Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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Fitting parameters given an inverse function. (Orosi, 2015)

In trying to replicate Orosi's (2015) 5-parameter implied volatility model, but I can't wrap my head around the parameter fitting procedure Orosi proposes. My main goal is to calibrate the model to my ...
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4 votes
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Characteristic function of the Bates model

I have a misunderstanding concerning the derivation of the SVJ model : Firsty,I understand how to reach the final differential equation from : \begin{gather} dS_t = (r - q - \lambda t (e^{m-\frac{\nu}{...
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Price difference digital option : constant vol vs local vol

I got the following interview question: Consider a digital option, it will be priced by using two approaches: 1)constant volatility; 2)local volatility. At the strike, both volatilities are equal. (...
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2 votes
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Do single name stock option volatility surfaces exhibit steeper volatility smiles after stock price crash episodes?

In index options, there was not much of a smile (on the put-side) until the 1987 market crash. I'm wondering if the same applies to single name stocks? That is, do price crashes in individual stocks ...
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5 votes
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Integrated Delta does not seem to be smooth (ATM, Heston)

I am interested in an integrated call option that removes the dependence on time, $$I(S)=\int_0^\infty C(S,t)\text{d}t.$$ Because the value of a call option is a smooth function, I expect this ...
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-1 votes
2 answers
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One touch UP no touch DOWN, One touch DOWN no touch UP [closed]

I was reading about exotic options and I came across something new. One touch down no touch up option and the other one I saw was One touch up no touch down option. I would like to understand how it ...
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2 votes
1 answer
131 views

Probability of touching short call strike and not touching touching short put strike of a short strangle?

I just came across a blog post. I believe the answer is a correct approximation: http://tastytradenetwork.squarespace.com/tt/blog/probability-of-touching-both-sides I modified the question in the post ...
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2 votes
2 answers
157 views

Gaussian copula calibration to option price

I have an "exotic" option that is a function of two interest rates (say 3m Libor at 1y maturity and 2y maturity). I assume both the rates follow sabr model (already calibrated to vanillas), ...
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2 votes
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How to compute the Present Value of this path-dependent option?

I have an option whose payoff depends on its value at two times $T_1$ and $T_2$ as follows. $$V(t) = \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B} (S(T_2)-K)^+)],$$ where the stock price follows the GBM ...
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Valuing a call option that is issued today, exercisable after 2 years from the issue date and expires 3 years after the issue date

if we assume: Current price: $0.25 Exercise price: $0.25 life: 3 years Risk free rate p.a: 0.2% volatility p.a: 85% The option cannot be exercised within the first 2 years, after 2 years, it is ...
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Option pricing under Vasicek, CIR, H-L and BDT model

I have implemented and calibrated recombining trees on Excel for the Vasicek, the Cox-Ingersoll-Ross, the Ho-Lee and the Black-Derman-Toy model. I now would like to price some options with these ...
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Discrete geometric asian option, analytic vs MC

I am attempting to price a discrete geometric Asian option using both the closed form formula (can be found in section 3.2.2 of 'Monte Carlo methods in Financial Engineering' by Glasserman) and an MC ...
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1 answer
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Floating lookback put, MC vs analytic

I am attempting to price a floating lookback put using the analytic formula. (eg. can be found in Shreve's vol II stochastic calculus section 7.4 or on Wikipedia) and wish to confirm the result by ...
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3 votes
1 answer
241 views

Bergomi Volatility Model

I was studying on the Bergomi volatility model(using forward variance represented as $\xi_{t}^{T}$).However I don't understand how the author passes from the sde to the first step by only integrating ...
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HNGARCHFIT in R (No standard deviations or P values printed)

When I estimate an HN-GARCH model using the hngarchfit() from the fOptions package in R, only the coefficient estimates are printed. There are no standard deviations or P-values printed. Does anyone ...
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3 votes
2 answers
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How do you derive this Carr-Madan-like equation?

How do you derive equation (3) below? The equation is tagged as equation (11) in this paper: http://janroman.dhis.org/finance/IR/Heston%E2%80%93Hull%E2%80%93White%20Model%20Part%20I.pdf There are ...
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Correlation effect in Quanto options

My question will probably be stupid but here it is. I try to understand the effect of the correlation between exchange rate and underlying in a quanto option. And to have a non-precise understanding ...
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1 answer
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Valuing Conditional "All Or Nothing" Multi Asset Options

I would like some insight as to how to value modified rainbow options on multiple assets: For example: A multi asset option, Call GOOG with $S_t$ \$1600 that you may exercise if and only if you also ...
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1 vote
1 answer
193 views

Why does the price of an option increase with increasing Rho?

