Write the payoffs in Figure 3.8 as linear combination of call options and derive a closed form formula for the Black-Scholes price, the Delta, and the Gamma of them. All the Greeks of the option are also linear combination of these call option Greeks. For instance, Δ(t,S)=Φ(d1(τ,K1,S))−Φ(d1(τ,K2,S))−Φ(d1(τ,K3,S))+Φ(d1(τ,K4,S))
Partial Solution: For the strangle we have a pay off of (K−ST)++(ST−K)+ Therefore the closed form solution of B-S price of option is V(τ,S)=P(τ,K,S)+C(τ,K,S) and the delta of the position is Δ(τ,S)=−Φ(−d1(τ,K,S))+Φ(d1(τ,K,S)) Finally our gamma for this position is Γ(τ,S)=Φ′(d1(τ,K,S))+Φ′(d1(τ,K,S))Sσ√τ
I guess my professor made a mistake in regards to the B-S closed form price: for the strange it is V(τ,S)=(−S0Φ(−d1)+e−rTKΦ(−d2))+(S0Φ(d1)−e−rTKΦ(d2)) and similar for the straddle
where τ=T−t not sure why we use τ any explanation of that would be great.
Answer
To express such payoff in mathematical form, it is better to use indicator functions. I assume that the bottom of graphs (i.e., the vertex for the left one and the bottom segment for the right side one) represents zero.
For the left-hand one, the payoff is given by (K−ST)11ST≤K+(ST−K)11ST≥K=(K−ST)++(ST−K)+, that is, a straddle that involves both a European call and put with the same strike price and maturity date.
For the right-hand one, the payoff is given by (K1−ST)11ST≤K1+(ST−K2)11ST≥K2=(K1−ST)++(ST−K2)+, that is, a strangle that involves both a European call and put with the same maturity date, but different strikes.
For valuation, as an example, let's consider (1). According to the Black-Scholes' formula, the value of Payoff (1) is given by V=[K1e−rTΦ(−d12)−S0Φ(−d11)]+[S0Φ(d21)−K2e−rTΦ(d22)], where the first term is the value of the put option payoff (K1−ST)+ and the second is the value of the call option payoff (ST−K2)+. Here, d11=lnS0K1+(r+12σ2)Tσ√T,d12=d11−σ√T,d21=lnS0K2+(r+12σ2)Tσ√T,d22=d21−σ√T. The delta hedge ratio is the sum of deltas of the first put option and the second call options, that is, ∂V∂S0=−Φ(−d11)+Φ(d21)=Φ(d11)+Φ(d21)−1, and the gamma hedge ratio is the sum of gammas of the first put option and the second call options, that is, ∂2V∂S20=Φ′(d11)S0σ√T+Φ′(d21)S0σ√T=Φ′(d11)+Φ′(d21)S0σ√T.
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