Questions about models for the valuation of option contracts.

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Exercise 2.2 from the book “The concept and practice of Mathematical Finance”

I am a newbie. Please help me understand how to resolve the exercise 2.2 from the book "The concept and practice of Mathematical Finance". The solution from the book says that our super-replicating ...
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2answers
105 views

Derivation of Stochastic Vol PDE

A couple questions regarding stochastic vol PDE derivation. Following Gatheral, a general stochastic vol model is given by \begin{align*} dS(t) & = \mu(t) S(t) dt + \sqrt{v(t)}S(t) dW_1, \\ dv(t) ...
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1answer
24 views

Difference between Closing Price, Last traded price and Settlement Price for option contracts?

What is the difference between Closing price, Last traded price and settlement price ? I got the difference between Closing Price and Settlement price from previous post : The difference between ...
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1answer
31 views

Delta hedging cost of exotic options?

I'm simulating dynamic delta hedging for up-and-out call option. For plain vanilla call options, I heard that the option price is the expected value of the accumulated delta hedging cost. Does it also ...
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1answer
45 views

Connection between implied volatily and implied probability

I am reading some lecture notes about Black-Scholes (BS) option pricing. Since the BS-formula is not supported by observed data because of the dependence of the implied volatility on the strik and ...
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1answer
56 views

Pricing employee stock options

ESOs are typically priced using the black-scholes model, but with an additional parameter for for the employee turnover rates . An example ...
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43 views

why many option contract price less than minimum boundary price?

I downloaded data from NSE(National Stock Exchange) website regarding closing price of European Call Option written on Index. From standard textbook, I read that option contract must satisfy $C(t) ...
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7answers
4k views

Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while. We all know geometric brownian motions have drifts $\mu$: $dS / S = \mu dt + \sigma dW$ and different stocks have different drifts of $\mu$. Why would ...
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1answer
614 views

How to price a Swing Option?

I'm working in the commodity market and I've to price Swing Options with MATLAB, preferably with finite element. Has anyone already priced these kind of derivatives? I'm thinking about using the ...
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3answers
333 views

Why do people always seek finite-variance models for option pricing

For the purpose of getting fatter tails than the Guassian, I have seen people for example use $\alpha$-stable processes to model the stock. But in that case they end up using 'tempered' versions of ...
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2answers
125 views

how we can derive $PIDE$ of double exponential Jump-diffusion model (we know as kou model)?

I'm working in double exponential Jump-diffusion model (we know as kou model) with following form , under the physical probability measure $P$: \begin{equation} ‎\frac{dS(t)}{S(t-)}=\mu‎‏ ‎dt+\sigma ...
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1answer
108 views

Which distribution do I get?

Let's assume the stock moves according to a classic Black-Scholes model, and makes a proportional jump with an unknown proportion. Say, it is either +1% or -3% of the stock value, and we know for sure ...
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analytic formula for the value of an American put option

It seems to be a foolish question but I can't take my mind off from , Is it true that there is no analytic formula for the value of an American put option on a non-dividend-paying stock (or a divident ...
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1answer
104 views

Real world monte-carlo (P-measure)

Consider the 2 following approaches to pricing a security: Monte-carlo ($\mathbb{Q}$-measure) $\begin{equation} C = \frac{1}{N} \sum_{i=1}^{n} e^{-rT} max(S_i(t) - K, 0) \end{equation}$ Monte-carlo ...
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3answers
98 views

How free are we in risk-neutral distributions?

Suppose we do not have a particular pricing model, we have just a frictionless market with constant interest rate (say $0$), and some traded stock $S$ which does not pay dividends. For any expiry $T$ ...
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33 views

What's the risk-neutral expectation of the arithmetic average of stock price?

