Questions tagged [black-scholes-merton]

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3
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0answers
47 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:...
1
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0answers
41 views

Discounted self-financing portfolio still a self-financing portfolio?

Assume a self-financing portfolio $V_{t}=\theta_{t}^{0}S_{t}^{0}+\theta_{t}S_{t}$ with $S_{t}^{0}$ the value of the non-risky asset at time $t$ and $\theta_{t}^{0}$ the amount of shares of the non-...
2
votes
1answer
105 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 ...
1
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0answers
38 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
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0answers
197 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
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1answer
59 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
229 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
121 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
123 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
129 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
169 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
590 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
91 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
787 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
352 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 $ ...
1
vote
1answer
663 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
139 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
338 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
208 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 ...