Questions about models for the valuation of option contracts.

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Probability Density of Returns of Bonus Certificates

Could anyone please help me with the following? I need to generate a histogram (resp. probability density) of returns of a bonus-certificate. A bonus-certificate can be replicated by an underlying ...
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79 views

Any thoughts on how Warren Buffet's B of A warrants might be “marked-to-market” by either counterparty?

It's not too long since Berkshire Hathaway got its 10-year warrants in Bank of America alongside its \$5 billion purchase of preferred stock. At the time I saw some discussion about the value of ...
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135 views

How to find the upper bound of a digital option given some market data?

Given the price of a call equals to 5 with Strike 100, please find the upper bound (sup) of the digital option with strike 105. I am not sure about the solution, but I write the condition like this, ...
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204 views

Tian third moment-matching tree with smoothing - implementation

I was wondering if someone has an implementation of the Tian third moment-matching tree (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1030143) with smoothing in code (e.g. c++, vba, c#, etc.)? ...
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1answer
144 views

american option and cash dividends

Can someoe help with this : What is the precise arbitrage argument demonstrating that the price of an american option should be continuous around an ex-dividend date? Thanks
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3answers
146 views

How can put options be more expensive than call options in an efficient market?

I noticed that for some securities, puts were more expensive than calls (with same expiration). For example, suppose the underlying security is trading at 50. A put with a strike of 45 is more ...
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191 views

Option arbitrage with dividends?

If a stock pays a discrete dividend, the stock price falls by the amount of the dividend. There is no arbitrage opportunity from this predictable jump, because the investors receive the same amount of ...
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336 views

how to calculate more efficient volatility figure than historical volatility?

can we use alpha value to calculate option price instead of historical volatility. And if we can please explain how. I am doing my MMS in Finance and this for a project i am doing. the project is ...
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117 views

Estimate simple option price without a calculator

I have been to two different interviews for jobs related to option trading, and both time I have been asked a question, which is pretty basic, and still I could not answer it. If you have an European ...
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2answers
204 views

What does “convergence” in Monte Carlo simulation mean?

I have read about convergence in terms of MC simulation for derivative pricing, but I am not clear on what it exactly means. Let us suppose I price an option 100,000 paths twice and both result in the ...
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2answers
1k views

Calculate volatility from call option price

Given call option price, what is the simplest formula to get the volatility value ? Test Data: ...
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1answer
464 views

How to explain the path dependency in binomial tree model to price options?

I'm new to quantitative finance, so I'm confused with the so-called path dependency in binomial tree model. Originally I thought the path dependency exists because in binomial tree model, we will ...
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2answers
97 views

Is Trading in the Underlying Necessary for Replication?

In a simple one-period binomial model we have two possible payoffs: $f(S^u)$ and $f(S^d)$. To replicate this we must trade in two assets, usually the stock $S$ and the money market account (assumed ...
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1answer
98 views

Pricing call option

Question: The price of a stock is 100. With equal probabilities, it either goes up to 130 or down to 70. What is the price of a 1 year call option with exercise price 100. Risk free rate is 5%. ...
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2answers
143 views

Intuitive Reasoning for Using Risk-Neutral Measure

Although we thoroughly covered risk-neutral pricing in university I never fully understood it in the context of continuous-time processes. But first of all, lets consider a discrete time example: ...
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1answer
69 views

Pricing American with floating strike

Consider a American floating strike put option with maturity $T$, written on a non-dividend paying stock $S_t$. The strike of this option at time $t\leq T$ is $Ke^{-r (T-t )}$, where $r$ is the ...
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2answers
452 views

calculate gamma value using finite difference method

I try to use the finite difference method to get the approximately gamma value, but there is an issue I can't solve. First, I set $h$ to 1 basis point of underlying asset value, but the result is not ...
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2answers
95 views

Efficient numerical approaches for pricing American Options with multiple sources of noise

I am looking for efficient numerical approaches for pricing American options when two or more sources of noise are involved (the simplest case coming to mind would be the Heston Model) Eventhough I ...
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1answer
147 views

Effects of random-generator-choice on derivative's price

There is a plethora of pseudo-random-generators out there. Some of them are definetly better and some of them severily underperform. My standard tool is Mersenne Twister - when I need to generate ...
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2answers
350 views

Is it wrong to use 'real world' probabilities for option valuation?

