Winter 2024 CS **6/676

Assignment 1

Due Feb-11, 11:59pm, via Crowdmark.

1. [8] Consider a **period binomial model with T = 1 and assume the risk free interest rate r = 0. That is, the stock (currently priced at S0 = 10) can got up to STu = uS0 with probability pu or down to STd = dS0 with probability 1−pu where pu ∈ (0,1). We know that d < 1 < u, but we do not know u or d. Suppose the following two options are traded in the market, both with maturity T = 1:

• European Put with strike K = 9 and current price P (1) = 1, 10

• European Put with strike K = 8 and P (2) = 1/3. 20

Assume the market is arbitrage free.

(a) [3] What is the fair value of a European call option with a strike price of K3 = 7?

(b) [2]Let δ0 be the number of stocks and η0 the number of bonds (noting B0 = BT = 1) you hold at t = 0. Find δ0 and η0 so that your strategy exactly replicates the payout of a short position in this call.

(c) [3] Using the actual probability pu, what is the expected option payoff for the European call in (a)? What is wrong with pricing this call option at this expected payoff value? If this European call option is priced at the expected payoff using p which is different from the fair value computed in (a), how can you construct an arbitrage?

2. [4] Consider the N-period binomial lattice. Denote by Snj for 0 ≤ n ≤ N and 0 ≤ j ≤ n be the price of the underlying at time tn and state j (i.e., j ups). A European Straddle Option has payoff at time T

max{K −SN,0}+max{SN −K,0}.

Denote by V (Snj , K, tn) be the fair value of the straddle option at time tn in state j. Use induction (over n)

to show that for any constant λ > 0,

V(λSnj,λK,tn)=λV(Snj,K,tn), n=0,1...,N,j=0,1,...,n.

3. [8] Consider the N-period binomial lattice where, at time t , the stock price Sj can go up to Sj+1 = uSj n n n+1n

with probability pu and sown to Sj n+1

interest rate and denote by

= dSj with probability 1 − pu. Denote by r > 0 the constant risk free n

qu = er∆t − d u−d

the risk neutral probability.

(a) [3] Provide an expression for all possible stock prices at T = ∆t · N .

(b) [2] If S0 is given, what is the risk neutral probability that, at time T, the stock price has experienced exactly k up moves?

(c) [3] Using risk neutral pricing, provide the expression, in terms of T, qu, K, r for the fair time t = 0 value of a European straddle expiry T and strike price K. Recall from Question 3 that such an option has payout max{K − SN , 0} + max{SN − K, 0}. Justify your answer.

4. [8] In this exercise, we consider the problem of pricing a Parisian Up-and-In Call. Let (St)t≥0 be a geometric Brownian Motion with drift r being equal to the risk-free interest rate and volatility σ, i.e., St = S0 exp(Xt) where Xt = (r − σ2/2)t + σWt for a Brownian Motion (Wt)t≥0 and S0 > 0 is some constant (today’s stock price). Let K > 0 be the strike price and L > 0 be a barrier. A Parisian Up-and-In Call (PUIC) option is activated if the stock price process (St)t∈[0,T] exceeds L consecutively for a period of time at least D > 0. Hence, the payout of a PUIC option at time T is max {0, (ST − K ) · A}, where

(1, if (St)t∈[0,T] had a consecutive excursion above L that lasted at least D, A=.

0, otherwise.

Today’s price is the discounted expected payoff; hence, we are estimating μ = E ?e−rT max {(ST − K) · A, 0}? .

For simplicity, we assume throughout the remainder of this question that 0 < D < T, K,L > S0. 1

(a) [4] Give an algorithm, in pseudo-code, that computes a Monte Carlo estimator for μ based on n simu- lations.

(b) [4] Implement your algorithm from a). Let N = 250, r = 0.05, σ = 0.25, D = 0.1, T = 1, S0 = 100 and i)K=110,L=120andii)K=120,L=110. Foreachofi)andii),reportaMCestimateforμalong with a 95% confidence interval based on n = 100, 000 independent simulations.

Note. In order to count how long the stock price was above L, use the following:

• IfStk ≥LandStk+1 ≥L,addtk+1−tk totheclock.

• If Stk < L and Stk+1 < L, there is no excursion.

• If Stk < L and Stk+1 ≥ L, an excursion started between tk and tk+1; add 0.5(tk+1 − tk) to the clock. • If Stk ≥ L and Stk+1 < L, an excursion ended between tk and tk+1; add 0.5(tk+1 − tk) to the clock.

