Questions tagged [option-pricing]

Questions about models for the valuation of option contracts.

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2
votes
2answers
72 views

What is the best book to learn about local vs. stochastic volatility, modelling and pricing of Exotics?

I am starting to delve into the world of Exotics and I am trying to find a rigorous yet understandable book that covers both mathematically and qualitatively (especially mathematically) the following ...
2
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1answer
40 views

Is the Non-discounted Bachelier call option price a Martingale?

My math finance professor once said someting that I can't make sense of. Hope you can answer: For a foward process the non-discounted price for a European call option under Bachelier is $$C_t = \...
2
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0answers
56 views

Model-Free Option Pricing

From Breeden and Litzenberger (1978) and subsequent work, we may find the risk-neutral density $q_{S_T}$ of $S_T$ from European option prices - assuming there are enough traded options (e.g. SPX) via ...
48
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8answers
50k 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 $...
2
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1answer
25 views

Valuation of Cash-Or-Nothing option

Studying options pricing, I'm stuck with the following problem: The price of a stock is described by the dynamic: $$dS_t = \mu\, dt + \sigma\,dW_t$$ Compute the fair price of a Cash or Nothing ...
2
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2answers
326 views

Black and Normal Model for Caplet using Python

I am able to Price Caplet using Black 76 model in Python. However, I am unable to price the same with Normal Model. Can anyone suggest what is missing ? I am valuing caplet that caps interest rate on ...
1
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1answer
105 views

Simulating assets of different currencies

I have a situation as follows: One year call option on a Euro stock with a Euro denominated strike. Knock in feature as follows - The option can only pay out if the growth in the Euro stock over ...
1
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1answer
51 views

Pricing with local volatility for derivatives beside options

Say I have calibrated an local volatility mode to market data on a forward on stock X. Say I want to price a derivative Y that is NOT a call/put option. What is the (or one of many) general strategy ...
6
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1answer
47 views

Control variate for pricing a best of assets option : $\mathop{{}\mathbb{E}}[ \max ( F^1_T,F^2_T, …,F^N_T )]$

I want to use Monte Carlo to price a best of assets derivative : $$\mathop{{}\mathbb{E}}[ \max ( F^1_T,F^2_T, ...,F^N_T )]$$ where the $F^i_T$ is the forward of the ith asset observed at expiry ...
1
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0answers
24 views

Swaption pricing and strategies

I am looking for resources (books, papers, websites, etc.) that deal with Vanilla and Exotic swaptions from a more advanced and quantitative perspective. I am interested in both the pricing side (e.g. ...
0
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1answer
46 views

Black Scholes Replication If Underlying Does Not Move?

Let's say you are long a call and want to replicate that call buy being short underlying and long bonds. If the underlying moves up in the next period but not enough to cover theta, the option ...
0
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0answers
21 views

Realized Option Return as a Function of Moneyness at Expiration

Main question: How does realized option return vary as a function of moneyness at expiration? Alternatively, if you have a crystal ball and see the exact price of a stock at some future date, what ...
0
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0answers
44 views

Simple Relation between Put Price and Zero Coupon Bond Price

Consider your standard European Put Option, with strike price $K$ and maturity $T$, and denote by $P_t(K,T)$ the price of this option at time $t$. Moreover, consider a standard Zero-Coupon bond with ...
1
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0answers
47 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 ...
2
votes
1answer
78 views

Does high levels of vol-of-vol parameter in SABR lead to Arbitrage? (Something seems wrong with Hagans formula)

Main question: Do we need to restrict the vol-of-vol parameter in SABR further than $\text{vol-of-vol}>0$ and how do we determine the interval of vol-vol which the model is arbitragefree? ...
4
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1answer
189 views

How to determine the risk-neutral measure in a Heston model?

To clarify, I'm quite familiar with the risk-neutral pricing framework, and I know one can efficiently Monte-Carlo a Heston model via the non-central $\chi^2$ distribution approach. But so far we're ...
1
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0answers
27 views

Why do simulation schemes have difficulty in pricing options with low spots?

