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16 votes
5 answers
48k views

Bachelier model call option pricing formula

Does anybody have the Bachelier model call option pricing formula for $r > 0$? All the references I've read assume $r = 0$. I don't speak French, so I can't read Bachelier's original paper.
Galsunja's user avatar
  • 161
9 votes
4 answers
24k views

What does it mean to be "long or short in volatility"?

I've heard a question regarding pricing of european calls. The question is: Is the call long or short in volatility when it is (deep) OTM? What is the profile of the implied volatility? I know ...
Cooper's user avatar
  • 91
7 votes
3 answers
2k views

Which is riskier: a call option or the underlying?

From Joshi's Quant Interview Questions and Answers: What is riskier: a call option or the underlying? (Consider a one day time horizon and compute which has bigger Delta as a fraction of value). I ...
Trajan's user avatar
  • 2,492
7 votes
1 answer
912 views

Why is the Put-Call Symmetry model dependent?

The put-call symmetry states that C(S,t;X,r,q) = P(X,t;S,q,r), and that this works for American options. According to my notes, this is 'model dependent' because it ...
Twilight Sparkle's user avatar
5 votes
1 answer
637 views

Equivalent form of Black-Scholes Equation (to transform to heat equation)

I am trying to understand the transformation of the Black-Scholes equation to the one-dimensional heat equation from Joshi, M. (2011). The Concepts and practice of mathematical finance. 2nd ed. ...
LeptoSq's user avatar
  • 53
5 votes
1 answer
722 views

At-the-money Call Spread approximation

In a trading manual I got during a course, the value of the ATM Call-Spread is approximated by $CS_{ATM}=\frac{1}{2}StrD+(F-m)\times\Delta CS$ The lecturer skipped the part where he derived this ...
Cindy88's user avatar
  • 431
5 votes
1 answer
250 views

FX Call under stochastic rates and deterministic volatility

Lets denote $S_t$, $r^d_t$,$r^f_t$ respectively the FX spot, the domestic rate and the foreign rate at time $t$. Lets $\mathbb{Q}^d$ , $\mathbb{Q}^f$ respectively be the domestic and foreign mesures,...
DeepInTheQF's user avatar
5 votes
0 answers
1k views

Greeks of a Basket Option

I want to estimate delta, vega and gamma for a basket option. This option is a European Call option. The underlying is $S=\omega_1 S_1 +\omega_2 S_2$ Where: $S1$ = stock price of asset 1 $S2$ = ...
Separata's user avatar
4 votes
5 answers
10k views

Value of Call Option as Volatility goes to Infinity

Why would the value of a call option go infinity as volatility goes to infinity? I understand how you could solve this question by taking $\sigma \rightarrow \infty$ in the solution to the black ...
Trajan's user avatar
  • 2,492
4 votes
2 answers
2k views

How many monte carlo runs do I need for pricing a Call?

I have to price several calls using Monte Carlo. Obviously, there is a huge tradeoff between the number of runs and the fair price of the call option. I know I can check how the approximation changes ...
phdstudent's user avatar
  • 8,306
4 votes
5 answers
2k views

Black-Scholes formula proof, without stochastic integration

I've looked into many books at my academic library, and very often it goes like this: Brownian motion Then, stochastic integration (Itô's formula etc.) Application: Black-Scholes formula for price of ...
Basj's user avatar
  • 787
4 votes
3 answers
109 views

Abritrage when Put Option Greater then Strike Price?

I am having a tough time conceptualizing this question here: Let $P$= Price of European Option, $S$ = Present Price of Option and $K$ = Strike Price. If $P > K$, why does abritrage exist? Assuming $...
Efrain Olivenhain's user avatar
4 votes
1 answer
192 views

Is the vega of a portfolio of a long 0.5 delta and short two 0.25 delta calls positive or negative?

