Questions about models for the valuation of option contracts.

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29
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5answers
23k views

What are some useful approximations to the Black-Scholes formula?

Let the Black-Scholes formula be defined as the function $f(S, X, T, r, v)$. I'm curious about functions that are computationally simpler than the Black-Scholes that yields results that approximate $...
13
votes
4answers
8k views

How should I calculate the implied volatility of an American option in a real-time production environment?

There are many models available for calculating the implied volatility of an American option. The most popular method, employed by OptionMetrics and others, is probably the Cox-Ross-Rubinstein model. ...
33
votes
9answers
2k views

Are there any new Option pricing models?

Back in the mid 90's I used the Black-Scholes Model and the Cox-Ross-Rubenstein (Binomial) Model's to price Options. That was nearly 15 years ago and I was wondering if there are any new models being ...
16
votes
9answers
7k views

Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while. We all know geometric brownian motions have drifts $\mu$: $dS / S = \mu dt + \sigma dW$ and different stocks have different drifts of $\mu$. Why would ...
16
votes
8answers
7k views

Probability of touching

For a vanilla option, I know that the probability of the option expiring in the money is simply the delta of the option... but how would I calculate the probability, without doing monte carlo, of the ...
3
votes
5answers
391 views

Estimate probability of limit order execution over a large time frame

I have a negligible amount of money (\$5000) that I would like to invest in a stock. I would like to buy the stock at some point in the next year, and get the lowest possible price. I would like to ...
9
votes
2answers
5k views

What causes the call and put volatility surface to differ?

I currently have a local volatility model that uses the standard Black Scholes assumptions. When calculating the volatility surface, what causes the difference between the call volatility surface, ...
7
votes
5answers
4k views

How to get greeks using Monte-Carlo for arbitrary option?

Let's assume I have an arbitrary option that I can price using Monte-Carlo simulation. What is the general approach (i.e. without relying on specific option type) to calculating the greeks in this ...
4
votes
2answers
490 views

Black-Scholes under stochastic interest rates

I'm trying to implement the Black-Scholes formula to price a call option under stochastic interest rates. Following the book of McLeish (2005), the formula is given by (assuming interest rates are ...
5
votes
1answer
471 views

How to value a floor when a loan is callable?

Certain bank loans pay a spread above a floating-rate interest rate (typically LIBOR) subject to a floor. I would like to find the value of this floor to the investor. Assume for this example that ...
5
votes
1answer
860 views

Risk-neutral pricing in incomplete markets

I know that in order to use the risk-neutral valuation principle, that is, pricing options as their payoff function under a risk neutral measure, one has to have a complete market. But in the ...
3
votes
1answer
524 views

Option prices in Bates SVJ model?

In this [post] discussed the European put and call price formulas under the Heston Stochastic Volatility model. There exists an important extension of Heston model to include diffusion jumps, known ...
1
vote
1answer
1k views

Implied state price density (Question 1 - derivation of the formula)

I came upon the term "implied state price density" in a couple of papers. As far as I understand the concept one basically tries to extract the "pricing density" from the market data. For the sake ...
7
votes
2answers
232 views

How to price an option allowing to change a call into a put?

A recruiter asked me this question: Suppose you have the following contract: a call option with maturity T = 2 years the possibility to change this call into a put at t = 1 year What is the price ...
12
votes
8answers
7k views

Why does implied volatility show an inverse relation with strike price when examining option chains?

When looking at option chains, I often notice that the (broker calculated) implied volatility has an inverse relation to the strike price. This seems true both for calls and puts. As a current ...
5
votes
5answers
5k views

Risk Neutral Probability

I read that an option prices is the expected value of the payout under the risk neutral probability. Intuitively why is the expectation taken with respect to risk neutral as opposed to the actual ...
10
votes
3answers
3k views

How does volatility affect the price of binary options?

In theory, how should volatility affect the price of a binary option? A typical out the money option has more extrinsic value and therefore volatility plays a much more noticeable factor. Now let's ...
10
votes
2answers
624 views

What are important model and assumption-free no-arbitrage conditions in options trading?

In the paper "Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula" (Espen Gaarder Haug, Nassim Nicholas Taleb) a couple of model-free arbitrage conditions are mentioned which limits ...
7
votes
1answer
393 views

How to reduce variance in a Cox-Ingersoll-Ross Monte Carlo simulation?

I am working out a numerical integral for option pricing in which I'm simulating an interest rate process using a Cox-Ingersoll-Ross process. Each step in my Monte Carlo generated path is a ...
3
votes
1answer
247 views

Heston Model Option Price Formula

What is the formula for the vanilla option (Call/Put) price in the Heston model? I only found the bi-variate system of stochastic differential equations of Heston model but no expression for the ...
5
votes
3answers
517 views

Debunking risk premium via “hedging” argument? (or why even in the real world $\mu$ should equal $r$)

Since I began thinking about portfolio optimization and option pricing, I've struggled to get an intuition for the risk premium, i.e. that investors are only willing to buy risky instruments when they ...
5
votes
1answer
724 views

How to apply quasi-Monte Carlo to path-dependent options?

