Questions tagged [black-scholes]

Black-Scholes is a mathematical model used for pricing options.

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I have the following general SV model: $$dS = \sigma S dW_S$$ $$d\sigma = a(\sigma,t) dt + b (\sigma, t) dW_\sigma$$ $$dW_S dW_\sigma = \rho dt$$ where $a , b$ are deterministic functions of $\... 1answer 177 views Uniqueness of Risk-neutral measure: Probabilistic view Suppose we are working on the Black and Scholes Framework. There are only two assets, the risk-less bank account and a stock. The discounted process is a GBM under the physical measure with drift term ... 1answer 98 views Discounted asset price is martingale in BS model I want to verify that the discounted stock price process$\mathrm{e}^{-r(T-t)}V(S_t,t)$is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that $$\mathrm{d}\mathrm{e}^{-r(T-t)}... 1answer 93 views Using BS Delta to hedge in a LV Model Why do some people use a Black Scholes Delta instead of the delta given by the Local Volatility model? 1answer 170 views Simulate double exponential process with correlated jumps? So, I'm trying to simulate a correlated double exponential jump process for two assets, and I understand the pure exponential jump process (\eta_1 and \eta_2, the probability of an upward jump ... 2answers 1k views How are the two concepts No arbitrage & Risk neutral probability related? The title, and might I add, that this question is in relation to the Black-Scholes model and why the concepts are important for option pricing in general. 1answer 830 views What's the intuition behind the transformation of Black-Scholes into Heat equation? A sequence of transformations can be used to turn the Black-Scholes PDE into the heat equation. Let C(S, t) be the price of a vanilla European option at time t, maturing at time T, where the ... 2answers 297 views Tradable information from BS Implied volatility These are two follow up questions to: Implied volatility as price transform I understand that the BS model is used as a 'Blackbox' that takes a market price and maps it in a 1to1 fashion to a 'BS ... 1answer 780 views Beta between stock and option In Black Scholes model I would like to compute$$ \beta_K = \frac{\mathrm{cov}(C_{K,T},S_T)}{\mathrm{cov}(S_T,S_T)} = \frac{\mathrm{cov}((S_T - K)^+,S_T)}{\mathrm{cov}(S_T,S_T)} $$with respect to say ... 1answer 1k views Replicating strategy in the Black-Scholes model I have a two-asset Black-Scholes model for a financial market: dB_t=B_t r dt dS_t=S_t(\mu dt+\sigma dW_t) I introduce a European claim \xi=max(K,S_T) with maturity T, for some fixed K. I ... 1answer 755 views Black-Scholes American Put Option Here is my question: This is a question about Black-Scholes model, but it may be applicable to more complicated models. Throughout the discussion, the strike price K, interest rate r and ... 0answers 75 views Is there an arbitrage free option model that treats volatility as a deterministic function of strike? I am trying to get a good understanding of the different models out there, and thus be able to study hedging errors, and strengths and weaknesses. My understanding of the Local Volatility model in ... 0answers 264 views Finding the dynamics of a dividend paying asset under arbitrary numeraire Assuming I have a dividend paying asset S with dividend process D. Now I would like to use the bank account process B as numeraire and determine the dynamics of S under the the corresponding ... 0answers 350 views Black-Scholes PDE to heat equation, nonconstant coefficients Can someone provide me with details or a reference on how to transform the Black-Scholes PDE with nonconstant coefficients (i.e. r=r\left(S,t\right), \sigma=\sigma\left(S,t\right)) to the heat ... 2answers 355 views Black-Scholes and Fundamentals So basically dS_t=\mu S_tdt+\sigma S_tdWt and \mu=r-\frac12\sigma^2 I have just been thinking about this later equation. This is very interesting because it ties together risk-free rate, ... 2answers 203 views When should we delta hedge? Let's say I'm the seller of a European call option on a non-dividend paying stock. I pocket the premium c_0 of the call at t=0. If I start to delta-hedge right away, this is equivalent to ... 2answers 811 views Understanding the solution of this integral The following integral represents an expected value of a geometric brownian motion for S_T>K (i.e. part of the Black-Scholes call option price):$$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\... 1answer 194 views Where can I find a clear explanation (brief derivation) of N(d1) and N(d2)? Where can I find a good explanation (perhaps with a brief derivation) of N(d1) and N(d2) from Black-Scholes? Just trying to understand the general idea about these 2 probability functions and how they ... 3answers 456 views Derivation of BS PDE problem using Delta hedging I've always been confused with Delta hedging. It is well-known that for a (smooth enough) function of$(S,t)$we have, due to Ito's lemma, that: \begin{eqnarray*} dC = \left(\frac{\partial C}{\partial ... 2answers 246 views Is it possible that under Black-Scholes:$\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$I have a slide on which there is written that under Black-Scholes model: $$\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$$ Now, here there is a good explanation ... 2answers 771 views Gamma/delta dynamics in the Black Scholes model and it's relation to PnL (Basic of option theory) If we are in a Black Scholes setup and a I have a Call option and hedged it by shorting delta amount of its underlying. What does the second derivative of the call with respect to Price of the ... 2answers 277 views Black Scholes: How does it help to transform uncertainty and still not be able to calculate a fair price? Recapitulating the history of Black-Scholes: Nobody knows the fair price of options. Revolution: BS! You put in all the parameters and get a price -> A Nobel Prize for that one! Wait: Nobody knows ... 2answers 744 views C# - Using Black Scholes Newton returns NaN occasionally First caveat: I'm a programmer doing this for a client, and my knowledge of options probably has holes in it. So be a little forgiving here. =) The Issue: When I run Black Scholes Newton against ... 2answers 1k views Is vega of Black-Scholes European type option always positive? We assume we work in the risk-neural measure with a stock which pays no dividend and a continuous discount rate. For PUT and CALL only: can someone please clarify if what I said is correct? The ... 