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Proof of the put-call parity formula

I just learned about the put-call parity formula and read the proof of it, which goes as follows. Put-call parity formula: Let $C,P$ denote respectively the prices of a call and a put, both of ...
letmethinkaboutit's user avatar
5 votes
1 answer
627 views

How does a stock becoming hard to borrow affect puts and calls?

Here is my understanding from what I’ve gathered, but I want to confirm if this is correct (and/or if there’s something I’m missing). If a stock becomes hard to borrow, one can create a synthetic ...
rb612's user avatar
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Why does the risk-free rate implied by put-call parity vary with strike prices?

Suppose I do the following: buy one lot of some underlying stock currently trading at price $S$, write a call with strike price $K$, earning some premium $C$, and buy a put with the same strike $K$, ...
Jérémie Koenig's user avatar
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1 answer
143 views

Put-Call Parity; Time Value of Money

The intrinsic value of a call option is found by subtracting the discounted strike price from the current share price: $IV = S - X/e^{rT}$ Put-Call parity: $S + p = c + X/e^{rT}$ $c = p + (S - X/e^{rT}...
quantum's user avatar
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87 views

Satisfying put-call parity in Monte Carlo option valuation

I am trying to price European call and put options on a stock using the Monte Carlo method, given some dynamics for the underlying that may or may not have a closed-form solution (e.g. Black-Scholes, ...
SupSquark's user avatar
2 votes
2 answers
297 views

How to apply put-call parity in volatility surface construction?

How to make the volatility surface free of put-call parity arbitrage? If I bootstrapped the implied vol from a call price and plugged it into the BS model to have a put price, what if it violates the ...
Parting's user avatar
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2 votes
0 answers
40 views

Finding upper bound for portfolio made from European call / put options

I tried finding upper bounds for each component in terms of E_1 using the put call parity but couldn’t get the correct answer.
Alex Seitzy's user avatar
1 vote
0 answers
216 views

Put-call parity and different IVs for puts and calls

For vanilla options put-call parity $$ C - P = F $$ must hold. On the other hand, it can happen that puts can have different implied vols than calls, due to for example shorting restrictions, ...
Frido's user avatar
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6 votes
2 answers
2k views

Is it possible to have only one volatility surface for american options (that fits both calls and puts)?

Put-Call Parity does not hold for american options. Hence, I don't see how it would be possible to have one surface that would encompass both calls and puts. For example: Let pick a call lying in the ...
Rodrigo's user avatar
  • 310
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1 answer
66 views

How does a principal protected note pay the return on the market, on the whole principal? [closed]

A real life principal protected note pays exactly the index return with 50% participation and a max 3y return of 30%: total_principle*(end_price/start_price-1)*50% These are all of the features, ...
nemui's user avatar
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2 votes
1 answer
357 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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1 answer
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Use the put-call parity to get the implied swap rate of a YoYIIS cap(floor)

Solved As pointed by @dm63 in the comments, the implied swap rate can be derived by solving the caplet (floorlet) formula for the the interest rate, where you set the formula equal to the Swap NPV (...
peanut_butter_from_the_tub's user avatar
1 vote
2 answers
283 views

Put Call derivation using two approaches [ Some confusion of getting different results ]

I understand the put-call parity and am trying to derive the results based on a CME article in their education section and also the Wikipedia explanation in the Black Scholes model where Put-call ...
Anand Kulkarni's user avatar
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1 answer
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How to prove with put-call parity: return multiplier ratio of ILCD (index linked certificate deposit ) < 1

Index-linked CDs pay interests based on a specific index, and a guaranteed payment of all principal.     The return multiplier ratio (multiplier) is defined as the rate of return of the derivative ...
jimy's user avatar
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1 answer
294 views

Does the put-call-parity hold for the Heston model? [closed]

My question is quite simple: Does the Put-Call-Parity hold for the Heston model? My textbook handels the Black-Scholes model with the Put-Call-Parity being $$p_t = Ke^{-r(T-t)}+c_t-S_t.$$ However, it ...
Landscape's user avatar
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0 answers
158 views

Put call parity with real time tick data

I am working with some real time options tick data (mainly futures options and index options), and in many cases the quotes are single sided (as seen on bloomberg terminal). I will denote a quote as ...
des224's user avatar
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1 answer
319 views

Put-call parity on SPY

I'm currently trying to model the IV curve for calls and puts on SPY using the Black-Scholes model with dividends. I'm able to calibrate the risk-free rate and dividends so that both ATM IVs match, ...
Alex's user avatar
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2 votes
0 answers
79 views

Model independent (or reasonable assumption) bounds on OTM put price given an ATM call price

I am looking for model independent (or weak/reasonable assumption) bounds on price of a OTM vanilla put on strike $k1$, conditional on an observable price for a ATM call at some strike $k2$. I ...
Arshdeep's user avatar
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-1 votes
1 answer
175 views

Why does bull call spread shows higher payoff than bull put spread?