Following up on my recent question on variance reduction in a Cox-Ingersoll-Ross Monte Carlo simulation, I would like to learn more about using a quasi-random sequence, such as Sobol or Niederreiter, ...
4
votes
2answers
607 views

Price an option and find a replicating portfolio

I got stuck on the following question whilst learning about basic option pricing. A stock is valued at \$75 today. An option will pay \$1 the first time the stock reaches \$100 in value, which it ...
3
votes
4answers
3k views

Ways of treating time in the BS formula

The Black-scholes formula typically has time as $\sqrt{T-t}$ or some such. My questions: What is the granularity of this? If we treat $t$ as the number of days, then logically on the day of expiry, ...
3
votes
1answer
120 views

How to price this basket option?

Underlying assets are three global stock index : Eurostoxx 50, HSI, KOSPI 200 Maturity: 36 months with advanced redemption date in every 6 months if prices of indexes satisfy given conditions at each ...
3
votes
1answer
200 views

Data Selection for Empirical Pricing Kernel Estimation (Stochastic Discount Factor)

I want to estimate an empirical pricing kernel for an index. Hence, I need to estimate a physical and risk neutral density. For estimating the physical density, only the index data in an observed time ...
7
votes
4answers
3k views

How does an option's time value depend on moneyness?

How does an option's time value (also known as extrinsic or instrumental value) depend on how far it is in the money or out of the money? In other words, how does the time value change as the ...
4
votes
3answers
320 views

How to hedge a derivative that pays the reciprocal of the stock price?

1) Suppose S is the stock price, how to hedge a derivative that pays $1/S_t$ at time $t$? 2) Suppose there will be a dividend of amount $d$ between $t$ and $T$, how to hedge a derivative that pays $...
4
votes
1answer
736 views

SABR calibration: simple explanation and implementation

I would like to learn more about the SABR model and ho it is used in modeling smiles in equity, FX and rates markets. How would you explain the process and its implementation in simple steps? Any web ...
4
votes
5answers
1k views

Consensus on Cauchy distribution for stock prices

What is the general consensus for using a Cauchy distribution to model stock prices? I can't find much after researching online and wonder if it has been tried and discarded. My motivation is to find ...
2
votes
4answers
252 views

Is there any other way to measure option pricing model performance than proximity to market prices?

Short version Why do we take market prices as the prices to be estimated and predicted? The common answer is efficient markets hypothesis as in "Market agents do their best effort given their ...
2
votes
2answers
88 views

Importance Sampling for pricing options with longstaff and schwartz

I have been asking this similar question before. However, I really want to be concrete and get and concrete explanation. I have been reading the paper by Moreni and try to implement the same ...
2
votes
1answer
87 views

Pricing homogeneous Basket Default Swap

Consider a basket with $K=10$ names. Default times of the names, $\tau_k$, are i.i.d. random variables with distribution $P(\tau_k \leq t) = 1 - e^{-\lambda t}$. Suppose that each name in the basket ...
2
votes
1answer
649 views

Implied probability density (Question 2 - Applications and Interpretation)

Using the second derivative of the Call-Option-Price one can try to recover the pricing density. Formally: Assuming a constant interst rate $r$ and also not making any assumptions on the model ...
2
votes
1answer
481 views

American Swaption Pricing with Monte-Carlo method

I want to price an American swaption but I am not sure about what I am doing. Tree methods and PDE discretization seem difficult to adapt to a swaption. I am trying a Monte-Carlo approach. (in ...
1
vote
2answers
230 views

Put-Call relationship for Option on Forward

The forward price of a forward contract maturing at time T on an asset with price St at time t is, $$ F=S_te^{(r-q)(T-t)} $$ where $r$ is the risk free rate and $q$ is the continuous dividend rate ...
1
vote
1answer
172 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 ...
1
vote
2answers
167 views

Why is the term structure of the implied volatility surface non-monotonic?

Does this reflect expectations & uncertainty about interest rates (exposure to rho?), event driven concerns about the underlying, or something else?
1
vote
0answers
154 views

American Swaption Pricing with PDE discretization

So I am still trying to price an american swaption. (MC approach here: American Swaption Pricing with Monte-Carlo method) I've found in Paul Wilmott, The mathematics of financial derivatives, a PDE ...
6
votes
1answer
569 views

How to use binomial tree for portfolio of equity products

How can I use a binomial tree to price a European option that's based on a portfolio of equity products? I have volatility and correlation matrix of all underlying products? Looking for a formula ...
3
votes
2answers
249 views

Pricing when arbitrage is possible through Negative Probabilities or something else

Assume that we have a general one-period market model consisting of d+1 assets and N states. Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with ...
3
votes
1answer
112 views

Why are there two expressions for the Black-Scholes hedging portfolio

I am new to derivatives pricing and am trying to understand why there are two different expressions for the Black-Scholes hedging portfolio. The first approach, used in books like Hull, stipulates ...
2
votes
3answers
121 views

Greeks for binary option?

How to derive an analytic formula of greeks for binary option? We know a vanilla option can be constructed by an asset-or-nothing call and a cash-or-nothing call, does that help us? Wikipedia states ...
1
vote
2answers
114 views

Annual dividend yield using option prices

If I have only strike, call and put prices for European options, how do I work towards computing the continuous dividend yield?