# Binomial Tree

(2) − (1) : c2 − c1 ≥ S0 − Ke−rT2 − S0 + Ke−rT1 if r = const. ∧ T1 > T2 ⇔ c2 − c1 ≥ K(e−rT1 − e−rT2 ) > 0 ⇒ c2 > c 1 .

Suppose one observes c2 < c1 in markets, then the following arbitrage strategy provides a risk free proﬁt with zero net investments: Position Sell call c1 Buy call c2 Buy stock S0 Short sell stock S0 Borrow Ke−rT1 Lend Ke−rT2 proﬁt t0 T1 c1 −(S1 − K) −c2 0 −S0 S1 S0 0 −rT1 Ke −K −rT2 −Ke 0 c1 − c2 + K(e−rT1 − e−rT2 ) 0 >0 >0

T2 0 S2 − K 0 −S2 0 K 0

Suppose now that c1 has a strike K1 = erT1 and c2 a strike K2 = erT2 , where T1 < T2 ⇒ K1 < K2 . Then the above diﬀerence (2)-(1) reveals c2 − c1 > 0 directly. Hence, the arbitrage proﬁt in t0 merely is equal to the price diﬀerence c1 − c2 .

ii) Downside insurance against losing the initial investment S0 leads to the following payoﬀ at T : max{ST , K} = max{ST − K, 0} + K = ST + max{K − ST , 0} . The strategy either consists of a long call on the fund with price c and a risk free investment with a present value of e−rT · K = S0 , or in the second case in an investment in the fund S0 and a long put on the fund with price p. The second strategy is called protective put. In the following the insurance premium c (ﬁrst strategy) for a capital guarantee (including risk free interest) is derived. c = S0 N (d1 ) − Ke−rT N (d2 ) d1 = d2 ln(S0 /K) + (r + σ 2 /2)T √ σ T ln(S0 /K) + (r − σ 2 /2)T √ = , where K = S0 · erT σ T

⇒ d1 =

⇒ d2

ln(S0 ) − ln(S0 · erT ) + (r + σ 2 /2)T √ σ T √ −rT + (r + σ 2 /2)T σ T √ = = 2 σ T √ √ σ T = d1 − σ T = − = −d1 2 S0 N (d1 ) − Ke−rT N (d2 ) S0 N (d1 ) − S0 erT · e−rT N (−d1 ) S0 · (N (d1 ) − N (−d1 )) S0 · (N (d1 ) − (1 − N (−d1 ))) S0 · (2N (d1 ) − 1)

⇒ c = = = = = σ = 0.2 ∧ S0 = 1:

T 1 3 5 10 30 √ σ T d1 = −d2 = 2 0.1 0.1732 0.2236 0.3162 0.5477 N (d1 ) 0.5398 0.5675 0.5871 0.6217 0.7054 c = S0 · (2N (d1 ) − 1) 0.0796 0.1350 0.1742 0.2434 0.4108

The higher the maturity the more valuable the option. In fact a capital guarantee is quite expensive in general. The costs are equal to 1 + c, whereby the minimum return of a $1 investment after T years is erT in case the fund didn’t do well. For the strategy to be proﬁtable 1 + c < erT has 1 to hold, which equals a risk free interest rate of r > T ln(1 + c) decreasing in T . For instance, a maturity of T = 1 would then require a continuous compounded annual rate of r1 = 7.66%, whereas r30 = 1.15%.

Q4 In node Su at t1 = 0.5 create a portfolio consisting of u units of the underlying asset and some borrowing/lending Bu which replicates the payoﬀ of the option f ∈ {C, P } in either ﬁnal nodes Suu and Sud , i.e. u Suu

+ Bu er r u Sud + Bu e

t t

= fuu = fud .

u Su

(1) (2)

The value of this portfolio at t1 = 0.5 is given by

+ Bu = fu .

In node Sd at t1 = 0.5 create a portfolio consisting of d units of the underlying asset and some borrowing/lending Bd which replicates the payoﬀ of the option f ∈ {C, P } in either ﬁnal nodes Sud and Sdd , i.e. d Sud

+ Bd er r d Sdd + Bd e

t t

= fud = fdd . + Bd = fd .

(3) (4)

The value of this portﬁlio at t1 = 0.5 is given by

d Sd

Today create a portfolio consisting of 0 units of the underlying asset and some borrowing/lending B0 which replicates the price of the option in either nodes Su and Sd , i.e. 0 Su

+ B0 er r 0 Sd + B0 e

t t

= fu = fd .

(5) (6)

The value of this portﬁlio f0 = 0 S0 + B0 is equivalent to the present value of the expected ﬁnal payoﬀ in n + 1(here 3) stages of the world.

From (1) and (2):

u

=

fuu − fud , Suu − Sud fud − fdd , Sud − Sdd fu − fd , Su − Sd

Bu = (fuu −

u Suu )e

−r t

(7)

From (3) and (4):

d

=

Bd = (fud −

−r t d Sud )e

(8)

From (5) and (6):

0

=

B0 = (fu −

0 Su )e

−r t

(9)

Replicator Portfolio vs. Risk-Neutral...

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