1.  (a) (i) Boris is a British investor who wants to sell 40 Billion EURGBP in one year’s time. The EURGBP spot rate is 0.85, the EUR interest rate is 0.00% and the GBP interest rate is 0.75%.   Economists estimate that  GBP will weaken by  10% within one year.  Using covered rate arbitrage, calculate the value of the one year forward on EURGBP and determine how much GBP Boris will deliver.

First note that the statement about GBP weakening by 10% is a red herring and has no bearing on the value of the forward today.

We can deliver the GBP in two ways:

Time t = 0 :     EUR    Spot = 0.85    GBP

________→

rEUR  = 0% t                    t rGBP  = 0.75%

Time t = 1y :    EUR GBP

To deliver a fwd in one year’s time, either,

● Invest EUR and earn EUR interest rEUR  = 0%. Convert to GBP at fwd value in 1 year’s time, or

● Convert to GBP today and earn GBP interest rGBP  = 0.75% for

1 year.

These must be equivalent or there is an arbitrage. Denote the notional amount of EUR by NEUR . Then

● Spot * NEUR * (1 + rGBP  * T)

● NEUR * (1 + rEUR * T) * Forward

By arbitrage, these are equivalent so

(1 + rGBP T)

Fwd = Spot *

Plugging in the values here we obtain

Fwd = 0.85 *                    = 0.856375

(ii) Suppose the European Central Bank decides to ban short- selling of EUR by British investors from the start of 2021. What effect will this have on the value of the one year for- ward on EURGBP and what will Boris have to do in order to deliver GBP in one year’s time?

In this case we cannot short-sell the EUR to obtain GBP in one year’s time.  Therefore to deliver GBP we need to short-sell EUR today and convert to GBP using the spot rate. We will earn GBP interest for one year on the GBP notional. The amount of GBP delivered in one year’s time is:

GBP notional = 0.85 * (1 + 0.75%) * 40B = 0.856375 * 40B = 34.255B

Note that the forward rate hasn’t changed here because the EUR in- terest is 0% so in this case the ban on short-selling has no effect on forward prices.

(b) (i) Define an arbitrage opportunity and a sure-thing arbi- trage.

Definition 1. We say that a portfolio H is a sure-thing arbitrage if V0 (H) = 0

and, for every ω e Ω ,

VT (H, ω) > 0.

Definition 2. We say that a portfolio H is an arbitrage opportunity if

V0 (H) = 0,

VT (H) > 0        for every ω e Ω

and

VT (H, ω) > 0        for some ω e Ω .

If no sure-thing arbitrage exists, show that for European call and put options called C and P respectively with the same strike K and expiry date T we have

C(0) = P (0) + S(0) _ K* ,

where K* is the discounted strike price.

Theorem 3 (Put-Call Parity). Let C be a European call option and P a European put option on a stock S, and suppose that both options have the same strike price K  and expiry date T .  If there is no sure-thing arbitrage then

P (0) _ C(0) = K* _ S(0).

Proof. We define two portfolios as follows. Let H1  consist of one unit of stock and one put option, and let H2  consist of K*  of the riskless asset and one call option.  Recall that, at time T, the call option is worth

C(T) = max{S(T) _ K, 0}

and the put option is worth

P (T) = max{K _ S(T), 0}.

At time T, the value of H1  is

S(T) + P (T) = S(T) + max{K _ S(T), 0} = max{S(T), K}, and the value of H2  is

K + C(T) = K + max{S(T) _ K, 0} = max{S(T), K}

Thus H1  and H2  have the same value at time T (in every state). The assumption of no sure-thing arbitrage implies that at time 0 we have

p(H1 ) = p(H2 ), and so S(0) + P (0) = K* + C(0).

(c)(i) Write down the definition for Arrow-Debreu securities and explain what is meant by a complete market.

Definition 4. We say that a market with Ω = {ω1 , . . . , ωm } and assets S1 , . . . , SN  is complete if

lin{S1 , . . . , SN } = Rm .

Thus a complete market is one in which every possible contingent claim can be replicated by a suitable portfolio.   Note that we could have N > m, in which case the replicating portfolio will not be unique.

Definition 5. We write ei  for the asset with contingent claim ei  = (0, . . . , 0, 1, 0, . . . , 0),

where ei (ωi ) = 1 and ei (ωj ) = 0 for j i .  The assets ei  are known as Arrow-Debreu securities .

(ii) For a complete market with no arbitrage opportunities, construct the risk-neutral measure Q using Arrow-Debreu se- curities.