I am trying to compare bull call spread and bull put spread for equity index option. For the options where the put call parity holds, I am getting a different payoff for bull call spread and bull put ...
Sumit's user avatar
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1 answer
234 views

Maximum value of a call option proof [closed]

I'm reading Sinclair's Option Pricing and am confused by the proof for the maximum value of a call. It makes sense logically that a call can't be worth more than the underlying, and so: c <= S The ...
James's user avatar
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1 answer
702 views

Forward Index Level in VIX calculation

The VIX white paper (https://cdn.cboe.com/resources/vix/vixwhite.pdf) step #1 (page 6) says the the Forward Index Price is calculated as: F = Strike Price + e^RT x (Call Price - Put Price). Why doesn'...
MikeD's user avatar
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0 answers
242 views

VIX options underlying: Can I safely use Put-Call parity instead of VIX futures

Couple of basic questions: 1- I'd like to calculate the implied of VIX options intraday, without access to intraday VIX futures. In the absence of VIX futures as underlyings, what would be the ...
annon's user avatar
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0 answers
831 views

calculating risk free interest rate from put call parity

I'm trying to calculate the interest rate $r$ from the put-call parity. As per hull, put-call parity is given by the below equation. $c + Ke^{-rT} = p + S_{0}$ where: $c$ = current call option price ...
akshay's user avatar
  • 101
1 vote
2 answers
2k views

How to prove no-arbitrage when a long butterfly is strictly positive?

I want to prove why there are no arbitrage opportunities when a long butterfly is strictly positive. I know there is a similar topic out there, but it seems it doesn't solve my question: Prove that ...
will_www's user avatar
  • 113
2 votes
1 answer
1k views

How do borrow rates in single-stock options affect their prices

Would following approach be suitable: First calculate European option price (does it even make sense to do so, if we are talking about less than 30 dte?), take the diff between European and American ...
Bython's user avatar
  • 121
0 votes
1 answer
951 views

Put-Call Parity with dividends

In which book will I find the exact proof of put-call parity in the case when asset pays continuous dividend? I need a book to cite this result
Johhn White's user avatar
1 vote
0 answers
185 views

How to Show an Arbitrage Opportunity Exist From a Market-Linked CD?

A bank issues a market-linked CD that guarantees the original principal with an interest at an effective annual rate of 2%, plus 70% of the percentage gain on the ABC Inc. non-dividend-paying stock ...
Matthew Farant's user avatar
1 vote
0 answers
71 views

Put-call parity under a regime-switching model

I need some help. I'm given $J$ different regimes, each one characterized by its own parameters $(r_i, \delta_i,\sigma_i,...)$ with $i\in \mathcal{J}= \{1,2,...,J\}$ ($r$ = risk-free interest rate, $...
james34's user avatar
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5 votes
1 answer
904 views

Intuitive explanation of put option pricing based on put-call parity

Assuming no dividends, the put-call parity equation says: $c + Ke^{-rT} = p + S$ where $c$ is the price of the European call, $p$ is the price of the European put, $S$ is the current stock price, $K$ ...
Flux's user avatar
  • 531
2 votes
1 answer
881 views

Under Put-Call Parity, why do we add the cost of carry to Call prices but subtract them from the Stock price and Put prices?

In Natenberg (1994) Chapter 11 he outlines the Put-Call parity relationships. ...
user avatar
1 vote
0 answers
73 views

The wider bid-ask spread of in-the-forward American option

Why is the bid-ask spread of a in-the-forward/money American call (put) much larger than the out-of-the-forward/money American put (call)? I suppose the answer to the same corresponding question ...
Hans's user avatar
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0 votes
0 answers
105 views

Black-Scholes pricing of european call option

I am really confused on the usage of the greeks and the Black-Scholes model for option pricing. To gain some more understanding I am attempting to see if I can price a european call option under the ...
lambdaepsilon's user avatar
1 vote
3 answers
97 views

Equivalent combination of puts

Suppose that a certain stock is currently worth $S_0=\$61$. Consider an investor that buys one call with a strike price equal to $K_1=\$55$, that costs $c_1=\$10$, buys another call with strike price ...
Papagaio_da_Fauna's user avatar
0 votes
0 answers
26 views

Insured Portfolio via call + cash: how much cash?

