Questions about models for the valuation of option contracts.

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Expected option return in MATLAB

The expected return of an option is given by its expected payoff under $P$ over its market price under $Q$. For the Black-Scholes model, expected call option return is given as (see here): $$ E(R)=\...
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21 views

Reference for option pricing, binomial multi-period model using martingales and conditional expectations

The title basically says it all. I am looking for a reference text on the pricing of options in a binomial multi-period model. It should be mathemathically rigorous using martingales and conditional ...
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1answer
165 views

Calculating historical implied volatility

I know that each individual option has it's own implied volatility, but how do you go about calculating the overall implied volatility for an underlying? For example when someone sais the IV of a ...
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1answer
123 views

Monte Carlo and PDE results are different for a Call Option!

Okay so this might be a fairly trivial question but I'm having an issue with valuing a call option using both a Monte Carlo method and a PDE method. When I started I first used the parameters: Spot =...
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1answer
71 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 ...
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4answers
376 views

How to price an exchange option using B&S framework?

Consider a market composed by two stocks whose prices $X$ and $Y$ are given by B&S diffusion: $$dX_t= \mu X_t dt+ \sigma X_tdW_t$$ $$dY_t= \mu Y_t dt+ \sigma Y_tdB_t$$ Supposing the market is ...
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70 views

why Implied Vol (VIX) increase with decrease in Stock Price or vice versa?

why Implied Vol (VIX) increase with decrease in Stock Price or vice versa? whereas Vega is positively related with change in option price to change in stock price.
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2answers
65 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 ...
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1answer
62 views

Importance Sampling for Least Square Monte Carlo

I am currently trying to implement and model an Importance Sampling estimator for Longstaff and Schwartz algorithm for pricing American put options. It is used such that more paths are in-the-money ...
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1answer
59 views

Implementation of an option tail-hedging strategy

This question directly refers to the paper "Capital Asset Pricing Mistakes: The Consistent Opportunities in Tail Hedged Equities", http://www.universa.net/Universa_SpitznagelResearch_201501.pdf. Very ...
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1answer
305 views

A question about pricing convertible bond with two different underlying assets

I have a question regarding the pricing of convertible bond. If I value the convertible bond with two different underlying assets, how can I incorporate two volatility and the correlation in the ...
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2answers
115 views

Solution for american perpetual put

I have been attempting an exercise in which I have to determine the value of an american perpetual put, $P$ in terms of the asset value $S$. The solution to the exercise says: When $S>S_f$ (the ...
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Use of real-world probabilities in options pricing: binary event with continuous effect

Let's say I have to price options on instrument X with a multitude of strikes. For simplicity, assume that X only makes one move during the options' lifetime, and this move is affected by some binary "...
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1answer
390 views

Which distribution do I get?

Let's assume the stock moves according to a classic Black-Scholes model, and makes a proportional jump with an unknown proportion. Say, it is either +1% or -3% of the stock value, and we know for sure ...
3
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1answer
130 views

Delta of an option derived from the binomial model

I have the following function $V=V(S,t)$, $V^- = V(vS,t+\delta t)$, $V^+ = V(uS, t +\delta t)$. The book proceeds to explain that if we use Taylor series expansion on the above we will confirm that $\...
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2answers
309 views

Derivation of Stochastic Vol PDE

A couple questions regarding stochastic vol PDE derivation. Following Gatheral, a general stochastic vol model is given by \begin{align*} dS(t) & = \mu(t) S(t) dt + \sqrt{v(t)}S(t) dW_1, \\ dv(t) ...
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3answers
102 views

How do you check your option calculations?

I'm implementing a bunch of different algorithms to price options/find Greeks: finite difference, Monte Carlo, binomial... I'm not really sure how to check my calculations. I tried using QuantLib to ...
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2answers
91 views

Option pricing, origin of formula $\Pi( t,X)= E^{\mathbb{Q}}\left[e^{-\int_{t}^{T}r_s\,ds} X| \mathcal{F}_t\right]$

Imagine a model with stock prices and dividends of these stocks, as well as a market bond with associated short rate process. It is known that this model is arbitrage-free if there exists an ...
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2answers
130 views

Is it possible that under Black-Scholes: $\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$

I have a slide on which there is written that under Black-Scholes model: $$\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$$ Now, here there is a good explanation ...
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1answer
31 views

Pricing Barrier Options with Rebates

How are rebates factored into the Black-Scholes analytical solutions to pricing barrier options? In Hull's book, he does not have rebates factored into the formulas. Can someone point me to a paper ...
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1answer
33 views

Find the parameter $d$ of the Affine Option Pricing Model in Duffie, Pan and Singleton (2000)

According to Duffie, Pan and Singleton (2000) for any real number $y$ and any $a$ and $b \in \mathbb{R}^n$, the price of a security that pays $\exp(aX_t)$ at time $T$ in the event that $bX_t \leq y$ ...
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2answers
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Price and constant hedging portfolio for straddle: $X=|S(T)-K|$

wondering if somebody could check my answer for a homework question! Given a straddle, characterized by its pay-off at maturity $X=|S(T)-K|$, I am asked to find the price of the (simple) claim at any ...
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1answer
218 views

Hedging - calculating option prices using implied volatility surface

To hedge a strategy is it accurate "enough" to price an option using an implied vol curve vs moneyness (strike/spot) assuming sticky delta? The moneyness can be read off the chart, its corresponding ...
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1answer
124 views

Complicated American style option contract with numerous non-standard features (simultanous exercise, additional premium, etc.)

