with scipy.optimize module. I have tried various scipy optimizers, but they all return the error "TypeError: can ...
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Quant Interview - Options pricing [closed]
I am fairly new to all this, merely read the first few chapters of "The Concepts and Practice of Mathematical Finance". I recently had a job interview and was asked a simple question about ...
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Bachelier Spread formula - input spread volatility question
I don't have much experience using Bachelier's single factor spread option formula, but I know it takes a dollar volatility of the spread as an input. What I don't know, is that just the standard ...
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How does $(d_2/\sigma) = (1-d_1)$ while deriving the Vanna Formula from BSM? [closed]
Just realized there was a quant finance board, so I figured I'd post it here instead.
I'm trying to derive Vanna from the Black-Scholes Model (BSM) equation, but had a hook up on one of the ...
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Assymetric Rate Distribution
The pandemic has disavowed any notion of nominal rate distributions to being truncated at 0%. However, if Central Banks at Debtor nations are conflicted in that they are incented to suppress interest ...
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Noob Question - Monte Carlo vs BAPM European Option Pricing Discrepancy
It's winter break (happy new year!), and I'm trying implement a few options pricing models (bapm, tapm, monte carlo, Fast Fourier etc.) for practice.
The issue: My BAPM CRR model converges to 8....
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Why conversion shows $\frac{\partial C}{\partial T} > 0$?
I'm reading Dupire's "Pricing and Hedging with smiles" (1993). After arriving at
$$\frac12 b^2 \frac{\partial^2 C}{\partial x^2}=\frac{\partial C}{\partial t} , $$
(note: here $C$ is the ...
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How to find the risk neutral valuation of $P(T_{1})$ und the measure $\mathbb Q^{P(T_{2})}$
How do I find the risk neutral valuation of $P(T_{1})$ und the measure $\mathbb Q^{P(T_{2})}$, where $P(T_{1})$ and $P(T_{2})$ refer to the $T_{1}$ and $T_{2}$ zero coupon bond with $0 < T_{1} < ...
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How to price OTC swaps to hedge non-economic cashflow variability
Suppose we have a stochastic cashflow $X_t$ from a portfolio of contracts with clients. We can simulate from $X_t$ and can calculate $E[X_t], \forall t \in [1,n]$ where $n$ represents the longest ...
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Gamma PnL when hedging with implied volatility - where is the mark to market PnL?
It is well known that hedging with implied volatility involves a PnL:
$0.5*(σ^{2}_r−σ^{2}_i)S^{2}*Γ_{i}dt$
In the Wilmott paper (http://web.math.ku.dk/~rolf/Wilmott_WhichFreeLunch.pdf), they imply ...
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Limit of digital call and put price when volatility goes to infinity
The price a digital call and put in the Black-Scholes model is given by
$$c^d = \Phi (d_-), \qquad p^d = \Phi (-d_-), \qquad \text{with} \qquad d_- = \dfrac{\log S_t / K}{\sigma \sqrt{T}} - \dfrac{1}{...
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Pricing a contract
I'm currently trying to price some different kinds of contracts. I'm stuck on this following exercise, which I can't seems to find a good solution for. The following is assumed:
We are in a standard ...
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Integral of brownian increments
I'm stuck at a problem and I'm not sure on how to proceed. My question is how would one go about and integrate the following
$$\sigma\int_{t}^{T}\mathrm{e}^{a\cdot u}\cdot (W_{u}-W_{t})du.$$
I've been ...
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Assymptotic behaviors of European options
In some of the numerical works on Black-Scholes generalized models, the boundary conditions on the truncated domain taken from the asymptotic behaviors of European call options, which is given by
$$\...
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How do you price an option on fresh corn?
I'm preparing for quant interviews, and I had this question for myself. I'm not actually trading corn options. My goal here is just to better understand how to deal with these kinds of options.
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Can call price increase in falling markets
Say SPX falls so much that there is panic and implied volatility(iv) increases so greatly that OTM call prices are increased during the fall due to high iv
In my observation in historical data this ...
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Pricing of European options on two underlying assets
Is anybody able to give the solution to the following problem?
Suppose we have two assets, each of which follows a GBM process, and where $dW_S$ and $dW_X$ are correlated $(dW_SdW_X=\rho)$.
$dS=\mu_s ...
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Pricing (Single and Double) Window Barrier FX Options
recently I have been trying to understand how to price FX options with single and double window barriers.
Could someone please recommend a source (e.g., book, article, etc.), where I can find the ...
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Continuation value definition in Longstaff and Schwartz
I am going through the paper by Longstaff and Schwartz (2001) on American-options pricing, and something got me confused.
There, in equation $(1)$ the continuation value at time $t_k$, $F(\omega; t_k)$...
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American Option Valuation - Induction algorithm
The price of an American put option is given by
$$V_k = \sup_{\tau\in\mathcal{T}, \tau\ge t_K} E\{e^{-\int_{t_k}^\tau r_sds} (K-S_{\tau})^+|\mathcal{F}_{t_k}\}$$
I found in one book the following:
$$\...
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Unable to link volatility structure to swaption pricing engine
Good morning,
I am trying to link the volatility surface to my swaption pricing engine.
...
