Hi guys
An overpriced put option means u sell put option and sell underlying shares. Can someone pls show a numerical example of how the arbitrage works? Pls make it simple and understandable. Many thanks
There’s more to it than that; you have to use put-call parity:
S0 + p0 = c0 + PV(X)
If the put option is overpriced, you want to sell the (real) put and buy a synthetic put. A synthetic put is:
p0 = c0 + PV(X) − S0
So, you:
- Sell the put option
- Buy a call option
- Buy a bond
- (Short) sell a share of stock
For example, suppose that S0 = $100, X = $105, T = 1 (i.e., one year), c0 = $10.68, p0 = $14.25, rrf = 2%. Then,
PV(X) = $105 / 1.02 = $102.94
and the price of the put option should be:
p0 = c0 + PV(X) − S0 = $10.68 + $102.94 − $100 = $13.62
Therefore, you sell the put option for $14.25 and buy the synthetic put for $13.62; your profit is $0.63.
I’ll leave it to you to show that in one year – when the options expire and the bond matures – the net cash flow will be $0 no matter what the spot price of the stock is.
Thanks s2000 this does help and I understand put call parity but the reason I was asking this question is bc in the schweser books page 215 in my book 4 they hAve a calculation of call options being overpriced and you sell call and buy shares and how In 1 year u make profit no matter what the payoff om the call option is or the stock price is
That one is easy. I was trying to replicate the logic for the put and somehow I’m not making sense of that. I’m tying rhis from my phone so I c ant paste the text here but do u know what I’m referring to?
Unless you also buy put options and sell a bond, their example won’t work. You can’t do it merely with calls and stock.
In which case I’m getting confused. Are you saying that the example given in schweser is wrong?
If they’re using only call options and shares of stock, yes.
Ah sorry for reposting the above comment. Typing on phone is tough. I’ll review this and come back to you.
Tha k you for your help.
S2000 I don’t think I phrased my question correctly. I’m going to have to re ask this again. There are 2 ways to arbitrage a mispricing in options. One is by put call parity which is the easier one to understand. The othe way is by delta hedging. Now the example I referred to on schweser page 215 on book 4 shows an example with making an arbitrage profit using delta hedging for calls which is easy for me to understand I’m struggling to replicate this for puts.
These are the details for the call one
Current price is $30
up move of stock is 40 and down move is 22.5
using binomial model they arrived at pv of call price of 5.14
delta hedge is 0.5714
risk free rate is 7%
Let’s say call is over priced at 6.50
so u sell call buy shares. Easy enough. Cost of portfolio is (57.14*$30)- (100*6.50)=1064
We borrow 1064 at rf rate so we gotta pay 1138.48 at endnof 1 year. Price goes upto 40 you do 57.14*40 -100*10=1286. U will make the same payoff when price goes down to 22.5.
For put same details as above
Put option price is 3.154. Now let’s say put option is overpriced at $4. How do I use similar logic to illustrate the arbiyrage profits. I’m really struggling with this. I umderstand the concept behind selling a put and stock but just cannot figure out the the math in any logical way. Any help from ur side would be very appreciated
You cannot make an arbitrage profit with delta hedging. Arbitrage is, by definition, risk-free. Delta hedging is not risk-free.
What if the price goes up to $50? $60? $200?
What if the price goes down to $20? $10? $2?
The market doesn’t know that your model had $40 and $22.50 for the prices. And if it did know that it wouldn’t care.
If Schweser says that this is an arbitrage profit, they’re wrong. There’s definitely risk in delta hedging.
I agree with most ur points but u should still be able make a profit right? Let’s say call is underpriced so you buy call and sell shares. As time goes by and call options return to correct levels can’t u close ou the position by selling those calls and locking In a profit?
Let me think.
No.
Imagine in your scenario that the price of the stock goes to $300/share tomorrow.
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I get that the overall concept is that the purpose of delta hedging is to do just that to hedge. But let’s say call option goes up by $2 but stock price only goes up $1 then there is a profit to be made rin the above scenario right? U can sell the call for $2 higher and buy shares that are now only a $1 more expensive.
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In which case now thay I think abt it ur delta would change and u wud have to re balance ur portfolio again but an investor could easily just as change their objective and think that there is a small profit to be made here?