Suppose that our model is complete and has no arbitrage opportunities. It follows that the time t = 0 price of each ei  is positive, i.e.

p(ei ) > 0.

Now suppose the interest rate is r, so

p(1) = 1

Let us define

qi  = (1 + r)p(ei ) > 0.

We can replicate the riskless asset by Arrow-Debreu securities:

1 =

m

ei .

i=1

So, by our assumption of no-arbitrage, we have

m

p(1) =      p(ei ).

i=1

Thus

qi  = 1

i=1

and

qi  > 0      Ai.

In other words, Q = (q1 , . . . , qm ) gives a probability measure on Ω .

2. Write an essay to explain the concept of risk-neutrality and how it is used to price financial derivatives. You should define risk-neutrality, demonstrate how to construct the risk-neutral probability for a European call option, and state and prove a version of the no-arbitrage theorem.

Solution guide:

Define risk-neutral probability

Example constructing risk-neutral probability

State and prove no arb theorem

Risk-neutral definition

Risk-neutral probabilities are synthetic probabilities constructed using the replicating portfolio of an option.  They can be used to value an option directly without explicitly constructing the replicating portfolio. Note that they are NOT the actual or predicted probabilites of any move in the underlying asset.

Consider a one-period model of a market with Ω = {ω1 , . . . , ωm } and assets S1 , . . . , SN . The current price of the ith asset is Si (0), while the vector of prices at time t = 1 is ¯(S)i  = (Si (ω1 , 1), . . . , Si (ωm , 1)).

Definition 6. A probability measure Q on Ω is risk-neutral if

Q(ωi ) > 0       Ai

and

匝Q [Sn (1)*] = Sn (0)       An.

or, alternatively,

Definition 7. S is a martingale if 匝 (St+1|Pt ) = St  for every t . H(t) is previsible with respect to P (t)  if H(t)is Pt_1 -measurable for every t .  An arbitrage opportunity H is  a self-financing, previsible trading strategy such that

V0 (H) = 0 VT (H, ω) > 0 VT (H, ω) > 0

for all ω

for at least one ω

A probability measure Q is a risk-neutral measure for the market with assets S1 , . . . , SN  if

(a) Q(ω) > 0 for every ω e Ω; and

(b)  For each asset Sn , the discounted price process Sn(*)  is a martingale with respect to Q and the filtration (Pi ) given by the information tree

Example 8 (Valuation of a call).  Consider the two-state model of a call option.  A stock is worth S(0) today, and either goes up to SH   or down to SL   at time T .  Consider a call option X with strike price K, where SL   < K < SH , and so claim (S(T) _ K)+ at time T . Using replication of the call one can obtain a no- arbitrage valuation of

S(0) _ SL(*)

SH(*)  _ SL(*)

Suppose that SL(*)  < S(0) < SH(*):

Asset

SH

一

S(0)  一

!

!

qL(!)!!!(    SL

Call

SH  _ K

0

We can define ‘high’ and ‘low’ probabilities by

S(0) _ SL(*)

SH(*)  _ SL(*)

and

qL  = 1 _ qH  =

Theorem 9 (No-Arbitrage Theorem). In any one-period model with a finite Ω, the following are equivalent:

(a)  There is a risk neutral measure.

(b)  There are no arbitrage opportunities.

Proof.  Suppose that Ω = {ω1 , . . . , ωk }, we have assets S1 , . . . , SN , and there is no arbitrage opportunity. We show that there is a risk-neutral measure.

Let

W = {G(H)*  : H e RN+1}

be the subspace of Rk  given by gain vectors of portfolios.  Let X+  be the subset of Rk  given by

X+  = {x e Rk  : xi  > 0 Vi,x 0},

and let P be the subset of Rk  defined by

P = {x e Rk  : x1 + . . . + xk  = 1, xi  > 0 Vi}.

[Note that P can be thought of as the set of all probability measures on Ω .]

We have

W n X+  = 0,

since any vector in W n X+  is an arbitrage opportunity.   [Any such

vector would be the gain vector of some portfolio, with every coordinate

positive and some coordinate strictly positive.] Since P c X+ , we have

W n P = 0.

Since P is closed and convex, it follows from the Separating Hyperplane Theorem that there is a vector u such that

u e Wl

and

(u, x) > 0 Vx e P.

As ei  e P for every i, we have ui  = (u, ei ) > 0 for every i. So, defining

ui

qi  =

we see that

k

qi  = 1

i=1

and

qi  > 0 Ai.