If you apply a simulation Scheme (log-Euler discretization, Euler discretization and even more advanced ones) on for instance SABR and other models, then they price a call option (where we can easy ...
0
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0answers
45 views

Most efficient way to find Option IV using binomial/BS model

I have a python script set up to run a loop to plug in different values for IV into the binomial model to get an option price as close as possible to the market price. My issue is that at the moment ...
0
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1answer
77 views

Barrier option on a basket with arbitrary stochastic process

Suppose I want to price a Down-and-out European call, barrier option. However, the stochastic process is not a gBm or any other Levy process with known structure. Practically, I want a barrier option ...
1
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1answer
28 views

Simulation scheme for SABR beside the standard Euler discretization

QUESTION: Beside Euler Scheme, is there another more robust (and preferably easy to implement) way to simulate asset path with SABR dynamics? Simulation that will withstand even for high volatilities....
1
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1answer
55 views

Example of complex structured products on FX market?

Lately I have been working a lot with the vol smile and different stochastic volatility models with FX forwards data. Now I want to work with pricing examples through simulations. Can you suggest some ...
0
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0answers
38 views

Put-call parity for equity share and debt share

Considering Merton's structural approach" for credit risk modeling, we arrive to prove that the pricing formules are $S_t=V_t\phi(d_{T,1})-Fe^{-r(T-t)}\phi(d_{T,2})$ for equity share and $F_t=FP_0(t,T)...
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0answers
74 views

“BTP ITALIA” Inflation Linker Pricing

I have some issue with pricing of Italian linker bonds (http://www.dt.tesoro.it/export/sites/sitodt/modules/documenti_en/debito_pubblico/titoli_di_stato/BTP_Italia.pdf) . The issue is their specific ...
1
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1answer
40 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%...
1
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1answer
32 views

Risk-neutral pricing the “un”guaranteed benefits of an insurance policy

I'd love to know if the model of Black-Scholes-Merton could be used to anything that replicates the payoff of a call or option, for example: An insurance contract with participation ( meaning that ...
4
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1answer
77 views

Equivalence of Put Pricing Formulas

I have to show that: \begin{equation} P_{t,T}(K)=e^{-r(T-t)} \int_0^{\infty}\left(K-S\right)^+ q_T^S(S)dS \end{equation} is equivalent to: \begin{equation} P_{t,T}(K)=e^{-r(T-t)}\int_{-\infty}^{...
0
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1answer
51 views

Computing option price with rates only

Hi I am learning about options and came across this example: The spot FX rate AUD/USD is 0.6868, the 6 month ATM implied volatility for AUD/USD is 7.7% p.a., for the 6 month USD deposit rate is 2.28% ...
0
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0answers
17 views

Pricing barrier option under Levy process: Biased estimate?

I want to price a down and out call, barrier option, with the underlying asset following a Levy process. I am interest on the Kou double exponential model or the NIG process, to capture asymmetric ...
1
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0answers
40 views

Why can't we create a “magic” basket of options to sell for no-arbitrage pricing in SVJ model?

I am learning how to price SVJ options and am reading some stuff on no-arbitrage pricing for SVJ model using the typical approach you would use (like in BSM option pricing) of creating a risk free ...
3
votes
1answer
73 views

Constant Maturity Swap dates and conventions

Let's note $L(t,T_i,T_{i+1})$ the libor rate observed at $t$, fixing at $T_i$ with delivery at $T_{i+1}$. The natural delivery date for this rate is $T_{i+1}$, so a vanilla swap with no pay lag would ...
3
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1answer
79 views

How to interpret the (expected) exposure and CVA of an option or a single share

I have a quick (hopefully simple) question regarding the interpretation of the expected exposure of a call option and a single share. I've done some computations on the formula for the expected ...
2
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1answer
52 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 ...
9
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0answers
278 views

Jim Gatheral's ansatz

In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes $$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$ where $\hat w_t:=\int_0^t \hat v_s ds$ is the expected total ...
3
votes
1answer
107 views

What is upper left vol?