I want to value the following contract for times $0<t<T$, i.e. determine $V(t,\cdot)$ where $\cdot$ refers to all other dependences (strike, spot, volatility, etc.). The contract is long and ...
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A question on option pricing [closed]

Calculate the value of 9-month American call option to buy 1 million units of a foreign currency using a three-step binomial tree. The current exchange rate is 0.79 and the strike price is 0.80 (both ...
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2answers
458 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 ...
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0answers
27 views

Replicating portfolio: initial portfolio?

I have a bit of trouble understanding how to determine the replicating portfolio of a call using just a stock and the riskfree asset. I have times $t = 0,1,2$, and at time $2$, we have $3$ payoffs ($...
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1answer
71 views

What models / methods are used in practice in derivative pricing?

I wrote my bachelor thesis about European Option Pricing under Stochastic Volatility and Jump Diffusion and am now near the end of my MSc in Quant Finance. As i want to write a "potential job"-...
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2answers
166 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 ...
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2answers
337 views

“Hedging” a put option, question on exercise

I have a question on the following exercise from S. Shreve: Stochastic Calculus for Finance, I: Exercise 4.2. In Example 4.2.1, we computed the time-zero value of the American put with strike ...
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160 views

Extrapolating SVI

In his paper Gatheral presents the following parametrization of the implied total variance $w(k,T) = \sigma_{BS}(k,T)^2T$ $$ w(k) = a + b\{\rho (k-m) + \sqrt{(k-m)^2 + \sigma^2} \}.$$ Assuming that ...
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1answer
64 views

Portfolio with a certain pay-off curve

I would like to find a relevant optimization option's portfolio models which can describe a certain pay-off curve (objective function) under same assumptions. For example, assumptions on how to limit ...
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2answers
79 views

How is the fundamental theorem of asset pricing used?

I know that a multi-period market model is complete and arbitrage free if there's a unique equivalent martingale measure. The thing is, I have absolutely no clue how to apply this theorem to a simple ...
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Option based approach to real capital structures

Has anyone made a serious attempt to apply option theory to real assets and capital structures, taking into account all the messy details ?
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Is $(1,0,0,0,…,0)$ a legitimate dividend stream?

A book I am reading defines a positive linear functional as a "price functional" from a set of adapted processes to the real numbers. Specifically, it defines a "consistent price functional" as one ...
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31 views

Estimate Option Price Given X% Move N Days in the Future

I was wondering if someone could recommend a method to estimate the price of an option N days from now given an X% move in the underlying. I have fitted a volatility surface but where I am running ...
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1answer
36 views

Does a Call Spread always need to be symmetric?

I have a plot of a Call Spread Option at time $t ={0}$ but the graph of the call spread is not completely symmetric. My question is: does it have to be? Here is the plot I'm referring to: I'm just ...
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1answer
106 views

Boundary Conditions for Call Spread

I was just wondering if someone could verify whether these are the two boundary conditions for a Call Spread Black-Scholes PDE. The first one I have is: $max(S_{T} - K_{1}, 0) - max(S_{T}-K_{2},0)$ ...
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43 views

Comparison of Implied Vol Models

My goal is to evaluate a collection of implied volatility models for accuracy supporting real time theoretical pricing of listed equity option. My current research approach is to define a set of ...
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1answer
124 views

Pricing a log-contract using Monte Carlo

Having a payoff of log-contract defined as $$ \Pi_T = \ln \left(\frac{S_T}{S_0} \right) $$ How would you express the MC-estimator for the price of this contract? The stock price dynamics here is ...
3
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1answer
76 views

Carr-Madan european contingent claim payoff decomposition formula - application

Looking for some clarification to the values of the parameters used in the Carr-Madan payoff decomposition formula. $$f(S_T)=f(\kappa) + f'(\kappa) (S_T - \kappa) + \int_0^{\kappa} f''(K) (K-S_T)^+ ...
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68 views

Is this formula correct to estimate a knock out option price using monte-carlo?

I have a knock-out option with barrier $L>0$ and strike $K$ that pays at maturity $(S-K)_+$. So, positive payoff occurs only in case the price stays below the barrier over life of the option. I am ...
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Pricing with-profit/smoothed bonus annuity using Black-Scholes

Would this be possible? Subsequently, would the pricing of such an annuity be somewhat similar to pricing a lookback option?
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1answer
25 views

Is it possible to find / estimate the volatility surface of non-listed index options?

I have 3 QNET options (european, 2 puts, 1 call, all same expiry, different strikes) that the broker is pricing clearly off a volatility surface. Bloomberg only carries historical volatility and I ...
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Numerical Methods for Merton Model

The stochastic differential equation for an underlying with jumps in Merton model is: $$d{{S}_{t}}=\mu \,{{S}_{t}}dt+\sigma \,{{S}_{t}}\,d{{W}_{t}}^{P}+(J-1){{S}_{t}}d{{q}_{t}}$$ where $t \quad\,\,\, ...
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Open source code based on quandl for security analysis and options priming

Quandl seems to be an excellent source of wide range of free/open financial data. But is there an open source code or platform that uses the quandl datasets to perform security analysis and option ...
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Generating process for stock price paths in this paper?

I am reading Longstaff and Schwartz Valuing Aerican Options by Simulation because monte carlo simulations, especially their use in option pricing, is interesting to me. However, I am having some ...
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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 ...
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2answers
85 views

Accuracy Rebonato Swaption Approximation Formula among Different Strikes

Can somebody explain me if the Rebonato swaption volatility approximation formula is accurate for only ATM strikes, and if yes why? Can it also be used for ITM and OTM strikes? My foundings: Let $0 &...