Is it wrong to use 'real world' probabilities for option valuation, even when the market is not liquid enough to delta hedge the option? My instinct is that it is wrong, because the time value of ...
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1answer
110 views

Calculating deltas of call options?

From a continuous standpoint, I understand why an ATM call has delta = 0.5 and for ITM call, the delta approaches 1 since each move in the underlying corresponds to same unit of value change in call ...
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1answer
225 views

Testing Black Scholes Analytical Options Pricer

I've written some code to calculate European option prices using the Black-Scholes analytical method. Can somebody recommend a good way to test that code? I have looked at option pricers online like ...
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1answer
318 views

Good Model Calibration Books/Papers for Common Option Pricing Models

I am trying to find a good book which focuses on the model calibration. I just want to know generally, what are the most common methods of model calibration(such as Black-Scholes Model, Stochastic ...
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404 views

Calculating Theta assuming other variables remain the same

Is there any way to calculate theta at X day in future based solely on knowing 1) Total Current Option Price 2) Days Till Expiration How would this be done? Thank you
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114 views

Linear combination of geometric Brownian motion

Let $X_t= e^{\left(\mu-\sigma^2/2 \right)t+\sigma W_t}$ be a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. I am trying to find an analytical solution to $$\mathbb{E}\left[ ...
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81 views

Hedging portfolio and extraction PDE of SV model with stochastic interest rate

How can I extraction this PDE \begin{align*} 0 =& P_t+P_SS(r-\delta)+P_\sigma a(\sigma)+P_r\alpha (r,t) \\ +& \frac{1}{2}P_{SS}S^2\sigma ^2 + \frac{1}{2}P_{\sigma ...
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1answer
74 views

Tradeable => Satisfies pricing equation?

In Wilmott's third volume, on p. 857, he tries giving an insight into the market price of risk by showing what it is for traded assets. For this he constructs a portfolio of two different options: ...
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1answer
81 views

Option Pricing under Jump Diffusion Models

I was wondering what the overall approach/intuition behind how to price options under Jump Diffusion Models. My understanding is under Diffusion models such as Geometric Brownian Motion (Black ...
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2answers
103 views

Delta and gamma neutral

A financial institution currently has a portfolio with delta of 450 and gamma of 6,000. A traded option is available with a delta of 0.6 and a gamma of 1.5. How could the portfolio be made both delta ...
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1answer
69 views

Determining swaption prices using the characteristic function

There exist multiple techniques to determine call option prices that make use of the characteristic function. These techniques boil down to some integral expression of the option price in terms of the ...
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1answer
98 views

Effect of vol smile on risk neutral probability of ITM

I was asked in an interview about how the vol smile affect the price of a binary option, which is essentially the Prob(ITM) under risk neutral measure. My thought is that the implied vol at spot ...
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1answer
45 views

Does a call calendar lose its entire value if underlying increases well past the strike?

