Lecture 18 - More on option pricingarmerin/FinInsMathRwanda/Lecture18.pdf · Lecture 18 6 / 21. Perpetual American options (1) When the derivative is of American type, then we are

Lecture 18More on option pricing

Lecture 18 1 / 21

Introduction

In this lecture we will see more applications of option pricing theory.

Lecture 18 2 / 21

Greeks (1)

The price f of a derivative depends on several parameters.

The sensitivity of the price with respect to parameter is called a greek.

Some of the most important greeks are

∆ =∂f

∂S

Γ =∂2f

∂S2

Θ =∂f

∂t

V =∂f

∂σ(‘Vega’)

Lecture 18 3 / 21

Greeks (1)




∆ =∂f

∂S

Γ =∂2f

∂S2

Θ =∂f

∂t

V =∂f

∂σ(‘Vega’)

Lecture 18 3 / 21

Greeks (1)




∆ =∂f

∂S

Γ =∂2f

∂S2

Θ =∂f

∂t

V =∂f

∂σ(‘Vega’)

Lecture 18 3 / 21

Greeks (1)




∆ =∂f

∂S

Γ =∂2f

∂S2

Θ =∂f

∂t

V =∂f

∂σ(‘Vega’)

Lecture 18 3 / 21

Greeks (1)




∆ =∂f

∂S

Γ =∂2f

∂S2

Θ =∂f

∂t

V =∂f

∂σ(‘Vega’)

Lecture 18 3 / 21

Greeks (1)




∆ =∂f

∂S

Γ =∂2f

∂S2

Θ =∂f

∂t

V =∂f

∂σ(‘Vega’)

Lecture 18 3 / 21

Greeks (2)

The greeks are used in risk management.

Note that

∆ =∂f

∂S,

which means that xt = ∆ in the hedging portfolio of the derivative.

In the Black-Scholes model we have

∆call = Φ(d1),

where again

d1 =ln(S/K ) + (rf + σ2/2)(T − t)

σ√T − t

.

Lecture 18 4 / 21

Greeks (2)


Note that

∆ =∂f

∂S,



∆call = Φ(d1),

where again

d1 =ln(S/K ) + (rf + σ2/2)(T − t)

σ√T − t

.

Lecture 18 4 / 21

Greeks (2)


Note that

∆ =∂f

∂S,



∆call = Φ(d1),

where again

d1 =ln(S/K ) + (rf + σ2/2)(T − t)

σ√T − t

.

Lecture 18 4 / 21

Greeks (3)

What is the Delta of a put?

Use the put-call parity:

Price of put + S = Price of call + Ke−rf (T−t).

Taking ∂/∂S on the LHS and the RHS yields

∆put + 1 = ∆call ⇔ ∆put = ∆call − 1.

In the Black-Scholes model we get

∆put = N(d1)− 1 = −N(−d1).

Lecture 18 5 / 21

Greeks (3)





∆put + 1 = ∆call ⇔ ∆put = ∆call − 1.


∆put = N(d1)− 1 = −N(−d1).

Lecture 18 5 / 21

Greeks (3)





∆put + 1 = ∆call

⇔ ∆put = ∆call − 1.


∆put = N(d1)− 1 = −N(−d1).

Lecture 18 5 / 21

Greeks (3)





∆put + 1 = ∆call ⇔ ∆put = ∆call − 1.


∆put = N(d1)− 1 = −N(−d1).

Lecture 18 5 / 21

Greeks (3)





∆put + 1 = ∆call ⇔ ∆put = ∆call − 1.


∆put = N(d1)− 1

= −N(−d1).

Lecture 18 5 / 21

Greeks (3)





∆put + 1 = ∆call ⇔ ∆put = ∆call − 1.


∆put = N(d1)− 1 = −N(−d1).

Lecture 18 5 / 21

Perpetual American options (1)

When the derivative is of American type, then we are allowed to choosethe time at which we want to get the derivatives payoff.

In general it is a very hard problem to determine the price and the optimaltime at which an American option should be exercised.

There is however one case when the problem is simplified, and that iswhen the option is of perpetual type.

A perpetual derivative has T =∞.

Lecture 18 6 / 21






Lecture 18 6 / 21






Lecture 18 6 / 21






Lecture 18 6 / 21


Let us consider the Black-Scholes model.

Recall Black-Scholes equation:

∂f

∂t+ rf S

∂f

∂S+

1

2σ2S2 ∂

2f

∂S2− rf f = 0.

