Questions about models for the valuation of option contracts.

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3
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1answer
187 views

How can one value a Bermuda option?

A Bermuda option allows early exercise at predefined dates, e.g. at maturity equal to $t_1$, $t_2$, $t_3$,...; hence , would its value be the sum of 3 discounted European options with 1-year ...
3
votes
1answer
223 views

Numerical example of how to calculate local vol surface from IV surface

I'm looking for an excel example (not a copy of Dupire's eqn) of how to convert an IV surface to a local vol surface. If unsuccessful I'll work through Dupire's eqn but would be helpful to look at an ...
4
votes
1answer
143 views

Need for Binomial Representation Theorem

In some texts (e.g. Baxter & Rennie, Shreve I) the binomial model is first constructed using the usual backward induction argument, and it is concluded that by no-arbitrage the time $t$ value of a ...
7
votes
3answers
860 views

Option Pricing Model Calibration In Practice

I'm curious how an option pricing model like the Heston model is calibrated in practice. Here's how I imagine it happens: Let's say I have access to the most recent option prices on a given stock ...
3
votes
1answer
109 views

Boundary conditions of PDE from SV model with stochastic interest rate

The PDE for the American put option price $P(S,\sigma ,r,t)$ is \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 + ...
0
votes
1answer
83 views

Local volatility parametrization using the spot

Is it possible to estimate the local volatility using the spot price S at time t instead of the strike price K and the expiry date T ? Any help would be appreciated.
3
votes
3answers
178 views

Arbitrage bounds for Black-Scholes

In some implied volatility code I came across, there is a check to ensure there is no violation of the arbitrage bounds based on the inputs to the method. For the call option, if $$P < 0.99 * ...
2
votes
2answers
89 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 ...
1
vote
0answers
94 views

Price of an American call option [closed]

I'm working through revision questions at the moment and we are asked to compute the price of an American call option. Suppose that $dS_t = \sigma S_t dW^*_t, S_0 >0$ Let $0<U<T$ be fixed ...
4
votes
2answers
109 views

Importance Sampling - where to center the sampling distribution?

Consider a Monte Carlo (MC) approximation to a European call with BS parameters $r = 0.05, \sigma = 0.4, T = 10, S_0 = 50$ and $K = 95$. Consider the following results, each using 1M points: plain ...
0
votes
0answers
45 views

Benchmarking option pricing under stochastic interest rates

I priced a long-term option (10 or 20 years) using two different models: one assumes constant interest rates, the other assumes stochastic interest rates. Is there a way (e.g. a benchmark) to ...
2
votes
0answers
52 views

replicating strategy three step binomial

I am having some trouble setting up a replicating strategy for a call option with a three step binomial model (discrete). I have no trouble doing this in a two step binomial model by backward ...
3
votes
1answer
144 views

Distribution of Black Scholes call option price at time 0<t <T

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $? (Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t ...
3
votes
1answer
235 views

Value of European Call equals Value of American Call, Question on Explanation/Proof

I am reading S. Shreve, Stochastic Calculus for Finance, Vol. I. There he proves that American Call Options have the same value as European Call Options. In the proof he uses that for a Call option ...
5
votes
5answers
2k views

How to price a calendar spread option?

How do you price calendar spread options, that is, options on the same underlying and the same strike but different times to maturity? Clarification: I'm interested in the pricing of a a CSO ...
1
vote
2answers
297 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 ...
8
votes
4answers
913 views

From Fourier Transforms to Option Values

I am trying to understand how Fourier transforms & Characteristics functions can be used to calculate option values. However, I am having difficulty following the process that is used in several ...
5
votes
3answers
496 views

Greeks: Why does my Monte Carlo give correct delta but incorrect gamma?

For a vanilla European call, my Monte Carlo method gives the right option price and delta but the wrong gamma. In particular, the value of gamma varies wildly each time I run the method. I estimate ...
1
vote
2answers
485 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 ...
1
vote
1answer
76 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: ...
1
vote
2answers
138 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 ...
7
votes
1answer
253 views

Speeding up computations: when to use Quasi and standard Monte-Carlo in pricing

I am familiar with the theory of Monte-Carlo techniques in the numerical integration, and recently I have started my experiments with these methods applied to derivatives pricing. I am using ...
0
votes
1answer
151 views

The State-Price Deflator in a Binomial pricing model

This question comes from a Financial Economics exam and I'm very confused about a state-price deflator which doesn't seem to exist. I've included the whole question for completeness, but my actual ...
0
votes
1answer
91 views

Binomial pricing model: When the Cox-Ross-Rubinstein assumption is not arbitrage-free

