# Questions tagged [black-scholes-merton]

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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 \... 0 votes 0 answers 30 views ### How to price an american put option on a dividend-paying stock? [duplicate] There is no Black Scholes formula for the value of an American put option on dividend paying stock eithe has been produced ? Should I use the binomial model ? 1 vote 1 answer 939 views ### 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 ... 1 vote 0 answers 66 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 ... 2 votes 1 answer 326 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\\... 2 votes 1 answer 214 views ### 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 ... -4 votes 1 answer 123 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 ... 0 votes 2 answers 73 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 1 answer 440 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 1 answer 725 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 0 answers 203 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 1 answer 305 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 0 answers 102 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 1 answer 406 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 0 answers 220 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 2 answers 2k 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 0 answers 161 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 1 answer 179 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 ... 3 votes 1 answer 4k 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 1 answer 379 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 1 answer 62 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 0 answers 103 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:... 3 votes 1 answer 277 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 0 answers 179 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 0 answers 703 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)$$... 1 vote 1 answer 200 views ### Proper maturity in the Merton's model I am working on a credit rating project using Merton's model. Basically it adopts Black-Scholes that equity value can be viewed as a call option with a strike price of face value of debts. Since the ... 2 votes 1 answer 480 views ### How to estimate Black Scholes parameters using Maximum Likelihood estimate method It might be a naive question but I'm new to finance. I've been trying to get my head around this question from a long time and still totally clueless about this. Suppose that the observed jumps in ... 1 vote 1 answer 307 views ### Dividend yield on ASX 200 (XJO) index options I'm trying to understand how to calculate the price and Greeks of XJO options. XJO options are European, the underlying is an index and they don't pay a dividend. However the underlying drops when ... 1 vote 0 answers 267 views ### Constant volatility and risk-free rate assumptions of Black Scholes I'm studying the risk-neutral derivation of Black-Scholes formula and feel confused about the requirement for the volatility of the underlying asset and the risk-free rate to be constant. It seems ... 2 votes 1 answer 231 views ### Problems in understanding BSM formula I'm currently learning Black-Scholes-Merton partial differential equation, and there are some confusions I can't work out. Under the Black-Scholes assumption, we have:$$df=\left(\frac{\partial f}{\...
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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 ...