When T =∞ in the Black-Scholes mode, the term ∂f /∂t disappears fromthe equation due to time-homogeneity, and we get an ODE instead of aPDE:

rf Sf′ +

1

2σ2S2f ′′ − rf f = 0.

To solve this equation, we look for solutions on the form

f (S) = Sa

with a ∈ R.

Lecture 18 7 / 21




∂f

∂t+ rf S

∂f

∂S+

1

2σ2S2 ∂

2f

∂S2− rf f = 0.


rf Sf′ +

1

2σ2S2f ′′ − rf f = 0.


f (S) = Sa

with a ∈ R.

Lecture 18 7 / 21




∂f

∂t+ rf S

∂f

∂S+

1

2σ2S2 ∂

2f

∂S2− rf f = 0.

When T =∞ in the Black-Scholes mode, the term ∂f /∂t disappears fromthe equation due to time-homogeneity

, and we get an ODE instead of aPDE:

rf Sf′ +

1

2σ2S2f ′′ − rf f = 0.


f (S) = Sa

with a ∈ R.

Lecture 18 7 / 21




∂f

∂t+ rf S

∂f

∂S+

1

2σ2S2 ∂

2f

∂S2− rf f = 0.


rf Sf′ +

1

2σ2S2f ′′ − rf f = 0.


f (S) = Sa

with a ∈ R.

Lecture 18 7 / 21




∂f

∂t+ rf S

∂f

∂S+

1

2σ2S2 ∂

2f

∂S2− rf f = 0.


rf Sf′ +

1

2σ2S2f ′′ − rf f = 0.


f (S) = Sa

with a ∈ R.

Lecture 18 7 / 21




∂f

∂t+ rf S

∂f

∂S+

1

2σ2S2 ∂

2f

∂S2− rf f = 0.


rf Sf′ +

1

2σ2S2f ′′ − rf f = 0.


f (S) = Sa

with a ∈ R.

Lecture 18 7 / 21


Inserting this into the ODE yields

rf SaSa−1 +

1

2σ2S2a(a− 1)Sa−2 − rf S

a = 0

⇔

Sa

[rf a +

1

2σ2a(a− 1)− rf

]= 0.

⇔

a2 +

(2rfσ2− 1

)a− 2rf

σ2= 0.

The solution to this equation is

a1 = 1 and a2 = −2rfσ2

Lecture 18 8 / 21



rf SaSa−1 +

1

2σ2S2a(a− 1)Sa−2 − rf S

a = 0

⇔

Sa

[rf a +

1

2σ2a(a− 1)− rf

]= 0.

⇔

a2 +

(2rfσ2− 1

)a− 2rf

σ2= 0.


a1 = 1 and a2 = −2rfσ2

Lecture 18 8 / 21



rf SaSa−1 +

1

2σ2S2a(a− 1)Sa−2 − rf S

a = 0

⇔

Sa

[rf a +

1

2σ2a(a− 1)− rf

]= 0.

⇔

a2 +

(2rfσ2− 1

)a− 2rf

σ2= 0.


a1 = 1 and a2 = −2rfσ2

Lecture 18 8 / 21



rf SaSa−1 +

1

2σ2S2a(a− 1)Sa−2 − rf S

a = 0

⇔

Sa

[rf a +

1

2σ2a(a− 1)− rf

]= 0.

⇔

a2 +

(2rfσ2− 1

)a− 2rf

σ2= 0.


a1 = 1 and a2 = −2rfσ2

Lecture 18 8 / 21


Hence, the solution to the time-homogenuous Black-Scholes equation isgiven by

f (S) = A1S + A2S−2rf /σ2

.

for A1,A2 ∈ R.

How do we determine the constants A1 and A2?

They will depend on the derivative’s payoff.

Lecture 18 9 / 21



f (S) = A1S + A2S−2rf /σ2

.

for A1,A2 ∈ R.



Lecture 18 9 / 21



f (S) = A1S + A2S−2rf /σ2

.

for A1,A2 ∈ R.



Lecture 18 9 / 21

The perpetual American put option (1)

Let us consider a put option with strike price K :

F (S(t)) = max(K − S(t), 0).

Note that there is no given exercise time – we have to choose that time aswell.

Now consider the general solution

f (S) = A1S + A2S−2rf /σ2

.

Sincelim

S→∞F (S) = 0,

we must havelim

S→∞f (S) = 0 ⇒ A1 = 0.

Lecture 18 10 / 21



F (S(t)) = max(K − S(t), 0).



f (S) = A1S + A2S−2rf /σ2

.

