Questions tagged [european-options]

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11
votes
2answers
4k views

Carr-Madan Formula

Really new to financial Maths. I am currently having problems with the Carr-Madan Formula. $$f(S_T)=f(F_t) + f'(F_t) (S_T - F_t) + \int_0^{F_t} f''(K) (K-S_T)^+ \ d K + \int_{F_t}^{\infty} f''(K)...
8
votes
1answer
3k views

Arbitrage opportunity interview question

I have seen this interview question mentioned in a couple of places: There are three call options on the market, with the same expiry and with strikes 10, 20, and 30. Suppose the call option with ...
8
votes
1answer
673 views

Option pricing and mean reversion

In different books one can find a formula for option pricing when we assume that $\ln(S)$ follows a mean reversion process $$ dS_t/S_t=\kappa(\theta-\ln(S_t))dt+\sigma dZ$$ If we calculate an ...
5
votes
0answers
106 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 ...
4
votes
1answer
142 views

How does financial institutions value European options in practice?

I am a little bit confused, or uninformed more truthfully, regarding how option pricing (Europeans only in this case) are handled in real life. Up to now I have acquired some theoretical knowledge of ...
4
votes
1answer
212 views

Pricing the European counterpart from American Options

I have American option prices for SPY and need to calculate the equivalent European option price to use in further calculations. What does it (formally) mean to price the equivalent European option ...
4
votes
1answer
195 views

How do I compute Value at Risk of a European call option?

Consider a European call option on a non-dividend paying stock, where the option has strike K = 100 and expiry T = 0.25, i.e. the option expires 3 months from now. The option is on a single share. The ...
4
votes
1answer
119 views

Pricing a call option with pay-off function max{$S_T - S_{T/2}, 0$}

Pricing a call option with payoff function $C=\max\{S_T - S_{T/2}, 0\}$, where $S_T$ is geometric brownian motion. I appreciate any help! Please close this question if this is a duplicated question. ...
4
votes
0answers
52 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, ...
3
votes
1answer
157 views

Dependency of an option price on time till expiry

I am trying to seek satisfaction when it comes to understanding why the price of an option is dependent on the time until expiry. I have read that the longer till expiration, the more time available ...
3
votes
2answers
183 views

How to calculate implied correlation via observed market price (Margrabe option)

I can't seem to figure out how to do the following: compute the implied correlation $ρ_{imp}$ by using the observed market price $M_{quote}$ of a Margrabe option, and solving the non-linear equation ...
3
votes
2answers
157 views

Conceptual explanation of the relationship between gamma and vega plotted against delta for a European call option

I recently plotted Gamma and Vega against Delta for a European call option and found that the graphs look very similar. This makes sense to me mathematically since the two formulas are pretty much the ...
3
votes
2answers
214 views

Is the European call option delta an increasing function of the spot?

In the Black-Scholes' setting, the delta hedge ratio of a European call option is given by $N(d_1)$, which is an increasing function of the underlying equity spot $S_0$. Does this property hold ...
3
votes
2answers
248 views

What is the Brownian motion in the model for the return of a stock price trying to capture?

I have read that in the derivation of the Black-Scholes PDE, we assume that the return of a stock $S$ is given by $$\frac{dS}{S}=\mu dt+\sigma dB$$ where $\mu$ is the average growth of $S$, $\sigma$ ...
3
votes
0answers
59 views

Deriving the black-scholes formula for the European asset-or-nothing call option

I would like to find out what boundary/final conditions i should be using to find the formula for a European asset-or-nothing call option, as i feel that is where I'm making my mistake. I've read ...
3
votes
1answer
296 views

Option Strategy: Python Implementation Advice

I've been tasked to create and backtest an option strategy. The strategy, in vague terms, is to essentially write call options on securities in a universe, i.e., selling insurance. I have an idea of ...
3
votes
0answers
91 views

Can the vega of ITM call-options be negative when the distribution of the underlyings returns is negatively skewed?