More specifically what I am trying to find out is whether the following relationship is always true or not. Same underlying for the calls, assume the most simplistic assumptions (interest rate = ...
mebiles's user avatar
  • 41
4 votes
0 answers
186 views

Zero-rebate barrier option pricing under the Heston model

I'm trying to derive an approximation for the zero-rebate barrier option under the Heston model: $$dS_t=\mu S_tdt+\sqrt{v_t}S_tdW^S_t$$ $$dv_t=\kappa(\bar{v}-v_t)dt+\eta\sqrt{v_t}dW^v_t,\quad d\langle ...
FunnyBuzer's user avatar
  • 1,012
3 votes
2 answers
1k views

Flaw in the following argument with Binary Options and Skew

A Binary option is ATM and expires tomorrow. If the skew of the vanilla options steepens (left side up, right side down) what happens to the price of the Binary Option. I know that using a ...
Trajan's user avatar
  • 2,492
3 votes
1 answer
238 views

Given $\mathbb{E}[X]$, $\mathbb{E}[\max(0,X)]$, and $\mathbb{E}[\min(0,X)]$, what is $\mathbb{E}[f(X)]$

Let $X$ be any random variable with any distribution. Given that we know $\mathbb{E}[X]$, $\mathbb{E}[\max(0,X)]$, and $\mathbb{E}[\min(0,X)]$, can you write a formula for $\mathbb{E}[f(X)]$ where $f$ ...
iluvmath's user avatar
  • 143
3 votes
1 answer
451 views

Can we trade theta?

The context is this theoretical result from Black-Scholes-Merton differential equation that the effects of theta and gamma cancel each other. Equation (5.23) from the book titled "Option Trading&...
TryingHardToBecomeAGoodPrSlvr's user avatar
3 votes
1 answer
145 views

Qualitative properties of call

I have read somewhere that we can show by using arbitrage argument the following relationship for call option : $$\frac{\partial{C_t(T,K)}}{\partial{K}}\leq0$$ $$\frac{\partial^2{C_t(T,K)}}{\...
ashu24's user avatar
  • 41
3 votes
2 answers
5k views

Estimate simple option price without a calculator

I have been to two different interviews for jobs related to option trading, and both time I have been asked a question, which is pretty basic, and still I could not answer it. If you have an European ...
Sithered's user avatar
  • 808
3 votes
1 answer
3k views

Delta Hedging/ Exchange for Currency Options

I'm looking at 2 cases of hedging EURUSD, using call spread or range forward. Lets say spot is 1.1300 and my buy call is at 1.1300 and sell call is at 1.1500. Hypothetically I'm assuming that this is ...
Shyam's user avatar
  • 71
3 votes
2 answers
2k views

Calculating Greeks in Covered Calls?

Just want to confirm whether Delta, Gamma, Theta, Vega will be calculated in the following way? Since we own 100 shares of stock while selling a call we need to subtract greek value from one? right? ...
Vtech's user avatar
  • 355
3 votes
1 answer
231 views

Digital call under Ornstein-Uhlenbeck dynamics

I am trying to price a digital option with payoff $\mathbb{I}_{S_T>K}$, where $S_t$ follows the Ornstein-Uhlenbeck dynamics $\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma\mathrm{d}W^{\mathbb{Q}}_t$ in the ...
user107224's user avatar
3 votes
1 answer
539 views

Boundaries for Call Spread

I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts regarding part of its solution and highlighted them in bold: Question: What are the ...
M00000001's user avatar
  • 647
3 votes
1 answer
604 views

Derivation for call option upper bound

In Euan Sinclair's book, Option Trading, he writes that $c <= S$, the price of a European call must be lower than the price of the underlying stock. To prove it, he applies the principle of no ...
Dürenand's user avatar
3 votes
1 answer
248 views

Call option prices in terms of maturity with negative interest rates

let's assume that interest rates are constant, $r$. When $r\geq 0$, we can see that if $T_1<T_2$ and $C_1$ (resp. $C_2$) is the price of a call option on a non-dividend paying stock with maturity $...
dbluesk's user avatar
  • 175
3 votes
2 answers
221 views

arbitrage opportunity in a two period model

I have a little problem evaluating an european call. I Suppose the following: in $$t=0 : S_0 = 10$$ $$t = 1 : S_1 = \{10,11\}~with ~p=0.5$$ riskless rate : $(1+r)=\beta=1.049$ Strike ...
Dachser's user avatar
  • 76
2 votes
4 answers
327 views

How to short an option?