Sincelim

S→∞F (S) = 0,

we must havelim

S→∞f (S) = 0 ⇒ A1 = 0.

Lecture 18 10 / 21



F (S(t)) = max(K − S(t), 0).



f (S) = A1S + A2S−2rf /σ2

.

Sincelim

S→∞F (S) = 0,

we must havelim

S→∞f (S) = 0 ⇒ A1 = 0.

Lecture 18 10 / 21



F (S(t)) = max(K − S(t), 0).



f (S) = A1S + A2S−2rf /σ2

.

Sincelim

S→∞F (S) = 0

,

we must havelim

S→∞f (S) = 0 ⇒ A1 = 0.

Lecture 18 10 / 21



F (S(t)) = max(K − S(t), 0).



f (S) = A1S + A2S−2rf /σ2

.

Sincelim

S→∞F (S) = 0,

we must havelim

S→∞f (S) = 0

⇒ A1 = 0.

Lecture 18 10 / 21



F (S(t)) = max(K − S(t), 0).



f (S) = A1S + A2S−2rf /σ2

.

Sincelim

S→∞F (S) = 0,

we must havelim

S→∞f (S) = 0 ⇒ A1 = 0.

Lecture 18 10 / 21


We want to exercise the put option at the first time t such that

max(K − S(t), 0)︸︷︷︸value of exercising at t

≥ f (t, S(t))︸︷︷︸value of not exercising at t

.

But it is never optimal to exercise the option when

max(K − S(t), 0) = 0.

Lecture 18 11 / 21


We want to exercise the put option at the first time t such that

max(K − S(t), 0)︸︷︷︸value of exercising at t

≥ f (t, S(t))︸︷︷︸value of not exercising at t

.

But it is never optimal to exercise the option when

max(K − S(t), 0) = 0.

Lecture 18 11 / 21


Let Sc solve the equation

K − Sc = f (Sc) = A2S−2rf /σ2

c .

Here ‘c ’ stands for critical

and Sc is the critical level at which we chooseto use the option when the stock price equals (or is less than) Sc .

Note that we need to find Sc ; this is a part of the solution.

One can show that we also have smooth fit. This means that the valuefunction f (S) meets the payoff function max(K − S , 0) tangentially.

Lecture 18 12 / 21



K − Sc = f (Sc) = A2S−2rf /σ2

c .

Here ‘c ’ stands for critical and Sc is the critical level at which we chooseto use the option when the stock price equals (or is less than) Sc .



Lecture 18 12 / 21



K − Sc = f (Sc) = A2S−2rf /σ2

c .




Lecture 18 12 / 21



K − Sc = f (Sc) = A2S−2rf /σ2

c .




Lecture 18 12 / 21


In formulas the smooth fit condition means that

f ′(Sc) =d

dS(K − S)

∣∣∣∣S=Sc

⇔

−2rfσ2

A2S−2rf /σ2−1c = −1.

We now have a system of equations with two equations and two unknowns:

K − Sc = A2S−2rf /σ2

c

1 =2rfσ2

A2S−2rf /σ2−1c

Lecture 18 13 / 21



f ′(Sc) =d

dS(K − S)

∣∣∣∣S=Sc

⇔

−2rfσ2

A2S−2rf /σ2−1c = −1.



c

1 =2rfσ2

A2S−2rf /σ2−1c

Lecture 18 13 / 21



f ′(Sc) =d

dS(K − S)

∣∣∣∣S=Sc

⇔

−2rfσ2

A2S−2rf /σ2−1c = −1.

We now have a system of equations with two equations and two unknowns

:


c

1 =2rfσ2

A2S−2rf /σ2−1c

Lecture 18 13 / 21



f ′(Sc) =d

dS(K − S)

∣∣∣∣S=Sc

⇔

−2rfσ2

A2S−2rf /σ2−1c = −1.



c

1 =2rfσ2

A2S−2rf /σ2−1c

Lecture 18 13 / 21


It is not hard to solve the system of equations, and doing this we get

f (S) = (K − Sc)

(S

Sc

)−2rf /σ2

,

where

Sc = K · 2rf2rf + σ2

.

Lecture 18 14 / 21

The perpetual American call option

Why didn’t we study the perpetual American call option?

The reason is that one can show that it is never optimal to exercise thisoption.

In fact, using the general solution

f (S) = A1S + A2S−2rf /σ2

we must have A2 = 0 in this case – otherwise f (0) = ±∞.

The smooth fit condition will then imply

f (S) = S

andSc =∞.

Lecture 18 15 / 21





f (S) = A1S + A2S−2rf /σ2



f (S) = S

andSc =∞.