I was wondering why the price of an option increases with Rho (price change for a derivative relative to a change in the risk-free rate of interest). I found this explanation on a website: "Each ...
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2 votes
1 answer
131 views

European call option lower bound derivation by Black-Scholes formula [closed]

Derive the lower bound of european call options: $$C(S, t)\geq[S-e^{-r(T-t)}K]^+$$ I know how to derive it using put-call parity, but is there any way to derive from Black-Scholes formula?
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Hedging Options assuming a non-constant Yield Curve

I have read most of Shreve's Stochastic Calculus for Finance II. In it, the author prices various option types assuming an interest rate that is constant with respect to time. We can expand this model ...
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What exactly are the “bounds” in arbitrage bounds?

Wikipedia’s article on arbitrage bounds is loaded with jargon, and thus requires a lot of prerequisite knowledge to understand what should be a basic definition. What exactly are the “bounds” in ...
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2 votes
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130 views

GARCH Option Pricing in R

I am trying to code a GARCH option pricing model in R. I am still new to R so this does seem a bit complicated. I want to estimate an asymmetric GARCH model as well as an EGARCH model. This I have ...
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4 votes
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Bates Model on Quantlib

I am actively trying to price an option using bates model on Quantlib.However,when I input my volatility I find the same Black Prices with the basic Heston Model.I wanted to know if my code was right. ...
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1 answer
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option pricing formula for $S_{t}=S_{0}+\mu t+\sigma B_{t}$ where r = 0

I have been on this for hours and it's not getting me anywhere. Any help is so highly and deeply appreciated. A call option with strike $K$ and expiration $T$ pays $C_{T}=\left(S_{T}-K\right)^{+}$ at ...
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2 votes
0 answers
122 views

HNGARCH Option Pricing in R (How to loop)

I am having difficulties when using the HNGOption program in R. The program will only run for 1 specific option price, meaning that I would have to manually insert strike price etc. and this would ...
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2 votes
1 answer
100 views

How to approximate a delta using monte carlo methods and finite differences via Higham's book?

I'm currently taking a Mathematical Finance module at University and one of the recommended texts is “An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation” by D.J. ...
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-2 votes
1 answer
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One-Period Binomial Model

So, I'm required to consider the one-period Binomial market model for a particular question. We're told that the savings account is \$1 at time 0 and \$β at time 1. The stock price is given by S0 = 1 ...
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1 vote
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Calculating E^2[σ^2] where σ is a GARCH(1,1) Proces

Given that α =0,113079 β = 0,873884 ω = 0,0000081 Need the calculate a call price using garch volatility I alsa calculated the kurtosis = 235 enter image description here: https://www.researchgate.net/...
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3 votes
1 answer
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EMM for Bachelier model

The stock price is assumed to evolve as $S_{t}=S_{0}+\mu t+\sigma B_{t}$, where $S_{0}>0, \mu>0$ and the process $B_{t}$ is Brownian motion. The saving account is assumed to be $\beta_{t}=e^{r t}...
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1 vote
1 answer
193 views

Replicating Portfolio / Complete Market / Attainable Claim

Attempt So Far: 1) First Part: I have shown that the market is arbitrage-free since the only possible portfolio for which $V_1^h\geq0 \ $ given that $V_0^h=0 \ $ is $h=(0,0,0)$ and this clearly ...
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3 votes
2 answers
280 views

EPE for interest rate swap

Hey how to calculate Expected positive exposure in the case of interest rate swap? Assume that I simulate $M$ interest rate paths for time grid $0=t_0\le t_1 \le ... \le t_N = T.$ What is the ...
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1 vote
0 answers
164 views

Index CDS Option (Spread Quoted) - Black's Formula

I have looked at the question and answers here and I have read Chapter 11 of Dominic O'Kane's book Modelling Single-name and Multi-name Credit Derivatives. The book is very clear and has some in-depth ...
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8 votes
1 answer
532 views

Bermudan Swaptions - Payer vs. Receiver (LGM)

There is abundant literature discussing the pricing of Bermudan swaptions and the relevance of single-factor Markov-functional models (e.g. LGM) versus multi-factor market models (e.g. LMM). From a ...
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1 vote
1 answer
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Calibration of Heston model with stochastic short rate