If I buy a call calendar spread, and the underlying increases, both options are in the money by the expiry of the short call. So both options increase in value, but the short one increases less ...
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2answers
163 views

Dupire model and Local Volatility model

In the context of Option pricing model. Is there a difference between the Dupire Model and the Local volatility model ? Thanks Achal
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1answer
147 views

Implied volatility and pricing of vanilla options

As far as I understood, implied volatility (IV) is a lucky parametrization of the vanilla option's price. That is, instead of deciding how much the call worth now, you can decide on its IV and put ...
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3answers
399 views

forward implied volatility skew

I would like to calculate implied forward volatility skew. I have stochastic volatility monte carlo. What kind of payoff do I need to price and how to use Black() formula to calculate the implied ...
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121 views

Pricing Principle 1

In Tomas Björk's Arbitrage Theory in Continuous Time (or here), $\exists$ this Pricing Principle. Is the one in red supposed to be the proof of the Pricing Principle 1? Or merely an intuitive ...
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1answer
94 views

Simple pricing example confusion

This it taken from "Heard on the Street", Section B. Consider a market with $0$ risk-free rate, no transactions costs etc. The IBM stock costs \$75 and does not pay dividends. Design a security ...
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1answer
429 views

Price of a down-and-out call in terms of European call

If $EC(S_0, K, \sigma, r, T)$ represents the price of a European call option with strike $K$, expiry $T$, initial price $S_0$, volatility $\sigma$ and where the constant interest rate is $r$, then I ...
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3answers
73 views

When valuing a vanilla option on an index, should we take dividend into account?

When valuing a vanilla option on an index (eg FTSE 100), should we take index dividend yield into account? $$ c=Se^{-q\tau}N\left(d_1\right)-Ke^{-r\tau}N\left(d_2\right) $$ $$ ...
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1answer
54 views

Connection between implied volatily and implied probability

I am reading some lecture notes about Black-Scholes (BS) option pricing. Since the BS-formula is not supported by observed data because of the dependence of the implied volatility on the strik and ...
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1answer
75 views

Pricing employee stock options

ESOs are typically priced using the black-scholes model, but with an additional parameter for for the employee turnover rates . An example ...
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1answer
69 views

Why we consider second derivative w.rt price but only first derivative w.r.t time and volatility

What is the reason (better if it is intuitive, and not too math heavy), that when we talk of Greeks, we consider second derivative with respect to price (gamma), but only first derivative with respect ...
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1answer
49 views

Hedging behind the decomposition of american put options

Now I'm reading a paper:"alternative characterizations of american put options" , the authors are Carr,Jarrow,Myneni http://www.math.nyu.edu/research/carrp/papers/pdf/amerput7.pdf After theorem 1 ...
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1answer
72 views

binomial option pricing model - problem with risk-neutral probability

I have a little problem: in the binomial option pricing model, the price of a european derivative security $V_{n}$ satisfies: $V_{n}=[1/(1+r)]*[\tilde{p}*optionUp +\tilde{q}*optionDown]$ where: ...
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2answers
71 views

Numerical delta of Bond Options

I'm trying to calculate the delta for bond Call options. I'm using the vasicek model which gives the following solution for a Zero-coupon bond call option: $Z = N P(t,S) \Phi(d_1) - K P(t,T) ...
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125 views

How to price an European call on zero-coupon from the yield curve?

It is known that the price of an European call of maturity $T^*$ on zero-coupon of maturity $T$ is given by $$p(0,T)= B(0,T^*)\mathbb E ^{\mathbb Q_{T^*}}\left[ (B(T^*,T)-K)^+\right]$$ where ...
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1answer
58 views

Pricing rule shall be a martingale measure

In the book "Financial Modelling with jump processes" by Cont and Tankov there is a chapter that explains martingale pricing principles. It is not extremely formal, but gives the idea underlying the ...
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279 views

FX Delta Conventions

I'm currently reading Iain Clark's book Foreign Exchange Option Pricing and I got stuck at one sentence in the beginning of Section 3.3 that I feel is important to understand. He writes: FX ...
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288 views

Price of a composite option

how would you calculate the fair value of an option on a fx'ed underlying, e.g. a put on a USD-stock which is changed into EUR? How should I get, in practice, the fx spot vol/correl? Purpose is to ...
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151 views

Selling an American call option early

I understand it is never optimal to exercise an American call option early. [1] [2] However, here are my two contradictory thoughts about selling an American call option early. Assumptions I can ...