All Black-Scholes assumptions apply ($y$ is yield): what's $E(A_T), E(A_T^2)$ and $Var(A_T)$ where $A_T=\frac{\int_0^T S_tdt}{T}$ is the continuous-sampling arithmetic average of the stock price ...
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20 views

Risk neutral pricing formula justification in incomplete markets [duplicate]

I'm having trouble understanding how to justify the use of the risk-neutral pricing formula $V(t) = \mathbb{E}^{*}[e^{r(T-t)}H(S_{T})|\mathcal{F}_{t}]$ in models which are characterized by ...
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65 views

Black-Scholes formula with deterministic interest rate and dividend yield

Does any one have the Black-Scholes formula for a European call with time-dependent but deterministic interest rate and dividend yield ?
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1answer
38 views

Calculate put price with Black-Scholes and one discrete dividend

I try to solve this exercise: a) Calclculate the price of a 3-month European put option on a non-dividend-paying stock with a strike price of 45 when the current stock price is 40, the risk-free ...
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1answer
119 views

Time value of option not always leading to an increased option value

My understanding was that as you increase the time to expiry of an option, the value of the option increases. However, I have run a bunch of scenarios and have realized that if you assume a dividend ...
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2answers
74 views

Time 0 value of an American Put in Cox-Ross-Rubinstein model

This is a question from a problem sheet which I have handed in and have solutions for. The only examples of this in class I have seen are examples where the interest rate is 0. "Consider a ...
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1answer
53 views

Do I need simulink to model the risks of an option portfolio

I wish to buy Matlab Home and learn to model the risks of a derivatives portfolio and then stress test it. So I am guessing I will need : Stochastic calculus Linear algebra Stats/Probability Some ML ...
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1answer
111 views

Heston Model Option Price Formula

What is the formula for the vanilla option (Call/Put) price in the Heston model? I only found the bi-variate system of stochastic differential equations of Heston model but no expression for the ...
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Shorting a Synthetic Long [closed]

I have the following information: Call Premium: 0.30 Put Premium: 40.4 Strike: 130 1-Month Risk-Free Rate: 0% Market Price: $85.00 If I use the Synthetic Long ...
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2answers
184 views

good R package for vectorized option pricing

I am using for now the package fOptions but it doesn't allow for vectorized computation of black76 prices and delta. Which package can be used to do that? As noted ...
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156 views

Normalized price process $Z(t)=\frac{\Pi(t)}{B(t)}$

If an interest rate model with the following $P$-dynamics for the short rate. $$dr(t)=\mu(t,r(t))dt+\sigma(t,r(t))d\bar{W}(t)$$ Now consider a $T$-claim of the form $\chi = \Phi(r(T))$ with ...
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1answer
73 views

How can one value a Bermuda option?

A Bermuda option allows early exercise at predefined dates, e.g. at maturity equal to $t_1$, $t_2$, $t_3$,...; hence , would its value be the sum of 3 discounted European options with 1-year ...
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1answer
72 views

Numerical example of how to calculate local vol surface from IV surface

I'm looking for an excel example (not a copy of Dupire's eqn) of how to convert an IV surface to a local vol surface. If unsuccessful I'll work through Dupire's eqn but would be helpful to look at an ...
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49 views

Hedging - calculating option prices using implied volatility surface

To hedge a strategy is it accurate "enough" to price an option using an implied vol curve vs moneyness (strike/spot) assuming sticky delta? The moneyness can be read off the chart, its corresponding ...
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1answer
87 views

Need for Binomial Representation Theorem

In some texts (e.g. Baxter & Rennie, Shreve I) the binomial model is first constructed using the usual backward induction argument, and it is concluded that by no-arbitrage the time $t$ value of a ...
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3answers
265 views

Option Pricing Model Calibration In Practice

I'm curious how an option pricing model like the Heston model is calibrated in practice. Here's how I imagine it happens: Let's say I have access to the most recent option prices on a given stock ...
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1answer
96 views

Boundary conditions of PDE from SV model with stochastic interest rate

The PDE for the American put option price $P(S,\sigma ,r,t)$ is \begin{align*} 0 =& P_t+P_SS(r-\delta)+P_\sigma a(\sigma)+P_r\alpha (r,t) \\ +& \frac{1}{2}P_{SS}S^2\sigma ^2 + ...
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64 views