2answers 216 views How to calculate implied correlation via observed market price (Margrabe option) I can't seem to figure out how to do the following: compute the implied correlation$ρ_{imp}$by using the observed market price$M_{quote}$of a Margrabe option, and solving the non-linear equation ... 1answer 498 views Equivalent Martingale Measure(EMM) of Inverse of Stock Price I met this question says how to price a vanilla call option$C(St,t,T,K) = \frac{1}{S_T}$which pays the inverse of a stock$V_{t} = \frac{1}{S_{t}}$at maturity if the stock price follows a geometric ... 1answer 174 views Why is$N(d_2)$not needed for hedging? I'm trying to understand delta hedging. If I sell a plain vanilla call option, in order to delta hedge it, I have to buy delta amount of stocks. What I don't understand is that the BS price of the ... 1answer 283 views How to get Black Scholes' Geometric Brownian Motion differential form form the closed form? My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that: $$S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu dt + \... 2answers 2k views Relationship between European, American options volatility Suppose, if the price of a European option (say a put) can be shown to be monotone in volatility (say for any maturity), does it follow that American options has to be monotone in volatility? ... 2answers 236 views Conceptual explanation of the relationship between gamma and vega plotted against delta for a European call option I recently plotted Gamma and Vega against Delta for a European call option and found that the graphs look very similar. This makes sense to me mathematically since the two formulas are pretty much the ... 1answer 143 views Do *all* non-dividend paying assets have the risk-free instantaneous return rate under the risk-neutral measure? For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics dB_t. The risk-free rate is r (one may assume it as constant for the time being). ... 1answer 254 views Determine price of financial contract I wonder if some one can help me with the solution to this question from Björk's "Arbitrage theory in continuous time": At date of maturity T_2 the holder of a financial contract will obtain ... 2answers 518 views ultra-long tenor European call option valued using Black-Scholes For tenors > 100 years (i.e. 150 years), my Black-Scholes model is telling me the price of a call option will be equal to its current stock price. Can anyone intuitively explain this? Thank you 1answer 175 views Is the asset-or-nothing call option in this example valued incorrectly in the Black-Scholes framework? I understand the solution to the author's example below, but I can't help but notice that the implied volatility is an imaginary number: The time-t price of an All-or-nothing Asset Call is S_t e^{-... 2answers 119 views Price and constant hedging portfolio for straddle: X=|S(T)-K| wondering if somebody could check my answer for a homework question! Given a straddle, characterized by its pay-off at maturity X=|S(T)-K|, I am asked to find the price of the (simple) claim at any ... 1answer 338 views derivation of general black-scholes formula I would like to find a derivation for the Black-Scholes fomrula in the general case (i.e., where the volatility function \sigma : [0,T] \to \mathbb{R}^+ and the investment rate r: [0,T] \to \mathbb{... 1answer 275 views Basic practical question about Delta hedging I am trying to understand a simple thing about Delta hedging in the Black-Scholes world. I know I'm doing something blatantly wrong, I just can't see it now. Let's say I write a call option and sell ... 1answer 4k views derive vega for black schole call from this formula? Is it possible to get the right formula for vega of a call option under the black scholes model from this formula?$$\frac{\partial{C}}{\partial{\sigma}}=\frac{S_0}{\sqrt{2\pi}}{e^\frac{-d_+^2}{2}}(\... 1answer 263 views Continuity of Black-Scholes formula How to proof B&S pricing formula is continuous in time$t$(or it is not?). The general pricing formula is $$C_t = e^{-r(T-t)} \mathbb{E}^*[(S_T-K)^+ | \mathcal{F}_t] \hspace{1cm} 0\leq t\leq T ... 1answer 796 views Does Implied Volatility always exist? I am considering a simple Heston Model Market with one risky and one riskless asset. The dynamics of the riskless asset is simply dB_t=r*B_t*dt The dynamics of the risky asset is as follows, ... 1answer 937 views options pricing using vwap This is a question about why options prices do not take volume into account. The popular option valuation formula "black-scholes" certainly does not account for this and I don't suggest that it does. ... 1answer 73 views When is a numerical solution the only way to obtain a solution to BS? I am only now reading into Mathematical Finance, I understand the derivation of the BS equation with vanilla European options. On the next page of my book it starts to delve into obtaining exact ... 1answer 61 views Proof that we can price any derivative as the discounted value of its expected return under the risk neutral measure I am reading a paper which tries to convey the intuition behind the Black-Scholes pricing formula. In that paper, the author states the following two things without proof, and I would like to know why ... 1answer 53 views How can the BS riskless hedge break down when volatility changes, if a random walk can produce any price history? Supposedly, a Black-Scholes riskless hedge will break down if the volatility is non-constant. However, a random walk with any sigma could produce any price history with some non-zero probability. If ... 1answer 571 views Black-Scholes pricing of binary options I'm trying understand something basic about Black-Scholes pricing of binary options. In my example above, the current price is over the strike price. The volatility is extreme but I'm still having ... 1answer 112 views Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 20,21 Find the Black-Scholes price of an option paying$$(S_T^{\alpha} - K)_{+}$$at time T. Solution - The forward price is given by$$F_T(t) = e^{r(T-t)}S_t$$So,$$F_T(0) = e^{rT}S_0$$and$...
Assume $S_0(t)=\exp(\int_0^t r(s) ds)$. Then $\mathbb{Q}\sim \mathbb P$ is a martingale measure $\iff$ every asset price process $S_i$ has price dynamics under $\mathbb Q$ of the form \$dS_i(t)...