First time question, so please let me know if you have feedback for how I am asking. I am reading a market research piece and it makes reference to the performance of "vol, particularly the upper ...
1
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1answer
61 views

American Put Option Pricing

I am trying to solve a question of American Put Option pricing as below. Build a 15-period binomial model whose parameters should be calibrated to a Black-Scholes geometric Brownian motion model with:...
3
votes
1answer
119 views

What is the easiest way to learn Option pricing with PDE?

I was reading about Ito's formula and Girsanov theorem, but I am still struggling to grasp how in reality these are combined to compute the price of an option. What are the main source to understand ...
3
votes
1answer
196 views

Mixing Black Scholes with SABR

I am new to the whole concept of stochastic volatility so I am experimenting with option pricing. I think the concept is really difficult to understand / grasp. I was wondering if the following ...
7
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3answers
2k views

Forward implied volatility

Can one price accurately by only using vanilla options a derivative that is exposed/sensitive mainly to the forward volatility ? If it is impossible, why do we hear sometimes "being long a long ...
1
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1answer
98 views

Least Squares Monte Carlo

Could you explain to me in words (no formulas) the concept of the Least Squares Monte Carlo method to price an American style option?
1
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0answers
106 views

A crash course in pricing

I need to refresh all the pricing theory. Is there anything like a crash course with practical and intuitive explanations? I will provide any further information. I am a mathematical engineer. I am ...
0
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1answer
47 views

How underlying asset price variance is connected with time

I'm dealing with option pricing models and there is a statement that says the variance of underlying asset price is propotional with time $𝑉𝑎𝑟(𝑆_{𝑚+1})=𝑆_𝑚^2𝜎^2Δ𝑡$ where $\Delta t = \frac{T}{...
4
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0answers
76 views

How to price the american options using local volatility

I have given with a surface of american option prices $C_{am}(T, K)$. From these american option prices the implied volatility surface is deduced. Now I want to find the local volatility $\sigma(s,t)$...
1
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0answers
42 views

Difference between local volatility and implied volatility [duplicate]

I have read several articles about local volatility and implied volatility, but I am still confused with the difference between the two. Please correct me if I am wrong: Implied volatility is the ...
1
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0answers
32 views

Option price with underlying growth rate distinct from discount rate

Consider a European style option. The price equation is $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 \tag1$$ ...
0
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0answers
26 views

How to retrieve greeks and IVOL historically for listed options (using Bloomberg)?

Is there a way to get Prices, IVOL and greeks on historic option contracts (eg on the underlying RXM15) on Bloomberg, and follow this particular option with Strike K throughout its lifetime (eg on ...
10
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3answers
6k views

How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?

I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
3
votes
1answer
166 views

Pricing a callable bond

I have read the Lehman Brother's paper on OAS which I mostly understand, they outline how to find the OAS for a callable bond of which the formula is effectively (ignoring refinancing costs): Market ...
2
votes
1answer
158 views

Pricing under risk-neutral probabilities for weird derivatives?

I would really appreciate some help to value a weird derivative that I've found in an assignment: $$ X=(S_{T_1}-k)^{+} = \max(S_{T_{1}}-k;0) $$ which expires at time $T_{2}$ and uses the price at ...
0
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0answers
16 views

Longstaff Schwartz with future conditional coupons

I've implemented the L-S algorithm for a simple put option. I want to value a more complex derivative which has future conditional coupons which only occur if the option is in the money. How would I ...
1
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0answers
52 views

Is it necessary for $P(K, t) - P(K + s, t) \geq se^{-rt}$ to hold?

Let $P(K, t)$ be a put option with strike price $K$ and expiration time $t$. Let $s > 0$. Is it necessarily true that the inequality $$P(K, t) - P(K + s, t) \geq se^{-rt}$$ holds? I know that ...