It appears to me that retail investors can only buy calls and puts, but not short them through any standardized way (except maybe borrowing the option from a friend ;) ). Is that correct, or how can ...
emcor's user avatar
  • 5,795
2 votes
2 answers
561 views

How to make the arbitrage if intrinsic value is greater than European call value

It always says if the intrinsic value is greater than European call value, there will be a arbitrage opportunity,but how to construct the portfolio $(S_t - K)^+$ or how to make this arbitrage. By the ...
A.Oreo's user avatar
  • 1,243
2 votes
2 answers
2k views

Black-Scholes call option formula, which probability measure

The stock and bond under the Black-Scholes framework, no dividends: $$S_t=S_0e^{\sigma W_t+\mu t}=S_0e^{\sigma \tilde{W}_t +(r-\frac{1}{2}\sigma^2)t}$$ $$B_t=e^{rt}$$ where $\tilde{W}_t$ is $\mathbb{Q}...
none's user avatar
  • 365
2 votes
1 answer
345 views

Why is this inequality strict for arbitrage argument for European call?

in the notes about arbitrage arguments I am reading, I notice the statement We can also see that $$C^E_t>(S_t-K\mathrm{e}^{-r(T-t)})^+$$ Notice that the inequality holds STRICTLY! I don't ...
Ice Tea's user avatar
  • 185
2 votes
2 answers
2k views

Why it is not possible to price American perpetual call option using PDE approach?

Using a standard PDE approach to price an American perpetual put option I obtain that the price of such option has the following form: $$ V(S) = A S + B S^{-2r/\sigma^2}. $$ And then I need to find a ...
MMM's user avatar
  • 153
2 votes
1 answer
277 views

Greeks of portfolio in response to underlying price change

I'm trying to wrap my head around Greeks, and I'm getting a little bit confused. For example, let's say my portfolio holds a long 5 month ATM call with strike \$20, and short 2 month OTM call with ...
Ice Tea's user avatar
  • 185
2 votes
1 answer
1k views

Black-Scholes Formula under $T$-forward measure

The Black-Scholes price of a European call option is given by $$ C_0^{BS}(T, K) = \mathbb{E}_Q[e^{-rT}(S_T - K)_+] = S_0 \Phi(d_1) - Ke^{-rT}\Phi(d_2) ,$$ where $$ d_{1,2} = \frac{\log\big(\frac{S_0}{...
R. Rayl's user avatar
  • 466
2 votes
1 answer
630 views

How to derive and interpret the duration of a call option?

I read here that CFA students are taught that $$ D_{C} = \frac{\Delta_{C} D_{B} B}{C} $$ Where $D$ is the duration, $\Delta_{C}$ is the first derivative of the options price with regards to the ...
jthg's user avatar
  • 445
2 votes
4 answers
1k views

pricing american calls on non dividend paying stocks

It is never optimal to exercise an american call option early if it is written on a stock that doesn't pay dividends, yet when pricing such an option, using a binomial model, we check whether or not ...
WeakLearner's user avatar
2 votes
1 answer
473 views

How to price a call option which depends on two Wiener processes?

