An Introduction to Stochastic Calculus - Department of Mathematics

An Introduction to Stochastic Calculus Haijun Li [email protected] Department of Mathematics Washington State University

Week 11

Haijun Li

An Introduction to Stochastic Calculus

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Outline

1

The Black-Scholes Option Pricing Formula A Short Excursion into Finance What is an Option? A Mathematical Formulation of the Option Pricing Problem

2

Change of Measure Girsanov’s Theorem

3

Extensions and Limitations of the Model

Haijun Li


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A Short Excursion into Finance Let Xt denote the price of a risky asset (let’s call it a stock) at time t.

Haijun Li


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A Short Excursion into Finance Let Xt denote the price of a risky asset (let’s call it a stock) at time t. Assume that the relative return from the asset in the period of time [t, t + dt] has a linear trend c dt which is disturbed by a stochastic noise term σdBt . Xt+dt − Xt = cdt + σdBt , or dXt = cXt dt + σXt dBt . Xt

Haijun Li


Week 11

3 / 20

A Short Excursion into Finance Let Xt denote the price of a risky asset (let’s call it a stock) at time t. Assume that the relative return from the asset in the period of time [t, t + dt] has a linear trend c dt which is disturbed by a stochastic noise term σdBt . Xt+dt − Xt = cdt + σdBt , or dXt = cXt dt + σXt dBt . Xt The constant c > 0 is the so-called mean rate of return, and σ > 0 is the volatility.

Haijun Li


Week 11

3 / 20

A Short Excursion into Finance Let Xt denote the price of a risky asset (let’s call it a stock) at time t. Assume that the relative return from the asset in the period of time [t, t + dt] has a linear trend c dt which is disturbed by a stochastic noise term σdBt . Xt+dt − Xt = cdt + σdBt , or dXt = cXt dt + σXt dBt . Xt The constant c > 0 is the so-called mean rate of return, and σ > 0 is the volatility. Observe that this is a crude, first order approximation to a real price process. But people in economics believe in exponential growth and they are often happy with this model. Haijun Li


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Trading Strategy Assume that you have a non-risky asset such as a bank account, which can be called a bond. Let βt denote the bond yield at time t.

Haijun Li


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Trading Strategy Assume that you have a non-risky asset such as a bank account, which can be called a bond. Let βt denote the bond yield at time t. Let’s say your initial capital is β0 , which will be continuously compounded with a constant interest rate r > 0. That is, dβt = r βt dt, or βt = β0 ert . Note again that this is an idealization since the interest rate changes over time as well.

Haijun Li


Week 11

4 / 20

Trading Strategy Assume that you have a non-risky asset such as a bank account, which can be called a bond. Let βt denote the bond yield at time t. Let’s say your initial capital is β0 , which will be continuously compounded with a constant interest rate r > 0. That is, dβt = r βt dt, or βt = β0 ert . Note again that this is an idealization since the interest rate changes over time as well. If you have at shares in stock and bt shares in bond at time t, then your portfolio at time t can be represented by (at , bt ), t ∈ [0, T ], which is called a trading strategy.

Haijun Li


Week 11

4 / 20

Trading Strategy Assume that you have a non-risky asset such as a bank account, which can be called a bond. Let βt denote the bond yield at time t. Let’s say your initial capital is β0 , which will be continuously compounded with a constant interest rate r > 0. That is, dβt = r βt dt, or βt = β0 ert . Note again that this is an idealization since the interest rate changes over time as well. If you have at shares in stock and bt shares in bond at time t, then your portfolio at time t can be represented by (at , bt ), t ∈ [0, T ], which is called a trading strategy. You want to adjust your strategy according to information available to you at time t, as to maximize your wealth Vt = at Xt + bt βt (the value of your portfolio) at time t. So, It is reasonable to assume that at and bt are stochastic processes adapted to Brownian motion B. Haijun Li


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Self-Financing Condition at and bt can be positive or negative. A negative value of at means short sale of stock (i.e. you sell the stock at time t). A negative value of bt means that you borrow money at the bond’s riskless interest rate r .

Haijun Li


Week 11

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Self-Financing Condition at and bt can be positive or negative. A negative value of at means short sale of stock (i.e. you sell the stock at time t). A negative value of bt means that you borrow money at the bond’s riskless interest rate r . We neglect transaction costs for operations on stock and sale for simplicity.

Haijun Li


Week 11

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Self-Financing Condition at and bt can be positive or negative. A negative value of at means short sale of stock (i.e. you sell the stock at time t). A negative value of bt means that you borrow money at the bond’s riskless interest rate r . We neglect transaction costs for operations on stock and sale for simplicity. Assume that you spend no money on other purposes (such as food), i.e., you do not make your portfolio smaller by consumption.

