Stratégies Options - Binary Options

It's easy to see how binary options can be understood using vanilla ones.

→ In the money binary options provide a payoff of $1,
→ Out of the money binary options provide a payoff of $0.

I - Binary option approximation using Vanilla options

First of all, binary payoff is limited. It leads to think about vertical spreads in order to replicate its payoff.

First attempt
■ If one finds two vanilla options struck at K1 and K2, say calls, the very first idea which comes on mind is to try to replicate binary payoff struck at K2 using strike difference of $1.
That way, if the spot is above K2=K1+$1, then one would find oneself with the same payoff as a binary call one. If the spot is under strike K1, payoff would be 0 once again the same as a binary call one.
What if spot stays between K1and K2 ? That's where it matters.

Second attempt
■ If one finds two vanilla calls with tight strikes as much as possible, then it's possible to replicate a binary option in a very effective way.
If strike difference is $0.1, say K1=99.9 and K2=100, 10 calls spreads 99.9/100 are enough to reach the goal !
If strike difference is $0.01, say K1=99.99 and K2=100, 100 calls spreads 99.99/100 are enough.
If strike difference is $0.001, say K1=99.999 and K2=100, 1000 calls spreads 99.999/100 are enough.

Finally,

Binary Call = ( 1 / ( K2 - K1 ).( C(K1) -C(K2) )

If all possible strikes were available, one would have :
Binary Call = lim (C(K-h)-C(K))/h with h→0
Binary Call = lim - (C(K)-C(K+h))/h with h→0
Binary Call = - ∂C/∂K
The same way,
Binary Put = - ∂P/∂K
Where C is the vanilla Call, P the vanilla Put both struck at K.

II - Binary option in a Black&Scholes framework

Binary Call = - ∂C/∂K
Binary Call = ∂( - ∆S + (exp( -rT ).N(d2))/∂K = exp( -rT ).N(d2)

S spot, K strike
r riskfree rate
T maturity
exp(.) exponential function
N(.) Cumulative Normal Distribution
d2 = ((Ln(S/K)+(r-d-0.5σ²)T)/(σ√T))
Ln natural logarithm function
d annualized dividend rate
σ annualized volatility

Binary Call-put Parity
At expiry, Binary Call +Binary Put = 1, no matter where the spot would be.

Before expiry, Binary Call +Binary Put =exp(-rT) because money needs to be actualized.

Thus,
Binary Put = exp( -rT )-Binary Call
Binary Put = exp( -rT )-exp(-rT).N(d2)
Binary Put = exp(-rT).(1-N(d2))
Binary Put = exp( -rT ).(N(-d2))

III - Binary option graphs

Binary Call

Binary Put

Next : Binary Option : Delta
Previous : Binairy Option : A First Attempt

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Other Derivatives - INDEX
Other Derivatives - CHAPTER I
Other Derivatives - CHAPTER II
Other Derivatives - CHAPTER III
Other Derivatives - CHAPTER IV

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