5. [22] Consider the Black Scholes model, that is, let (St)t∈[0,T] be a geometric Brownian Motion with drift r and volatility σ, i.e., St = S0 exp(Xt) where Xt = (r − σ2/2)t + σWt for a Brownian Motion (Wt)t≥0 and S0 > 0 is some constant (today’s stock price). An Asian Option with maturity T and strike price K has

payout at time T given by max n0, 1 R T St dt − K o . Given time steps 0 ≤ t1 < · · · < tN = T for some N , we T0

consider the discretized Asian option with payout max n0, N1 PNj=1 Stj − Ko . Today’s price for this option

?−rT n 1PN o? is the discounted expected payoff. As such, we are estimating μ = E e max 0, N j=1 Stj − K

the remainder of this question, assume the time steps are given by tk = Nk T for k = 1,...,N. (a) [1] Explain why this option is a path dependent option.

. For

(b) [2] Write down an algorithm in pseudo-code to estimate a 95% confidence interval for μ using (naive) Monte Carlo.

(c) [2] Write down an algorithm in pseudo-code to estimate a 95% confidence interval for μ using antithetic variates.

(d) [2] There is no known analytical formula for μ. The reason is that the distribution of the sum of log-

normals is not known. However, if we replace the arithmetic average N1 PNj=1 Stj by the geometric

?−rT ? ?QN ?1/N ?? average, i.e., if we consider μg = E e max 0, j =1 S (tj ) − K

instead, the problem sim- plifies as the distribution of the product of log-normals is again log-normal. One can show (you don’t

need to) that μg = e−rT ?ea+b/2Φ(d1) − KΦ(d2)?, where a=log(S0)+(r−σ2/2)T(N+1)/(2N), b=σ2T(N+1)(2N+1),

√ √ 6N2 d1 =(−log(K)+a+b)/ b, d2 =d1 − b.

Explain why using the Asian Option with geometric averaging can be used as a control variable to price the Asian Option with arithmetic averaging.

(e) [3] Write down an algorithm in pseudo-code to estimate a 95% confidence interval for μ using the geometric Asian Option as a control variable. Estimate the internal β using a pilot run.

(f) [4] Implement your algorithms from b), c) and e) and call your functions with S0 = 100, K = 110, r = 0.01, σ = 0.1, T = 1, N = 260 and n = 105 and npil = 100. You should report 3 confidence intervals, one for each algorithm. Comment on your output.

Finally, suppose instead of the continuous time Black Scholes model, we assume an N-period binomial model √

with u = eσ

(g) [3] Write down an algorithm, in pseudo-code, to find the fair value V0 of an Asian Call Option.

T/N, d = 1/u and qu = 1/2 (and the same σ,r,S0,K,T as before).

(h) [3] Implement your algorithm and, for each N ∈ {5, 10, 20} report the output for S0 = 100, K = 110,

√

r=0.01,σ=0.1(justlikebefore),u=eσ T/N,d=1/u.

(i) [2] Discuss advantages and shortcomings of the MC method versus the approximation through the binomial model.

2

6. [5] Graduate Students Only. Consider the N-period binomial model, but assume that at each time n ≥ 1, the up factor un = un(ω1,...,ωn) and down factor dn = dn(ω1,...,ωn) and the risk free interest rate rn = rn(ω1,...,ωn) are allowed to depend on n and the first n outcomes ω1,...,ωn ∈ {up,down}. The initial values u0,d0,r0 at time 0 are given. The stock price at time 1 is

and, for n ≥ 1, the stock price at time n + 1 is (un(ω1,...,ωn)Sn(ω1,...,ωn),

(uS0, if ω1 = up, S1(ω1) = dS0, if ω2 = down.

if ωn+1 = up,

if ωn+1 = down. .

One dollar invested or borrowed from the bank account at time 0 grows to an investment or debt of er0∆t at time 1; for n ≥ 1, one dollar invested or borrowed at time n grows to an investment or debt of ern(ω1,...,ωn)∆t at time n + 1. We assume that the no-arbitrage condition

0 < dn(ω1,...,ωn) < exp{rn(ω1,...,ωn)∆t} < un(ω1,...,ωn)

for n ∈ N and ω1,...,ωn ∈ {up,down}. Similarly, assume that at time t = 0, 0 < d0 < er0∆t < u0.

Consider a derivative which, after N periods, pays off the random amount VN (ω1, . . . , ωN ).

(a) In the model just described, provide an algorithm for determining the price at time zero for the derivative.

(b) Construct a replicating portfolio in this general model. That is, provide a formulas for δn and ηn, n = 0, 1, . . . , so that if we hold δn stocks and ηn bonds, then this portfolio replicated the derivate payout VN at time N.

Sn+1(ω1,...,ωn,ωn+1) = dn(ω1,...,ωn)Sn(ω1,...,ωn),

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