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Continuous flows Perpetual maturity cap on Exchange Options PDE Change of variable

Im trying to do a change of variable on the following PDE Using the following change of variable $$ V(P^1,P^2) = P^2 W(C), C=\frac{P^1}{P^2} $$ I get this for the homogeneous part of the equation: ...
f4bby's user avatar
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How to access the Black Sholes Formula through the Distributive Law?

Recently I read a comment on how to interpret the Black Sholes Formula and more specifically how to wrap your head around the d1/d2. Although there were many good comments, this one stood out when one ...
Telefondemonen_se's user avatar
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110 views

interpret payoff [closed]

How can the payoff f = min(max(S - K1, 0), K2) be interpreted?
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Clarification on a Claim in Black-Scholes-Merton Derivation

In these notes: https://johnthickstun.com/docs/bscrr.pdf, towards the end of the proof of Proposition 5.2 on page 6, the author claims: $$ \log \lim_{n \to \infty} \Bbb{E}_\pi \left[\frac {S^*_n} S \...
Joe Shmo's user avatar
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Calculating the PnL of a delta-hedged option at a point in time

In a BS world (constant volatility, no transaction costs, continuous hedging) If I buy or sell an option and continuously delta-hedge, I know how to calculate the final expected PnL based on implied ...
JamesSmith12's user avatar
1 vote
0 answers
80 views

Mixing formula for SVJ models

I am trying to understand the mixing formula (Hull and White formula) for stochastic volatility models with jumps in the asset price. One article which discusses this is Lewis, The mixing approach to ...
p.sibuea's user avatar
2 votes
1 answer
363 views

Black and scholes option pricing

I have to solve the following problem in the Black and scholes model: find the price at anty $t\in[0,T)$ for an option whose payoff at the maturity is: \begin{equation} 0 \ \ \ \text{if} \ S_T<K_1\\...
Pefok's user avatar
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Why is call option value same as portfolio value at all times in Black Scholes model?

Following is a part of the text from Steven Shreve Stochastic Calculus for Finance II, for pricing the European Option in Black Scholes model. The argument is that today I start by selling a European ...
kishlaya's user avatar
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1 answer
142 views

Black Sholes Options Pricing Clarification Questions [closed]

I am interested in pricing American Call and Put Options using BSM and I am new to exploring options prcing. I have some questions here that would really remove the confusion I have on how to more ...
Shion's user avatar
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2 answers
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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.
Shayan Slaesi's user avatar
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1 answer
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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 ...
Xiaohuolong's user avatar
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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)^{+} \...
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246 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 $T$ ...
user39039's user avatar
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350 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 ...
Maria Oikon's user avatar
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106 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 ...
383930283423's user avatar
1 vote
1 answer
504 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 ...
user1157's user avatar
3 votes
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254 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.
chacha's user avatar
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4 votes
2 answers
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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 ...
pez's user avatar
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3 votes
0 answers
173 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 ...
Kirsten Caroline Abildgaard's user avatar
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1 answer
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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 ...
confused's user avatar
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1 answer
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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 ...
confused's user avatar
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1 answer
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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) ...
user6472523's user avatar
2 votes
1 answer
68 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 ...
Roger Bravus's user avatar
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113 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:...
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3 votes
1 answer
320 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 ...
user avatar
2 votes
0 answers
191 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] \...
godric97's user avatar
2 votes
0 answers
734 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: $$C(S,T)=\max(S-K,0)$$...
user1157's user avatar
1 vote
1 answer
226 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 ...
YellowRiver's user avatar
2 votes
1 answer
528 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 ...
rakesh's user avatar
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1 vote
1 answer
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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 ...
Zeus's user avatar
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1 vote
0 answers
275 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 ...
Yilie Ma's user avatar
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2 votes
1 answer
238 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}{\...
Bubblethan's user avatar
1 vote
0 answers
382 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 ...
sound wave's user avatar
2 votes
2 answers
2k 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-...
user avatar
3 votes
0 answers
185 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 ...
Frank Blythe's user avatar
1 vote
1 answer
2k 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, ...
manu's user avatar
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5 votes
2 answers
1k 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 $ ...
elbarto's user avatar
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2 votes
1 answer
1k 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, ...
ZHI's user avatar
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1 vote
2 answers
261 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 ...
PLui's user avatar
  • 113
4 votes
2 answers
495 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 ...
user3618375's user avatar
4 votes
1 answer
357 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 ...
Xun Bao's user avatar
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