I have following Heston model with stochastic short rate: \begin{eqnarray*}dS\left(t\right)&=&r\left(t\right)S\left(t\right)dt+\nu\left(t\right)S\left(t\right)dW^{S}\left(t\right)\\dr\left(t\...
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1 vote
3 answers
187 views

How to price a call option with long maturity (5 to 10 years)

I am trying to find the industry accepted method on how to price a long term American call option (maturities 5 to 10 years) on an underlying which is an accumulation fund (so no dividend payouts) ...
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2 votes
1 answer
118 views

Girsanov transform when drift coefficient is a function of the stock price

I'm working my way through an elementary stochastic calculus textbook. I'm having trouble with one of the questions: Bachelier type stock price dynamics. Let the SDE for stock price $S$ be given by $...
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0 answers
74 views

How smooth is Black-Scholes?

For each variable $(S,T,K,r,q,\sigma)$ in the Black-Scholes formula, how many times can you take a partial derivative? Adjacently, is the nth order greek for some variable a constant? Thanks
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1 vote
0 answers
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Valuing American Options using Tilley algorithm

Hey I want to implement Tilley's algorithm (Valuing American Options in a Path Simulation Model by JA Tilley, 1993) to price american options. Where can I find implementation of this method in any ...
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1 vote
1 answer
255 views

Least Square Monte Carlo Longstaff-Schwartz method implementation problem

While trying to implement the Least Square Monte Carlo (LSMC) method by Longstaff-Schwartz I came across an error I am not quite sure how to fix. The method uses a regression method (be it Multiple ...
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2 votes
1 answer
108 views

Variance swaps and the Log-Moment formula

I was looking at the paper of Raval and Jaquier The Log Moment Formula For Implied Volatility available here : https://arxiv.org/pdf/2101.08145.pdf On the page 4 they wrote(with $<logS>_T$ and $&...
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-4 votes
1 answer
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Black Sholes Options Pricing Clarification Questions [closed]

I am interested in pricing American Call and Put Options using BSM and I am new to exploring options prcing. I have some questions here that would really remove the confusion I have on how to more ...
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1 vote
0 answers
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Black 76 and Asian Style Options on Shaped Power Futures

I am attempting to price a monthly lookback option on the gen-weighted average price of power at a particular solar plant over a given month. If the option settles at hub H, am I right to shape the ...
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1 vote
0 answers
65 views

Any research paper further studying the conclusions given by Derman Regimes of Volatility

As we know Emanuel Derman mentioned 3 different market conditions where sticky delta, sticky strike, and sticky implied tree are relatively best suited. Are there any relevant research paper further ...
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0 votes
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Volatility for options pricing: fixed window or match maturity?

When calculating the volatility or covariance matrix of stock returns for the purpose of pricing a vanilla option on an underlying, it is difficult to choose the window over which the volatility ...
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6 votes
1 answer
173 views

Likelihood ratio and pathwise sensitivity method for coupled SDEs

I have two coupled SDEs \begin{align*} dS_t=rS_tdt+V_tdW_t^{(1)},\\ dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\ \end{align*} where $W_t^{(1)}$ and $W_t^{(2)}$ are independent Brownian motions, initial input data ...
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0 votes
0 answers
141 views

What is the relationship between Vanna and Gamma?

I'm trying to build a crude model for the effects of delta hedging on major indices like the S&P 500. My background is more in pure mathematics so a lot of this stuff is new to me. That said I ...
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2 votes
0 answers
205 views

Implementing a Variance Swap Hedging in R

I am trying to compute a hedge for a variance swap, in a simulation. Fo that I am using the following equation:\begin{align*} E^Q\bigg(\sum_{i=1}^n \bigg(\frac{S_{t_{i}}-S_{t_{i-1}}}{S_{t_{i-1}}}\bigg)...
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3 votes
1 answer
242 views

What is the Radon-Nikodym derivative in the Heston model?

It is clear to me that $$ \frac{dQ}{dP} = e^{-\lambda W_T-\frac{\lambda^2}{2}T}$$ is the Radon-Nikodym derivative that defines the change of measure in the framework described by Black and Sholes. But ...
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0 votes
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Is the market price of risk deterministic or stochastic in the Heston model?

I am recently digging into the Heston model and I have noticed that every author refers to the market price of risk simply as $\lambda$, or sometimes it is more clearly specified to be bi-dimensional ...
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