Questions tagged [european-options]

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Proof European call price is always less than stock price. (proof verification)

Let $C_K(t,T)$ be the value of a European call with strike $K$ and maturity $T$ on a stock with value $S_t$ at time $t$. Then for all $t\leq T$ we have $$C_K(t,T)\leq S_t.$$ $\textbf{Proof}$: We ...
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34 views

Calculation of upper stochastic dominance bound of an option

I’d like to calculate, for a call option on a stock, the upper stochastic dominance bound as proposed by Constantinides et al. in their 2002 paper 'Stochastic dominance bounds on derivatives prices in ...
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2answers
228 views

Can strike prices of options be negative?

I am trying to understand the stochastic model of a financial market in one period by [Föllmer, Schied]. They introduce call and put options for the primary assets, which are non-negative. They do not ...
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1answer
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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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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1answer
182 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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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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1answer
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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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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61 views

Delta Hedging Example

I was reading Dynamic Hedging by N. Taleb and in the chapter dedicated to the delta, there is this example of a trader position in options (one-month European call, flat yield curve, forward is ...
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1answer
123 views

Cap price as bond options

I am currently struggling with model calibration of the Hull-White (or Vasicek) model to Caps and Floors. My main problem is that I am confused about the notation. In Brigo & Mercurio (2006, p. ...
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306 views

Compute the price of a derivative

Consider the payoff function \begin{align*} f(x)=\begin{cases} 3 & \text{if }x\leq 30, \\ 33-x & \text{if }30<x<35, \\ -2 & \text{if } x\geq35. \end{cases} \end{align*} How would I ...
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1answer
89 views

Is the european put option an increasing function?

My question is to show that the function $K \rightarrow p(T,K)$ is increasing. T being maturity time,K being any strike and $p(T,K)$ is a european put option. My only approach to this question has ...
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1answer
92 views

Finding todays price of a derivative

Today's market prices for European call options $c(T;K)$ and put options $p(T;K)$ with maturity T and any strike K. Let $B_t = e^{rt}$ be the price of the risk-free bond and St the price of the stock. ...
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3answers
197 views

Can increase in volatility reduce the price of a deeply in-the-money European put?

Hull states that option prices increase with an increase in volatility. I think that statement could be false in a specific scenario: when we are considering a deeply in-the-money European put ...
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1answer
51 views

Sensitivity Approximation - Crank Nicolson

I am looking into a new method of calculating sensitivities starting off with a proof of concept with Black Scholes PDE. Suppose I want to calculate Rho and take the derivative of the PDE (heresy!!) ...
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1answer
40 views

Where can I find the formulas to compute the Greeks for European Call and Put Options Assuming no annual dividend yield?

Every formula I come across involves a $q$ (the annual dividend yield). Where Can I find the formulas to compute the greeks assuming no dividends?
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65 views

What is the reason that an American option has a lower volatility than an European counterpart?

I was researching some plain vanilla option American/Option data and I found some European option which are more expensive than there American counterpart (all other factors are equal, except for the ...
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1answer
51 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%...
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1answer
102 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 ...
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81 views

Valuation of Callable Bonds

Is there any way to price American Callable Bonds (those which can be called on any date before expiration) other than basic CRR interest rate trees, since they won't be accurate enough to give ...
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129 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 ...
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90 views

Kirk Spread Approximation, Greeks by Finite Difference

I am using finite difference on Kirk's Approximation for Spread Options to estimate greeks of the Spread Option. Now this is creating an problem in the estimation of gamma. For at the money options (...
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53 views

Multiple layer Monte Carlo Option pricing

I have simulated 10000 price paths from the SVCJ model under $\mathbb{Q}$ from $S_{t0}$ until $S_{tm}$ and have computed one discounted option price $C_t$. I want to compute the numerical simulated ...
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42 views

Free Call Option [duplicate]

Suppose we follow the assumptions of the Black-Scholes Model, including unlimited borrowing, continuous prices, and frictionless markets. For simplicity assume the risk-free rate is 0. In this world, ...
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1answer
232 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)...
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66 views

Construct a portfolio of European call options with a certain payoff function

My question is similar to Replicate a Portfolio with Given Payoff but I am not quite sure how to apply this to my problem. A portfolio of European call options on an asset $S_T$ has a payoff ...
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2answers
320 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 ...
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42 views

Historical options data for FX/FI

I know that my question is quite large and that quite a lot of questions already deal with the options data. However most of questions deal with options on American equity markets. Could you ...
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76 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, ...
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302 views

What is the second derivative with respect to price of a put option? [closed]

What is the reasoning/meaning behind the second derivative of a put option
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122 views

Is American option price lower than European option price?

I used to think under the same condition, the American option is always more expensive than the European option, because American option can be exercised at any time (has more rights than European ...
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1answer
134 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 ...
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375 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 ...
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1answer
61 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 ...
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1answer
423 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 ...
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51 views

If the value of a call option is not dependent on the drift of the stock, why does a higher stock price mean a higher call option price [duplicate]

I have read that the price of an option is not affected by the drift of the stock since the drift term doesn't appear in the Black Scholes PDE. I become confused because to me, this implies that the ...
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1answer
65 views

Why does a higher stock value imply a higher call option value [closed]

This may seem like a very dumb question, but if the underlying stock price is greater, then why should a call option be worth more. My reasoning is that, if the option price is not affected by the ...
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1answer
166 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 ...
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1answer
92 views

European put options

Why is it that for European Puts on Non-Dividend-Paying Stocks, the lower-bound for price is $$p=Ke^{-rT}-S_0?$$
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2answers
369 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 $S$, $\sigma$ ...
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1answer
83 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 ...
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1answer
127 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. ...
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1answer
412 views

Basic Replication of European Call Option

I am looking at the very basics of replicating an option with a portfolio of risky and risk free assets. As such we can define a portfolio of $x$ no. of shares, $y$ bonds & $z$ options at time $(T)...
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1answer
129 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 ...
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2answers
318 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,...
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1answer
383 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 ...
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3answers
380 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 ...
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1answer
119 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(...
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200 views

Best Way of Interpreting Black-Scholes Formula [duplicate]

I'm curious to know the best interpretation of the Black-Scholes formula for a European equity call option: $$C(S,t)=S_tN(d_1)-Ke^{-r(T-t)}N(d_2),$$ where $d_1=\frac{1}{\sigma\sqrt{T-t}}\big[\ln(\...