We claim that Q = (q1 , . . . , qk ) is a risk-neutral probability measure. All that remains is to check that

匝Q [Si (1)*] = Si (0) Ai,

or equivalently

匝Q [∆Si(*)] = 0 Ai.

But this is immediate, as if H is the portfolio consisting of one unit of Si , then

匝Q [∆Si(*)] = 匝Q [G(H)*] = (Q, G(H)* ) = 0,

since Q = u/(u1  + . . . + uk ) and u e Wl .  We conclude that Q is a risk-neutral measure.

For the converse, suppose that Q is a risk-neutral measure.  [Actually, we have proved the converse already, but here it is again in the language of gain vectors.]

If H is an arbitrage opportunity, then there is j such that

G(H, ω)*  > 0 Aω

G(H, ωj )*  > 0.

So, summing over states ωi ,

匝Q [G(H)*] =       qi G(H, ωi ) > qj G(H, ωj ) > 0.

i

But, summing over assets,

匝Q [G(H)*] = 匝Q [

Hn ∆Sn(*)]

=

n = 0,

which gives a contradiction.

n

Hn匝Q [∆Sn(*)]

3. A one-touch option pays out a fixed amount of currency at expiry if at any time a share price is above a pre-specified value B . Describe how we may apply dynamic programming to value one-touch options. Use this method to value a one- touch option which pays $100 above B  = $11 written on an asset where the asset prices in dollars are given below, the interest rate per period is zero, and no dividends are paid.

(a) In dynamic programming we value a one-touch option using a bi- nomial model where we step backwards through time and look at the one-period submodel.  At each node we determine if the level B has been exceeded in which case the value is 1. Otherwise we calculate the value using replication.

The risk-neutral measures can been calculated using

p(up) = ,  p(down) =

Asset:

0.5   尸一

尸 尸

尸(0.2尸)尸一 12|!

0.5 尸一

尸

尸

尸

尸

10 |!

16

12

8

3

First fill in the one -touch payoff to have value 1 when S > 11:

One-touch:                              0.5 尸尸一

尸 尸

尸(0.2尸)尸一 1 |!

V1 |!    V2尸尸0(尸)5(尸)尸尸一 ||0(|)5(|)|!

0.5 尸一

尸

尸

尸

尸

V3 |!

尸

尸

V4 |!

1

1

0

0

Using the risk-neutral probabilities V3  = 1 * 0.5 + 0 * 0.5 = 0.5 and V4 is clearly 0.

For V2 :

V2  = 0.5 * V3 + 0.5 * V4  = 0.5 * 0.5 = 0.25

Therefore V1  = 0.2 * 1 + 0.8 * 0.25 = 0.4.

So the value of the one-touch option at time 0 is 0.4 * 100 = $40

(b) A binary call option pays out a fixed amount of currency at expiry only if the share price is above a pre-specified value B  at e北piration. Compute the value of a binary call option where B = $11 using the above model. Explain why the value of the binary call option is different from the one-touch option.

The binary call option has a pay-off with value 1 when S > 11:

1

0.5 尸一

尸

尸

Binary call:                              尸(0.)尸(5)尸尸一 V6 |0|!

V1 尸尸一 V5|!    V3 尸尸尸05(尸)尸尸一 1 ||!    V2尸尸0(尸)5(尸)尸尸一 |||5|!     0

||0(|)5(|)|!    V4 尸尸尸2(尸)尸尸一 |||8(|)|!     0

Using the risk-neutral probabilities V3   = 1 * 0.5 + 0 * 0.5 = 0.5 and V4  = 0 as before.

Now V6  = 1 and V5  = 0.5 * 1 + 0.5 * 0.5 = 0.75.

For V2 :

V2  = 0.5 * V3 + 0.5 * V4  = 0.5 * 0.5 = 0.25

Therefore V1  = 0.2 * 0.75 + 0.8 * 0.25 = 0.35.

So the value of the binary call option at time 0 is 0.35 * 100 = $35

The value of the binary call option is lower than the value of the one- touch because the one-touch pays out for the path ω4  where the spot price goes above 11 triggering the one-touch option but ends up at 8 meaning the binary call has zero pay-out.

(c) A no-touch option pays out a fixed amount of currency only if the share price never trades above a given value B at any time.

(i) Use an arbitrage argument to value a no-touch option in the above model.

(ii) Write down general formulas governing the relationship between one-touch and no-touch options both in the absence of interest rates and assuming a fixed USD interest rate.

