Questions tagged [black-scholes-merton]
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32
questions
0
votes
2answers
66 views
Black Scholes informal derivation - question about a term in the equation [closed]
I am wondering what the term S means in the equation I have circled? I am not sure how to interpret it.
1
vote
1answer
162 views
Hull's book - Futures option's rho
In Hull's book (9th edition), on page 420, in table 19.6, it says rho of a European call on an asset with yield $q$ is
$$KTe^{-rT}N(d_2)$$
Below it says we can compute greeks of European options on ...
3
votes
1answer
164 views
Boundary conditions Heston's stochastic volatility model
I'm trying to derive the following boundary conditions for heston's stochastic volatility model.
This is p. 289 of Shreve's Stochastic calculus for finance
\begin{align}
c(T, s, v) &=(s-K)^{+} \...
2
votes
0answers
86 views
Stability of Finite Difference method for Breeden-Litzenberger
I am trying to derive a risk-neutral density from European call option prices using a second order finite difference scheme. Let $C(K,T)$ be the price of a European call with strike $K$ and expiry $T$ ...
1
vote
1answer
139 views
Black Scholes model without using Girsanov's theorem? It might happen?
We can calculate the stock price by the equation: $\frac{dS_t}{dt} = \mu dt + \sigma dB_t$,where $B_t$ is a Brownian motion.
First i create a portfolio that consists of $\Phi$ units of stock share ...
0
votes
0answers
60 views
Relationship between profit margins, historical volatility and implied volatility
In a scenario where no historic data exists in an option market, how would one come up with implied volatility for pricing of options, under the Black-Scholes-Merton model?
My professor has ...
1
vote
1answer
144 views
Nonlinear Black-Scholes model Vs linear Black-Scholes
I am working on a project related to Nonlinear BS partial differential equation, with terms for transaction costs and/or discrete hedging.
I have two questions:
Is there any exact solution to the ...
3
votes
0answers
126 views
How are Autocallables modelled?
What models are used to price autocallables ? Should we talk about Heston/SABR models which talking about this topic ? Any reference link is welcome.
4
votes
2answers
693 views
Deriving implied volatility programmatically
I'm working on a project to calculate the value of options using Python. I'm using the Black-Scholes model, and I can get accurate results by plugging in a given ...
3
votes
0answers
113 views
Alternative derivation of Black Scholes by Merton
I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I ...
0
votes
1answer
124 views
In literature, is IV constantly adjusted during option delta hedging?
In a lot of literature, they like to compare the performance of buying an option, and then delta hedging either at that options implied volatility (IV) or the true future volatility. This is under ...
2
votes
1answer
1k views
Option and probability of finishing in the money?
This seems to be another easy question but I am a bit confused. I know delta is a proxy for an option finishing ITM. Delta also happens to be N(d1) in the BSM pricing model. N(d1) usually is pretty ...
1
vote
1answer
228 views
Modifying Basic Black Scholes Equation For Time Dependent Variables - Per Wilmott?
I am reading Wilmott's book and don't understand why he makes the following step to re-write the PDE. I get equation 8.4, that's just the typical PDE for a dividend yielding stock where r(t), D(t) ...
2
votes
1answer
47 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 ...
3
votes
0answers
87 views
American Perpetual Put Option
I want to compute the expected payoff of a (classical) perpetual American put option in the Black-Scholes-Merton (BSM) framework with an optimal strategy of exercising the option at time $\tau=\inf\{t:...
2
votes
1answer
172 views
Why does Black Scholes formula give inconsistent dimensional analysis result?
For example, distance = speed * time, m = m/s * s.
But this technique gives wrong answer on the Black Scholes formula. The square root in the denominator gives wrong unit inside of the culumulative ...
2
votes
0answers
136 views
Poisson parameter in Merton's Jump-Diffusion Model to price call option
I've been taught the following European call valuation formula under jump-diffusion model:
\begin{equation}
price = E[e^{-rT}max(S_T-K,0)] =\sum_{j = 0}^\infty e^{-rT}P_j(\lambda)E[max(S_T-K,0)|J=j]
\...