While calculating european call option prices, using the variance-gamma model formula provided by Madan, Carr & Chang (1998), I noticed that, holding all other things constant, the value of an ITM ...
3
votes
0answers
65 views

European call option delta and maximum principle

From comments, the maximum principle for parabolic PDE can be used to show that the European call option delta cannot be greater than 1. I am looking forward to such derivations.
2
votes
4answers
2k views

Prove that the butterfly condition is always greater than zero

I need to prove that the butterfly condition is always positive under no arbitrage theorem. We are constructing a long butterfly using European call options ...
2
votes
2answers
2k views

A simple question: Cost of delta hedging when a call option is sold

Consider a vanilla European call option C, with underlying asset S, strike price K and time to maturity T. Assume that S follows a geometric Brownian motion with mean growth rate of μ and volatility σ....
2
votes
1answer
295 views

European option Vega with respect to expiry and implied volatility

I was told that the Vega of an European option always increases when its time to expiry increases (all else equal). I found this confusing and potentially wrong, but there doesn't seem to be relevant ...
2
votes
1answer
140 views

Is there any useful links for option pricing (american + asian + european) using R

I'm trying to evaluate option pricing mainly american, asian and european options in order to get a plot to measure option valuation in time. Is there any useful references to do that using R ?
2
votes
1answer
166 views

Payoff of European Call Option with Transactioncosts

I was wondering about the following scenario: assume that you have a underlying which trades under a positive bid-ask spread $S^B \leq S^A$ and that there is also a European Call-Option on this ...
2
votes
2answers
198 views

What is a Short Option Hedging Portfolio?

In his book 'Stochastic Calculus for Finance II' Shreve uses the term: 'Short Option Hedging Portfolio' on page.156 (4.5.3). Can someone please explain this term with some kind of an example? It is ...
2
votes
1answer
1k views

How to use the Feymann-Kac formula to solve the Black-Scholes equation

I have the Black-Scholes equation for European option with maturity $T$ and strike $K$ $$\begin{cases}\frac{\partial u}{\partial t} = ru - \frac{1}{2} \sigma^2 x^2 \frac{\partial^2 u}{\partial x^2}-r ...
2
votes
1answer
90 views

European put options

Why is it that for European Puts on Non-Dividend-Paying Stocks, the lower-bound for price is $$p=Ke^{-rT}-S_0?$$
2
votes
1answer
173 views

Finding the True Option Value

Many research papers use differing solution methods to attempt to find the 'true' value of an option whether it be Euro, American, etc. They never mention how they do find the true option value to ...
2
votes
1answer
554 views

How to approximate the Carr-Madan decomposition formula?

I have came across the excellent answer. I'm looking for a dicrete approximation of the Carr-Madan decomposition formula of the function $f(F_T)$ of the terminal futures price by taking a static ...
2
votes
1answer
92 views

zero-shift SABR vega and re-calibration of SABR

I have a zero-shifted SABR model, where I need to confirm if the model is generating the calibration and vega's correctly. The underlying model is the standard SABR lognormal (there is normal as well)...
2
votes
1answer
121 views

Arbitrage when risk-free portfolio earns less than riskless portfolio

I'm currently reading Paul Wilmott's excellent book on option pricing. Near the beginning, he constructs a risk-free portfolio using an option, and a short on the underlying to hedge the risk. I'm ...
2
votes
1answer
93 views

Perpetual Put vs European Put

I am looking at a perpetual put option where the strike price is initially the stock price $K(0)=S(0)$ (i.e. at the money), but the strike price grows at the constant risk-free rate $r$ [i.e. $K(t)=S(...
2
votes
1answer
67 views

How to handle bid-offer spread causing negative estimations of risk-neutral densities from option prices?