Sir: please write real words. I’m happy to help you, but this isn’t Twitter.
What’s going to happen if the stock price goes to $300? Will you still make a profit?
Let’s see some calculations; this is the stuff you (should) want to learn.
Hi s2000 sorry maybe what I’m asking is outside the syllabus and I don’t think I’m explaining myself well. What I was trying to really ask is if in the above scenario where stock goes up to $300 then u make a loss which i understand but what if it doesn’t go up by so much.
Call option is at $4 when it should be $6. Current stock price is $30. Delta hedge is 1. You buy 1 call option and sell 1 stock. When call option goes to $6 you sell call option. Lets say stock price has only gone up by $1. Haven’t u made a net gain of $1? I.e. $2 gain call price -$1 loss from selling share at $30.
If I’m over complicating it pls let me know and I’ll just be happy with the fact that a delta hedge doesn’t allow u to make profits.
In the scenario you describe you’ve made a profit, but it’s not an _ arbitrage _ profit: there is risk.
My point is that in an arbitrage transaction, you make money no matter what happens to the price of the underlying stock: no risk. What Schweser describes is not arbitrage.
Ok I think I finally got it. Thank you so much for persisting with me s2000.
My pleasure.
Continuing the discussion from overpriced put option:
I realize this is an old post but find it helpful to clear up some notions. The textbook explanation is quite bad.
The example is under a binomial model, which is why a mis priced option can be arb’d via the hedge ratio. Op forgot to mention this.
Let us say the put is $4.00. hedge ratio is then (0.4285714).
At Initiation, as you said you short the put and short the hedge ratio number of stock. Let’s use 100 put contracts (assume 1 share per contract). This means at initiation:
Prem received = 4.00 (market price put) * 100 = 400.00 dollars
Short funds received = 30 (market price stock) * 100 * .4285714 = 1,285…7143
You will invest these funds until expiration and earn the risk free rate, 7%.
At expiration, first assume price moves up to 40.
Put expires worthless. You must buy stock back to return on your short.
Cash outflow: 40*42.85714 = (1,714.2857)
Cash Inflow (from $ lent out) = 1,285.7143 * 1.07 = 1,803.7143.
Net: 89.23 (you can discount back to t0 if desired)
Now, assume price moves down to 22.50.
Put now has value. Below assumes cash settlement, but an almost identical work through will show stock settlement works the same, except you need to remember you put was short 100 and your stock short was 42.85714 units, so if stock settled, you will have 100-42.85714 stock units to sell in the market.
put value = 7.500.
Cash outflow for put settlement = 1007.50 = (750.00)
Cash outflow for stock buy back to cover short= 22.5042.85714 = (964.2857)
Net outflow = (1,714.2857)
Cash Inflow (from $ lent out) = 1,285.7143 * 1.07 = 1,803.7143.
Net Cash flow = =89.23.
So, you see, under the binomial model, you can apply the logic to a put and receive arb profit using the hedge ratio.
Put-call parity (assuming fair pricing) will give risk free pay-off in ALL final pricing environments.
A delta hedge invloves risk. if the share finishes at either 2 points of you binomial all well and good, but they may not. The price of share at expiry could be anything
If you write a put (bring in premium) and short a share (bring in cash)
You are effect borrowing money.
If price is 22.5 or 40 at expirry then the borrowing cost of borowing = 2% better than risk free. BUt if stock finshes at 10 you ineffect borrwong money at 44%
Do delta hedge you constantly need to change the number of shares in the hedge.
This binominal model is a good introduction to what is going on but especially over a 1 year period it is too crude.
The binomial gives a hedge ratio and fair price of option if only the 2 future prices are possible and nothing else. That may be realistic over short periods - minutes - but not long periods.
You need the balc scholes model.
Yes, I agree with all your points.
My illustration above was to demonstrate that we can still apply the logic that Schweser used, under the assumptions of the binomial model under an overpriced put scenario, not to justify the use of the binomial (or any other) model or delta ratio in general. Under the extreme assumptions outlined you can use the hedge ratio to find the correct # of shares to buy or sell to lock in an arb free profit
In other words, I was addressing the precise question OP asked