(c) (i) For any given path ωi  the level B is either breached or not.  If we consider a portfolio that is long a one-touch option and a no-touch option both with level B then one or other (but not both) must pay out. Therefore:

One _ touch(1) + No _ touch(11) = $100

Hence the no-touch must be worth $60.

(ii) More generally,

One _ touch(B) + No _ touch(B) = Payout

In the presence of interest rates the payout needs to be present-valued to today so

One _ touch(B) + No _ touch(B) = exp(_r * T) * Payout

where r is the risk-free rate for the payout currency and T is the expi- ration of the options.

4. (a) (i) Write down the four conditions required for a process (B(t))t>0 to be a standard Brownian motion.

Definition 10. A process ╱B(t)、t>0  is said to be a (standard) Brownian motion if the following four conditions are satisfied

(a)  B(0) = 0

(b)  For s, t > 0, the random variable

B(s + t) _ B(s)

is normally distributed with mean 0 and variance t

(c)  Whenever 0 < t0 < t1 < . . . < tn , the quantities

B(t1 ) _ B(t0 ), B(t2 ) _ B(t1 ), . . . , B(tn ) _ B(tn_1 )

are independent

(d)  With probability 1, B(t) is a continuous function of t

(ii) Show that if B(t) is a Brownian motion and c  > 0 is a constant then

W (t) = (cB(t/c2 ))t>0

is also a Brownian motion.

(ii) If B(0) = 0 then W (0) = cB(0/c2 ) = 0.

W (s + t) _ W (S) = c╱B ╱ 、 _ B ╱ 、、

has mean 0 and variance

c2 ╱ _ 、 = t

.

Clearly the W (t) increments are independent if B(t) are. Also if B(t) is continuous then multiplying by a constant it remains continuous.

(iii) Assume that an asset price S is given by a Brownian motion. Argue from the definition why it is not possible to predict future values of the asset based on the past values of S .

(iii) The fact that increments are independent (part (c) of the defini- tion) means that we cannot deduce any future increment from previous increments and therefore they have no predictive power. If an asset is represented by a Brownian motion then from its definition we cannot predict future returns based on previous returns.

(b) Let (B(t))t>0 be a standard Brownian motion and let (X(t))t>0 be a process defined by

Xt  = exp(Bt ).

Compute dXt .

(b) Note that we only need the time-independent form of Itˆo:

Theorem 11 (Itˆo’s Lemma:  time-independent version). Let f(S) be continuously twice- differentiable, and suppose that

dS = µdt + σdB .

Then

df = (µdt + σdB) + σ 2 dt.

Define S(t) = B(t) so that

dS = 0 . dt + 1 . dB

so µ = 0 and σ = 1. We can rephrase Itˆo’s Formula as

df(Bt ) = f/ (Bt )dBt + f// (Bt )dt.

Now choose f(t) = exp(t) to get

d exp(Bt ) = exp(Bt )dBt + exp(Bt )dt.

Substituting Xt  = exp Bt . we obtain the formulation for dXt :

Xt

(c) Let f (S, t) be a function of two variables (continuously twice differentiable in S and once in t). State Itˆo’s Formula for df (S(t), t), where S(t) is an asset price obeying the stochastic equation

dS = µdt + σdW

in which W = W (t) is standard Brownian motion and µ, σ are continuous functions of S and t.

Find an expression for

T

sin(W (t))dW (t)

0

that does not involve Itˆo integrals.

Ito’s formula:  let f (S, t) have continuous second partial derivatives, and suppose that

dS = µdt + σdW.

Then

df = ╱ µ + + σ 2 、dt + σdW.

Define S(t) = W (t) so that

dS = 0 . dt + 1 . dW

so µ = 0 and σ = 1. Now choose f\ (x) = sin(x) so that

f (x) = _ cos(x)

f\\ (x) = cos(x)

Then the time-dependent form of Ito’s formula gives:

df = sin(W (t))dW + cos(W (t))dt

Replacing S for W then in integral form this becomes

T                                              1    T

_ cos(W (T))+cos(W (0)) =       sin(W (t))dW (t)+ 2       cos(W (t))dW

rearranging

T                                                                                            1    T

sin(W (t))dW (t) = 1 _ cos(W (T)) _ 2       cos(W (t))dt.

5.  (a)By hedging the delta, derive the Black-Scholes equation satisfied by the function V (S, t)

Let us consider a portfolio with h(t) units of the asset, combined with a short position of 1 unit of the derivatives contract.

At time t, the value U(t) of the portfolio is

U(t) = h(t)S(t) _ V (t).