2
votes
0answers
499 views
Black-Scholes equation to Heat equation .(Boundary conditions)
I have been given a problem to code the heat equation which is transformed from B-S equation (European call option) .
Now the boundary conditions are for European call option:
$$C(S,T)=\max(S-K,0)$$...
1
vote
1answer
127 views
Proper maturity in the Merton's model
I am working on a credit rating project using Merton's model. Basically it adopts Black-Scholes that equity value can be viewed as a call option with a strike price of face value of debts. Since the ...
2
votes
1answer
348 views
How to estimate Black Scholes parameters using Maximum Likelihood estimate method
It might be a naive question but I'm new to finance. I've been trying to get my head around this question from a long time and still totally clueless about this.
Suppose that the observed jumps in ...
1
vote
1answer
228 views
Dividend yield on ASX 200 (XJO) index options
I'm trying to understand how to calculate the price and Greeks of XJO options.
XJO options are European, the underlying is an index and they don't pay a dividend. However the underlying drops when ...
1
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0answers
212 views
Constant volatility and risk-free rate assumptions of Black Scholes
I'm studying the risk-neutral derivation of Black-Scholes formula and feel confused about the requirement for the volatility of the underlying asset and the risk-free rate to be constant. It seems ...
2
votes
1answer
145 views
Problems in understanding BSM formula
I'm currently learning Black-Scholes-Merton partial differential equation, and there are some confusions I can't work out.
Under the Black-Scholes assumption, we have:
$$df=\left(\frac{\partial f}{\...
1
vote
0answers
279 views
Option pricing in Merton model, comparison between Merton series and Carr-Madan
I'm studying the Merton model for pricing an European call option. The jump-diffusion process is:
$$X_t=bt+\sigma W_t+\displaystyle\sum_{i=1}^{N_t}Y_i.$$
$N_t$ is the Poisson process,
$W_t$ is the ...
2
votes
1answer
950 views
Understanding Vega calculation in black Scholes model
I am attempting to calculate the Greeks, and I understand their derivation. However when it comes to actually implementing Vega I am a little lost. Vega is defined analytically as:
$$ SN'(d_1)\sqrt{T-...
3
votes
0answers
132 views
delta hedging with stochastic volatility
In my thesis I want to work with delta hedging with stochastic volatility using Black-Scholes model. How will you suggest I implement numerical solutions using data from the real world? Beside Monte ...
1
vote
1answer
1k views
How to compute the volatility for the Merton's Model for Private firm?
After one day of research i did not figured how to compute the input volatility for PRIVATE COMPANY in order to calculate the PD.
My goal is to compute the PD of each of my company in my portfolio, ...
5
votes
2answers
742 views
Why is the black-scholes model arbitrage free when σ>0?
I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
2
votes
1answer
946 views
Why is rate of return on the stock normally distributed under GBM?
Let us assume the geometric Brownian motion, and we have
$$dS_t= uS_tdt+\sigma S_tdz,$$ and $S_t$ follows a log-normal distribution, but why is $r_t$, the continuously compounded rate of return, ...
1
vote
2answers
165 views
Value a structured note with Black-Scholes
Apologies in advance if this seems like a straight forward question but I'm really unsure how to go about it. Say I have the payoff for a structured note benchmarked against an index and I have a ...
4
votes
2answers
413 views
How to interpret negative asset volatility numerical results in Merton model?
I am currently working on my thesis where I discuss the Merton default probability model. I have a huge sample of US firms for the period 1990-2010. I use both numerical and complex iterative approach ...
3
votes
1answer
245 views
derive black scholes greeks
I am reading a paper and get a problem here, the following terms are all from standard BS models.
the paper says using the well known fact
$$Se^{-q(T-t)}N^{'}(d1)=Ke^{-r(T-t)}N^{'}(d2)$$ here the ...