Haijun Li


Week 11

5 / 20

Self-Financing Condition at and bt can be positive or negative. A negative value of at means short sale of stock (i.e. you sell the stock at time t). A negative value of bt means that you borrow money at the bond’s riskless interest rate r . We neglect transaction costs for operations on stock and sale for simplicity. Assume that you spend no money on other purposes (such as food), i.e., you do not make your portfolio smaller by consumption. We assume finally that your trading strategy (at , bt ) is self-financing. That is, the increments of your wealth Vt result only from changes of the prices Xt and βt of your asserts: dVt = at dXt + bt dβt = (cat Xt + rbt βt )dt + σat Xt dBt . Z t Z t Vt = V0 + (cas Xs + rbs βs )ds + σas Xs dBs . 0 Haijun Li


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Option An option at time t = 0 is a “ticket” which entitles you to buy one share of stock until or at time T , the time of maturity or time of expiration of the option.

Haijun Li


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Option An option at time t = 0 is a “ticket” which entitles you to buy one share of stock until or at time T , the time of maturity or time of expiration of the option. If you can exercise this option (or exercise the call) at a fixed price K , called the exercise price or strike price of the option, only at time of maturity T , this is called a European call option. If you can exercise it until or at time T , it is called an American call option. There are many other kinds ....

Haijun Li


Week 11

6 / 20

Option An option at time t = 0 is a “ticket” which entitles you to buy one share of stock until or at time T , the time of maturity or time of expiration of the option. If you can exercise this option (or exercise the call) at a fixed price K , called the exercise price or strike price of the option, only at time of maturity T , this is called a European call option. If you can exercise it until or at time T , it is called an American call option. There are many other kinds .... The purchaser of a European call option is entitled to a payment of (XT − K )+ = max(0, XT − K ). We illustrate option pricing using European call options.

Haijun Li


Week 11

6 / 20

Option An option at time t = 0 is a “ticket” which entitles you to buy one share of stock until or at time T , the time of maturity or time of expiration of the option. If you can exercise this option (or exercise the call) at a fixed price K , called the exercise price or strike price of the option, only at time of maturity T , this is called a European call option. If you can exercise it until or at time T , it is called an American call option. There are many other kinds .... The purchaser of a European call option is entitled to a payment of (XT − K )+ = max(0, XT − K ). We illustrate option pricing using European call options. A put is an option to sell stock at a strike price K on or until a particular date of maturity T . A European put option is exercised only at time of maturity with profit (K − XT )+ , and an American put can be exercised until or at time T . Haijun Li


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Option Pricing Since you do not know the price XT at time t = 0 when you purchase the call, a natural question arises: How much would you be willing to pay for such a ticket, i.e. what is a rational price for this option at time t = 0?

Haijun Li


Week 11

7 / 20

Option Pricing Since you do not know the price XT at time t = 0 when you purchase the call, a natural question arises: How much would you be willing to pay for such a ticket, i.e. what is a rational price for this option at time t = 0? Black, Scholes and Merton responded as follows:

Haijun Li


Week 11

7 / 20

Option Pricing Since you do not know the price XT at time t = 0 when you purchase the call, a natural question arises: How much would you be willing to pay for such a ticket, i.e. what is a rational price for this option at time t = 0? Black, Scholes and Merton responded as follows: 1

You, after investing this rational value of money in stock and bond at time t = 0, can manage your portfolio according to a self-financing strategy so as to yield the same payoff (XT − K )+ as if the option had been purchased.

Haijun Li


Week 11

7 / 20

Option Pricing Since you do not know the price XT at time t = 0 when you purchase the call, a natural question arises: How much would you be willing to pay for such a ticket, i.e. what is a rational price for this option at time t = 0? Black, Scholes and Merton responded as follows: 1

You, after investing this rational value of money in stock and bond at time t = 0, can manage your portfolio according to a self-financing strategy so as to yield the same payoff (XT − K )+ as if the option had been purchased.

2

If the option were offered at any price other than this rational value, there would be an opportunity of arbitrage, i.e. for unbounded profits without an accompanying risk of loss.

Haijun Li


Week 11

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Hedging Against the Contingent Claim Goal: Find a self-financing strategy (at , bt ) and a wealth process Vt , such that Vt = at Xt + bt βt = u(T − t, Xt ), t ∈ [0, T ], for some smooth (a technical assumption) deterministic function u(t, x) with the terminal condition VT = u(0, XT ) = (XT − K )+ . That is, to hedge against the contingent claim (XT − K )+ .