Could someone explain to me why the regular call pricing formula works, just with $\sigma$ replaced by $\|\sigma\|$ in the case where the underlying asset depends on two Wiener processes? For example,...
Jin's user avatar
  • 21
2 votes
2 answers
3k views

difference between caplet and call

I wanted to know the difference between a caplet and a call. In my course (Interest rate models and curves) , we said that a caplet is a call option. Is it really true? Thanks
Sino's user avatar
  • 77
2 votes
2 answers
3k views

fair price for a call option

I am struggling with the following problem: An investor is considering a European call option, whose price $C_0$ is yet to be determined, on the shares of a company called XYZ. You know that : the ...
user7130's user avatar
2 votes
1 answer
285 views

Double Call Option

A double call option allows the holder to either exercise at time $T_{1}$ or time $T_{2}$, where $T_{2}$>$T_{1}$. With corresponding strike prices $K_{1}$ and $K_{2}$, it can be shown that it is never ...
cmcw's user avatar
  • 51
2 votes
1 answer
214 views

Explaining an Option product: SIX Discount Certificates

So I have the option with the important info above. I am trying to generate a portfolio that represents the option. However I am stuck on the first hurdle as I believe it is a call option as the ...
chocolatekeyboard's user avatar
2 votes
1 answer
154 views

How does gamma trading depend on $K$?

If we think realized vol > implied vol, then we might go ahead and delta hedge a call, hoping that profits from gamma outweigh the decay. Question: What should $K$ be on the call? ATM? If so, why? ...
Mane's user avatar
  • 21
2 votes
1 answer
206 views

Put call parity: when are the premiums the same?

Please explain why put call parity could be compared to the payoff of a long forward contract. ie. $C_E-P_E=V_X(0)$ where $C_E,P_E$ are the call/put premiums and $V_X(0)$ is the value of a long ...
foshizzle's user avatar
  • 432
2 votes
1 answer
82 views

Understanding the necessary and sufficient conditions for rational early exercise of a call option

I am self-studying for an actuarial exam, and I encountered the following in my text: The author states that if $PV_{t, T}\text{(Divs)} < K(1 - e^{-r(T - t)})$, early exercise is not rational. ...
user2521987's user avatar
2 votes
1 answer
163 views

Quick way to extrapolate call price as function of strike

Let's say I know the price of a call for two different values of strike. Is there a quick way to guess the price for another value of strike ? Actually, I know that C(100)=15 and C(90)=20 and I have ...
Dark's user avatar
  • 427
2 votes
1 answer
486 views

Gamma-neutral delta-neutral call ratio spread

I have been looking into options strategies that minimize risk via delta neutrality. One such strategy seems to be the gamma-neutral delta-neutral call ratio spread, in which the gamma is neutralized ...
Dhruv Kapu's user avatar
2 votes
1 answer
378 views

Risk Reversal quoting convention in FX market

How is RR bid offer quoted in market? For example: If a 25delta call and 25delta put is quoted as 5.5%/5.6% and 5.3%/5.5% respectively. What would be quote of a 25d RR with these call and Put?
Ussu's user avatar
  • 577
2 votes
1 answer
291 views

Fair value for a LEPO (Low Exercise Price Options)

In one of my lecture notes, I stumble across this exercise question: Consider Low Exercise Price Options, LEPOs, (with dividends) in Australia. Using the value at the outset, explain why such ...
Joshua's user avatar
  • 123
2 votes
1 answer
725 views

Will pricing a Bermudan option default to a value of a European option?

I have a call option with 2 expiry in two years. For the first 9 months I cannot excercise the option. After that the I can exercise at any time. I am pricing this option using a binomial tree using ...
PBD10017's user avatar
  • 623
2 votes
0 answers
99 views

Path integral approach to price call option on zero coupon bonds

I am given the following identities: $$ Z[J,t_1,t_2]=\int D W e^{\int_{t_1}^{t_2}dtJ(t)W(t)}e^{S}=e^{\frac{1}{2}\int_{t_1}^{t_2}dtJ(t)^2} $$ $$ \int_t^Tdx\alpha(t,x)=\frac{1}{2}\left[\int_t^Tdx\sigma(...
TheHunter's user avatar
  • 133
2 votes
0 answers
337 views

Callable Bond = long Bond - call on bond?

Can someone verify (maybe there is some literature around) the following relationships? Callable Bond= Long on Bond + short on a Call Position --> PV(CallableBond) = PV(Bond) - Call on Bond? or ...
Kosta S.'s user avatar
  • 209