I understand that in an arbitrage-free Binomial model, we assume that $S_{t+1} = S_t \cdot u$ in the event of an up-jump and $S_{t+1} = S_t \cdot d$ in the event of a down-jump. We call $u$ and $d$ ...
0
votes
0answers
143 views

Vanna-Volga Adjustment

I'm reading Uwe Wystup's "FX Options and Structured Products" to understand Vanna-Volga pricing, which, in his book Chapter $\S3.1$ is called "The Trader's Rule of Thumb". I generally got the idea ...
1
vote
2answers
104 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 ...
2
votes
1answer
413 views

Using FX ATM/RR/BF Volatility to Estimate Smile

Suppose $S$ is some FX rate, EUR/USD say, and $\sigma_{S}(K,T)$ is the implied volatility for some option written on $S$, sourced from the surface $\sigma_{S}(\cdot,\cdot)$ (alternatively, consider ...
2
votes
1answer
155 views
1
vote
3answers
676 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 ...
3
votes
0answers
98 views

How can a beginner trader make use of 'volatility of volatility'

For a beginner option trader in equity options, how can he use this metric that is provided by his broker/data vendor? How can he use this metric to gain an added understanding of the option ...
2
votes
3answers
82 views

Questions on the relationship between option price and maturity

From the plot of volatility surface, as maturity goes up, the implied volatility will decrease. Dose it mean that options with the same strike have higher value when maturity is larger. If so why ...
7
votes
2answers
217 views

What's the point of discounting in risk-neutral pricing?

Let $\phi$ be a self-financing strategy that replicates a time $T$ option payoff $X$ on stock $S$. By definition of a trading strategy, $\phi$ is previsible. Finally, let $V_t$ be the time $t$ value ...
2
votes
1answer
206 views

Black Scholes Formula, drift term

In the formula, the stock return is modelled as a brownian motion that is a drift + a stochastic term, ok I get that. But the drift term is then modelled as r - volatility ^ 2 / 2. I am not sure how ...
1
vote
1answer
116 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 ...
-2
votes
1answer
93 views

Show that the equation solves the Black-Scholes PDE

I have the solution as given Based on this, I have to show that this solves the Black-Scholes formula It means that I should take the partial derivatives of the solution above and then receive the ...
1
vote
1answer
134 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 ...
1
vote
2answers
401 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 ...
2
votes
1answer
115 views

The source of “Cost of hedging” in the Black Scholes model

I am trying to get some intuition for the fact that a Black-Scholes price for an option is equal to the cost of replicating the option. Say the interest is 0. The option is obviously still worth ...
10
votes
2answers
609 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 ...
1
vote
1answer
112 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%. ...
1
vote
2answers
161 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: ...
4
votes
0answers
49 views

How to price lookback american option when its payment is distributed during its life

I would like to price a floating strike american lookback with a particular feature: I don't want to charge upfront the client, rather I would like to insert a "running fee", some sort of a dividend. ...
2
votes
1answer
64 views

Solving a Non-Linear PDE using a Finite Difference Scheme

I have the following non-linear PDE and I have no idea how to go about solving it using a finite difference scheme in Python. Can someone get me started and/or point me to an algorithm for doing this? ...
8
votes
4answers
844 views

Replicating portfolio and risk-neutral pricing for interest rate options

For equity options, the pricing of options depends on the existence of a replicating portfolio, so you can price the option as the constituents of that replicating portfolio. However, I am not seeing ...
28
votes
5answers
22k 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 ...
3
votes
2answers
185 views

Stock Returns Distribution in Heston Model

There is a paper by Dragulescu and Yakovenko (DY) in 2002 proposing a pdf for the stock returns in the Heston model. However, in a paper by Daniel, Bree and Joseph, they actually perform statistical ...
2
votes
2answers
124 views

Braess's paradox in quantitative finance: When optionality leads to lower value…?

One of the standard tenets of quantitative finance is that options should have an intrinsic value because optionality as such (in the sense of having more choices) should bring about value. This ...
0
votes
1answer
742 views

Binomial tree vs trinomial tree in pricing options

Very new to pricing models. Is there a general guideline when to use binomial tree and when trinomial tree is preferred? As far as I know, unlike binomial tree, trinomial tree only gives a range ...
4
votes
2answers
171 views

Time-independent local volatility

Suppose somebody provides us with a surface of European call prices $C(\tau,K)$ where $\tau$ stands for time-to-maturity and $K$ for the strike. By Dupire's results, there is a unique local volatility ...
2
votes
1answer
342 views

Method for finding a arbitrage opportunity when market price of call is incorrect

The solution of the Black-scholes equation is the price of a European call. And the option price assumes the underlying stock is a geometric Brownian motion with volatility $\sigma_{1}>0$. ...