Lecture 18 15 / 21





f (S) = A1S + A2S−2rf /σ2



f (S) = S

andSc =∞.

Lecture 18 15 / 21





f (S) = A1S + A2S−2rf /σ2



f (S) = S

andSc =∞.

Lecture 18 15 / 21

Risk-neutral valuation

One can show that the principle of risk-neutral valuation (or risk-neutralpricing) holds generally.

For a simple European derivative this means that the price is given by

f (t,S(t)) = e−rf (T−t)E [F (S(T ))|St ].

When calculating expected values E (· · · ), we use the dynamics

dS(t) = rf S(t)dt + σS(t)dz(t).

Lecture 18 16 / 21







Lecture 18 16 / 21







Lecture 18 16 / 21

Interest rate derivatives (1)

In general, money in the bank grows according to

dB(t) = r(t)B(t)dt with B(0) = 1.

We can solve this equation:

B(t) = e∫ t0 r(s)ds .

So far we have assumed that r(t) = rf , i.e. a constant, but we can as wellassume that r(t) is a stochastic process.

As when valuing options on stocks we can use either use a discrete timemodel or a continuous time model.

Lecture 18 17 / 21





B(t) = e∫ t0 r(s)ds .



Lecture 18 17 / 21





B(t) = e∫ t0 r(s)ds .



Lecture 18 17 / 21





B(t) = e∫ t0 r(s)ds .



Lecture 18 17 / 21


When using discrete time models, it is usual to use trinomial trees

:

u · rt↗

rt → rt↘

d · rt

t t + 1

Lecture 18 18 / 21


When using discrete time models, it is usual to use trinomial trees:

u · rt↗

rt → rt↘

d · rt

t t + 1

Lecture 18 18 / 21


In continuous time models, we use Ito diffusions to model the interest rater(t).

One can show that the value at time t ∈ [0,T ] of a zero-coupon bondwith maturity time T is given by

P(t,T ) = E(e−

∫ Tt r(s)ds

∣∣∣ rt) .Note that we use risk-neutral probabilities in this formula.

Let us now lok at some examples of models.

Lecture 18 19 / 21




P(t,T ) = E(e−

∫ Tt r(s)ds

∣∣∣ rt) .

Note that we use risk-neutral probabilities in this formula.


Lecture 18 19 / 21




P(t,T ) = E(e−

∫ Tt r(s)ds



Lecture 18 19 / 21




P(t,T ) = E(e−

∫ Tt r(s)ds



Lecture 18 19 / 21


The Vasicek model:

dr(t) = a(b − r(t))dt + σdz(t)

The Cox-Ingersoll-Ross (CIR) model:

dr(t) = a(b − r(t))dt + σ√

r(t)dz(t).

The Ho-Lee model:

dr(t) = θ(t)dt + σdz(t)

The Hull-White model:

d(r) = [θ(t)− ar(t)]dt + σdz(t).

Lecture 18 20 / 21


The Vasicek model:



dr(t) = a(b − r(t))dt + σ√

r(t)dz(t).

The Ho-Lee model:




Lecture 18 20 / 21


The Vasicek model:



dr(t) = a(b − r(t))dt + σ√

r(t)dz(t).

The Ho-Lee model:




Lecture 18 20 / 21


The Vasicek model:



dr(t) = a(b − r(t))dt + σ√

r(t)dz(t).

The Ho-Lee model:




Lecture 18 20 / 21


The Vasicek model:



dr(t) = a(b − r(t))dt + σ√

r(t)dz(t).

The Ho-Lee model:




Lecture 18 20 / 21


There is also an equation similar to the Black-Scholes equation in theinterest rate derivatives case.

Ifdr(t) = µ(r(t), t)dt + σ(r(t), t)dz(t)

then the price f (r(t), t) of a derivative satifies the PDE

∂f

∂t+ µ(r , t)

∂f

∂r+

1

2σ2(r , t)

∂2f

∂r2− rf = 0.

Lecture 18 21 / 21





∂f

∂t+ µ(r , t)

∂f

∂r+

1

2σ2(r , t)

∂2f

∂r2− rf = 0.

Lecture 18 21 / 21





∂f

∂t+ µ(r , t)

∂f

∂r+

1

2σ2(r , t)

∂2f

∂r2− rf = 0.

Lecture 18 21 / 21

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Lecture 18 - More on option pricingarmerin/FinInsMathRwanda/Lecture18.pdf · Lecture 18 6 / 21. Perpetual American options (1) When the derivative is of American type, then we are