I have attempted to estimate the risk-neutral probability density, from CBOE options prices on S&P500 from 2010 to 2016, using the following approximation from Hull (2018). For call options on a ...
2
votes
1answer
69 views

Pricing Secured Barrier Call 2

EDIT: OK, I understand the reasoning for the initial answer now; however, I don't understand why we would need the digital call with a strike of 33 in this question. Is it just there to serve as a red ...
2
votes
1answer
182 views

Fair value for a LEPO (Low Exercise Price Options)

In one of my lecture notes, I stumble across this exercise question: Consider Low Exercise Price Options, LEPOs, (with dividends) in Australia. Using the value at the outset, explain why such ...
2
votes
1answer
3k views

Option pricing: Risk neutral probability calculation

Let $u=1.3$ $d=0.9$ $r=.05$ $S(0)=50, X = \text{strike} = 60$. Assume binomial model Why isn't the risk neutral probability found by solving the following for $p$: $$E[S(T)]=p65+(1-p)45=S(0)(1+r)^T=...
2
votes
1answer
55 views

Difference between modelValue from HestonModelHelper and NPV() from VanillaOption

I am trying to calibrate an Heston model and price vanilla option using Quantlib 1.15 and Python 2.7. I use the following code ...
2
votes
0answers
73 views

Forward spot calculation for a dividend paying no-short sell ETF

I am trying to fit an implied volatility curve for options on the SSE 50 etf that has no borrow (no short selling allowed) and pays a single annual dividend. I originally thought I could use the ...
2
votes
0answers
28 views

Use of second similar European Option as control variate to simulate a European option

I understand the idea and math behind the concept of control variate for the sake of variance reduction, but I struggle to apply it to option pricing. I need to simulate an European option of a stock ...
2
votes
0answers
101 views

Second order convergence for the Leisen-Reimer tree

I have a question about this paper "Achieving higher order convergence for the prices of European options in binomial trees" by Mark Joshi, (Link: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=...
2
votes
0answers
364 views

American Vs European Options behavior with fixed strikes and varying expiration

Following is from page 10 of Fengler (2005), "The prices of American calls for the same strikes must be nondecreasing, Merton (1973), and in the absence of dividends, this property translates to ...
1
vote
2answers
242 views

Why do we need to calibrate vega?

I was going through some paid video on options. The tutor in the video asked the following question: Person $A$ has the following portfolio at the start of April Portfolio of options with vega $20,...
1
vote
1answer
388 views

Valuation of Bermudan option as maximum of relevant European options

Assume I need to price a Bermudan option which can be exercised at following dates: $t_1$, $t_2$, ..., $t_n$. I think that the price of such an option will be maximum of the prices of European options ...
1
vote
3answers
287 views

From Butterfly Price to Probability of $S_T$ Falling within a Range

If a butterfly in the limit represents a probability (by the Breeden-Litzenberger result), what can be said about the relative likelihood of a random variable $S_0$ from the price of a vanilla-option ...
1
vote
1answer
250 views

Isn't Black's approximation for American options inconsistent?

I have came across a formula suggested by Fisher Black (Fact and fantasy in the use of options, FAJ, July–August 1975, pp.36) for approximating the price of an American call written on a dividend-...
1
vote
1answer
392 views

Where are the prices of real European Call options listed?

In order to solve an exercise, I need data from real European Call Options (on the same underlying). It sounds definitely trivial, but actually I feel a bit lost...do you mind giving a link/suggestion ...
1
vote
1answer
58 views

Iron condor with positive vega

I am backtesting this Iron Condor before earnings. In the position summary Vega (Mid Quote) is -3.04\$ but in the chart below (IV vs Profit $) it's clearly shown that a decrease in volatility will ...
1
vote
1answer
112 views

Why futures pricing not calculated like options?

I have read about futures and options ( from online resources ). I only have the basic understanding,not math heavy ( for eg. for Black Scholes I know only the intuitive idea from the khan academy ...
1
vote
1answer
40 views

Domestic and foreign interest rate; dividends?

The spot price AUD/USD is 0.6868, strike price is 0.6915,the 6 month ATM implied volatility for AUD/USD is 7.7% p.a., for the 6 month USD deposit rate is 2.28% and the 6 month AUD deposit rate is 1.45%...
1
vote
1answer
53 views

European put price when stock price is 0 before maturity

According this answer, https://quant.stackexchange.com/a/39298/29108, the European put price (with maturity $T$) at time $t$ for a stock whose current price is $0$ should be the strike $K$ discounted ...
1
vote
1answer
165 views

Upper bound option price in volatility dimension

All, I have a theoretical question about the value of an option when spot price goes to infinity as a function of volatility going to infinity. I know that for a call option: The option value ...