Haijun Li


Week 11

8 / 20

Hedging Against the Contingent Claim Goal: Find a self-financing strategy (at , bt ) and a wealth process Vt , such that Vt = at Xt + bt βt = u(T − t, Xt ), t ∈ [0, T ], for some smooth (a technical assumption) deterministic function u(t, x) with the terminal condition VT = u(0, XT ) = (XT − K )+ . That is, to hedge against the contingent claim (XT − K )+ . 1

Apply the Itô lemma to the wealth process Vt = u(T − t, Xt ) and we obtain an integral representation.

Haijun Li


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2

Apply the Itô lemma to the wealth process Vt = u(T − t, Xt ) and we obtain an integral representation. t Xt Plug bt = Vt −a into the self-financing condition, and we obtain βt another integral representation.

Haijun Li


Week 11

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2

3

Apply the Itô lemma to the wealth process Vt = u(T − t, Xt ) and we obtain an integral representation. t Xt Plug bt = Vt −a into the self-financing condition, and we obtain βt another integral representation.

From these two integral representations, we derive a PDE with condition u(0, x) = (x − K )+ , x > 0.

Haijun Li


Week 11

8 / 20

Black-Scholes PDE We obtain that 0.5σ 2 x 2 u22 (t, x) + rxu2 (t, x) + u1 (t, x) − ru(t, x) = 0 with boundary conditions u(t, 0) = 0, lim

x→∞

Haijun Li

u(t, x) = 1 ∀t ∈ [0, T ]; u(0, x) = (x − K )+ ∀x ≥ 0. x


Week 11

9 / 20

Black-Scholes PDE We obtain that 0.5σ 2 x 2 u22 (t, x) + rxu2 (t, x) + u1 (t, x) − ru(t, x) = 0 with boundary conditions u(t, x) = 1 ∀t ∈ [0, T ]; u(0, x) = (x − K )+ ∀x ≥ 0. x Transform the equation into a diffusion equation by using θ = T − t, y = log(x/K ) + (r − σ 2 /2)θ, w(θ, y ) = er θ u(t, x).

u(t, 0) = 0, lim

x→∞

Haijun Li


Week 11

9 / 20

Black-Scholes PDE We obtain that 0.5σ 2 x 2 u22 (t, x) + rxu2 (t, x) + u1 (t, x) − ru(t, x) = 0 with boundary conditions u(t, x) = 1 ∀t ∈ [0, T ]; u(0, x) = (x − K )+ ∀x ≥ 0. x Transform the equation into a diffusion equation by using θ = T − t, y = log(x/K ) + (r − σ 2 /2)θ, w(θ, y ) = er θ u(t, x). We arrive at a heat equation

u(t, 0) = 0, lim

x→∞

∂w σ2 ∂ 2w = ∂θ 2 ∂y 2 with an initial condition w(0, y ) = K (ey − 1)+ .

Haijun Li


Week 11

9 / 20

Black-Scholes PDE We obtain that 0.5σ 2 x 2 u22 (t, x) + rxu2 (t, x) + u1 (t, x) − ru(t, x) = 0 with boundary conditions u(t, x) = 1 ∀t ∈ [0, T ]; u(0, x) = (x − K )+ ∀x ≥ 0. x Transform the equation into a diffusion equation by using θ = T − t, y = log(x/K ) + (r − σ 2 /2)θ, w(θ, y ) = er θ u(t, x). We arrive at a heat equation

u(t, 0) = 0, lim

x→∞

∂w σ2 ∂ 2w = ∂θ 2 ∂y 2 with an initial condition w(0, y ) = K (ey − 1)+ . Use the heat kernel, we have Z ∞ 2 2 w(θ, y ) = (2πσ 2 θ)1/2 w(0, z)e−(y −z) /2σ θ dz. −∞ Haijun Li


Week 11

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Black-Scholes-Merton Approach The explicit solution can be simplified as u(t, x) = xΦ(g(t, x)) − Ke−rt Φ(h(t, x)), where Φ is the standard normal distribution function, and g(t, x) =

Haijun Li

ln(x/K ) + (r + 0.5σ 2 )t , h(t, x) = g(t, x) − σt 1/2 . σt 1/2


Week 11

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Black-Scholes Option Pricing Formula A rational price at time t = 0 for a European call option with exercise price K is V0 = X0 Φ(g(T , X0 )) − Ke−rT Φ(h(T , X0 )).