I am unsure about the quantities to keep in the risky asset, S, and the non-risky asset, M, when constructing an insured portfolio via Call + Cash (rather than Stock + Put). My understanding so far is ...
s5s's user avatar
  • 442
1 vote
1 answer
183 views

Put-Call parity arbitrage relationship

I would like to know what the relationship is between the time value of call/puts. From the put call parity formula $$C-P = S_{t} - PV(K)$$ and that value of call/put options is simply the sum of ...
sebzaant23's user avatar
0 votes
1 answer
545 views

Calculating risk free rates from risky options using put call parity

My questions relates to this post Implying risk-free rates using Put/Call parity , but I am using a different approach. Given: ODAX (Options on "DAX") Settlement prices across different maturities ...
C0deP0ntage's user avatar
0 votes
1 answer
1k views

Calculate forward price based on option chain

I've got historical data for a spy option chain which looks as follows ...
Hekri's user avatar
  • 3
2 votes
1 answer
2k views

Boundary for European Put Option

As an entry level financial engineer, I'm learning about call-put parity, which helps us to get the boundary for call option: $S-Ke^{-rT}\leq c\leq S$, what about put option? Should its upper bound be ...
M00000001's user avatar
  • 647
1 vote
0 answers
74 views

What is the effect of put call open Interest on price action

how option put call open Interest affects price actions as put sellers feel price when price goes down or call sellers feel pain when price goes up and how this affects price action. ie when price ...
frazier k's user avatar
1 vote
1 answer
6k views

Proving the put call parity

In my course notes on the put-call parity, the proof is presented by going over two inequalities, namely $\text{RHS} > \text{LHS}$ implies arbtirage and $\text{RHS} < \text{LHS}$ implies ...
Stijn D'hondt's user avatar
0 votes
2 answers
431 views

Why are put and call options worth the same despite that put has no upside whereas call has unlimited upsides?

The following is an interview question. All Black-Scholes assumptions hold. Assume no dividends. Consider a standard European call and a standard European put on the same stock. Assume that each ...
Idonknow's user avatar
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0 votes
1 answer
95 views

Time to Put or Call a Bond

I was studying putable bond and callable bond on my own, there is an exercise question that was a little confusing to me: I understand what the answer explains, but I am confused that, is a bond "...
Betty's user avatar
  • 171
1 vote
1 answer
631 views

Continuous Geometric Asian Options

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Let $c(t; ...
Anon's user avatar
  • 281
2 votes
1 answer
614 views

How to prove Gamma is the same for a European call and European put with the same inputs?

I saw from a text "From put-call parity, call and put with the same inputs have the same gamma", but I don't see how put-call parity is related to Gamma. Can someone explain? Thanks!
CPT's user avatar
  • 121
2 votes
1 answer
440 views

Why is put-call parity defined differently by CME and Wikipedia?

In general, Wikipedia defines Put-Call parity as: C - P = D(F - K) ---------------- C = call price P = put price F = *FORWARD* price K = strike which can be re-...
Denis's user avatar
  • 185
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0 answers
80 views

Put-call parity for equity share and debt share

Considering Merton's structural approach" for credit risk modeling, we arrive to prove that the pricing formules are $S_t=V_t\phi(d_{T,1})-Fe^{-r(T-t)}\phi(d_{T,2})$ for equity share and $F_t=FP_0(t,T)...
Marco Pittella's user avatar
1 vote
1 answer
82 views

Why do we need to borrow money in the call-put parity? [closed]

As I understand it, the call put parity is given by $$c = p + S - \frac{X}{(1 + r)^T}$$ I understand the rationale behind simultaneously buying the call, put and underlying asset for $S$, but why ...
Yuna 's user avatar
  • 11
3 votes
0 answers
582 views

What are the main problems for calculating the implied volatility of in the money American put options?

As stated in the question I have a problem with calculating the implied volatility for in the money put options I have a data set of 2.6 million American style plain-vanilla call and put options. For ...
10uss's user avatar
  • 156
6 votes
2 answers
389 views

Why do I get a curved line when I plot "implied interest rate" on the strike price?

Currently, I am working on my thesis (MSc. Finance) and I run into an interesting “phenomenon”. I have option data for a non-dividend paying stock. In class I have learned, how to calculate the ...
10uss's user avatar
  • 156
1 vote
0 answers
148 views

Is it necessary for $P(K, t) - P(K + s, t) \geq se^{-rt}$ to hold?

Let $P(K, t)$ be a put option with strike price $K$ and expiration time $t$. Let $s > 0$. Is it necessarily true that the inequality $$P(K, t) - P(K + s, t) \geq se^{-rt}$$ holds? I know that ...
Ekesh Kumar's user avatar