Local volatility parametrization using the spot

Is it possible to estimate the local volatility using the spot price S at time t instead of the strike price K and the expiry date T ? Any help would be appreciated.
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134 views

Arbitrage bounds for Black-Scholes

In some implied volatility code I came across, there is a check to ensure there is no violation of the arbitrage bounds based on the inputs to the method. For the call option, if $$P < 0.99 * ...
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2answers
81 views

Hedging portfolio and extraction PDE of SV model with stochastic interest rate

How can I extraction this PDE \begin{align*} 0 =& P_t+P_SS(r-\delta)+P_\sigma a(\sigma)+P_r\alpha (r,t) \\ +& \frac{1}{2}P_{SS}S^2\sigma ^2 + \frac{1}{2}P_{\sigma ...
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Price of an American call option [closed]

I'm working through revision questions at the moment and we are asked to compute the price of an American call option. Suppose that $dS_t = \sigma S_t dW^*_t, S_0 >0$ Let $0<U<T$ be fixed ...
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2answers
67 views

Importance Sampling - where to center the sampling distribution?

Consider a Monte Carlo (MC) approximation to a European call with BS parameters $r = 0.05, \sigma = 0.4, T = 10, S_0 = 50$ and $K = 95$. Consider the following results, each using 1M points: plain ...
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41 views

Benchmarking option pricing under stochastic interest rates

I priced a long-term option (10 or 20 years) using two different models: one assumes constant interest rates, the other assumes stochastic interest rates. Is there a way (e.g. a benchmark) to ...
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replicating strategy three step binomial

I am having some trouble setting up a replicating strategy for a call option with a three step binomial model (discrete). I have no trouble doing this in a two step binomial model by backward ...
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1answer
105 views

Distribution of Black Scholes call option price at time 0<t <T

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $? (Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t ...
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“Hedging” a put option, question on exercise

I have a question on the following exercise from S. Shreve: Stochastic Calculus for Finance, I: Exercise 4.2. In Example 4.2.1, we computed the time-zero value of the American put with strike ...
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1answer
63 views

Value of European Call equals Value of American Call, Question on Explanation/Proof

I am reading S. Shreve, Stochastic Calculus for Finance, Vol. I. There he proves that American Call Options have the same value as European Call Options. In the proof he uses that for a Call option ...
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1answer
102 views

Black-Scholes under stochastic interest rates

I'm trying to implement the Black-Scholes formula to price a call option under stochastic interest rates. Following the book of McLeish (2005), the formula is given by (assuming interest rates are ...
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5answers
2k views

How to price a calendar spread option?

How do you price calendar spread options, that is, options on the same underlying and the same strike but different times to maturity? Clarification: I'm interested in the pricing of a a CSO ...
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2answers
169 views

Option arbitrage with dividends?

If a stock pays a discrete dividend, the stock price falls by the amount of the dividend. There is no arbitrage opportunity from this predictable jump, because the investors receive the same amount of ...
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816 views

From Fourier Transforms to Option Values

I am trying to understand how Fourier transforms & Characteristics functions can be used to calculate option values. However, I am having difficulty following the process that is used in several ...
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3answers
215 views

Greeks: Why does my Monte Carlo give correct delta but incorrect gamma?

For a vanilla European call, my Monte Carlo method gives the right option price and delta but the wrong gamma. In particular, the value of gamma varies wildly each time I run the method. I estimate ...
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2answers
260 views

FX Delta Conventions

I'm currently reading Iain Clark's book Foreign Exchange Option Pricing and I got stuck at one sentence in the beginning of Section 3.3 that I feel is important to understand. He writes: FX ...
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1answer
73 views

Tradeable => Satisfies pricing equation?

In Wilmott's third volume, on p. 857, he tries giving an insight into the market price of risk by showing what it is for traded assets. For this he constructs a portfolio of two different options: ...
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Delta and gamma neutral

A financial institution currently has a portfolio with delta of 450 and gamma of 6,000. A traded option is available with a delta of 0.6 and a gamma of 1.5. How could the portfolio be made both delta ...