Haijun Li


Week 11

10 / 20



Black-Scholes Option Pricing Formula A rational price at time t = 0 for a European call option with exercise price K is V0 = X0 Φ(g(T , X0 )) − Ke−rT Φ(h(T , X0 )). The stochastic process Vt = u(T − t, Xt ) is the value of your self-financing portfolio with trading strategy at = u2 (T − t, Xt ) > 0, bt = Haijun Li

u(T − t, Xt ) − at Xt . βt


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The Radon-Nikodym Theorem Consider two measures µ and ν defined on a σ-field F on Ω. µ is said to be absolutely continuous with respect to ν (denoted by µ ν) if ν(A) = 0 implies µ(A) = 0, ∀A ∈ F. We say that µ and ν are equivalent measures if µ ν and ν µ.

Haijun Li


Week 11

11 / 20

The Radon-Nikodym Theorem Consider two measures µ and ν defined on a σ-field F on Ω. µ is said to be absolutely continuous with respect to ν (denoted by µ ν) if ν(A) = 0 implies µ(A) = 0, ∀A ∈ F. We say that µ and ν are equivalent measures if µ ν and ν µ.

Theorem Assume µ and ν are two σ-finite measures. Then µ ν holds if and only if there exists a non-negative measurable function f such that Z µ(A) = f (ω)dν(ω), ∀A ∈ F. A

Moreover, f is almost everywhere unique with respect to ν. The function f is called the (relative) density of µ with respect to ν, and denoted by f = dµ dν . Haijun Li


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Girsanov’s Theorem Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration. Consider ˜ t = Bt + qt, t ∈ [0, T ], for some constant q. B ˜ is not a standard Brownian motion under P for q 6= 0, B ˜ Although B can be shown to be a standard Brownian motion under the new probability measure Q.

Haijun Li


Week 11

12 / 20

Girsanov’s Theorem Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration. Consider ˜ t = Bt + qt, t ∈ [0, T ], for some constant q. B ˜ is not a standard Brownian motion under P for q 6= 0, B ˜ Although B can be shown to be a standard Brownian motion under the new probability measure Q.

Girsanov-Cameron-Martin Theorem 1. The stochastic process 1 Mt = exp{−qBt − q 2 t}, t ∈ [0, T ] 2 is a martingale with respect to the natural Brownian filtration under the probability measure P. Haijun Li


Week 11

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Eliminating the Drift Term Girsanov-Cameron-Martin Theorem R 2. Q(A) = A MT (ω)dP(ω), A ∈ F, defines a probability measure Q (called an equivalent martingale measure) on F that is equivalent to P.

Haijun Li


Week 11

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Eliminating the Drift Term Girsanov-Cameron-Martin Theorem R 2. Q(A) = A MT (ω)dP(ω), A ∈ F, defines a probability measure Q (called an equivalent martingale measure) on F that is equivalent to P. ˜ is a standard 3. Under the probability measure Q, the process B Brownian motion.

Haijun Li


Week 11

13 / 20

Eliminating the Drift Term Girsanov-Cameron-Martin Theorem R 2. Q(A) = A MT (ω)dP(ω), A ∈ F, defines a probability measure Q (called an equivalent martingale measure) on F that is equivalent to P. ˜ is a standard 3. Under the probability measure Q, the process B Brownian motion. ˜ is adapted to the filtration Ft . 4. The process B

Haijun Li


Week 11

13 / 20

Eliminating the Drift Term Girsanov-Cameron-Martin Theorem R 2. Q(A) = A MT (ω)dP(ω), A ∈ F, defines a probability measure Q (called an equivalent martingale measure) on F that is equivalent to P. ˜ is a standard 3. Under the probability measure Q, the process B Brownian motion. ˜ is adapted to the filtration Ft . 4. The process B Consider the linear stochastic differential equation dXt = cXt dt + σXt dBt , t ∈ [0, T ]. With a linear drift term, X is not a martingale under P.

Haijun Li


Week 11

13 / 20

Eliminating the Drift Term Girsanov-Cameron-Martin Theorem R 2. Q(A) = A MT (ω)dP(ω), A ∈ F, defines a probability measure Q (called an equivalent martingale measure) on F that is equivalent to P. ˜ is a standard 3. Under the probability measure Q, the process B Brownian motion. ˜ is adapted to the filtration Ft . 4. The process B Consider the linear stochastic differential equation dXt = cXt dt + σXt dBt , t ∈ [0, T ]. With a linear drift term, X is not a martingale under P. ˜ t = Bt + c t, and we have Define B σ ˜ t , t ∈ [0, T ]. dXt = σXt d(Bt + σc t) = σXt d B

Haijun Li


Week 11

13 / 20

Eliminating the Drift Term Girsanov-Cameron-Martin Theorem R 2. Q(A) = A MT (ω)dP(ω), A ∈ F, defines a probability measure Q (called an equivalent martingale measure) on F that is equivalent to P. ˜ is a standard 3. Under the probability measure Q, the process B Brownian motion. ˜ is adapted to the filtration Ft . 4. The process B Consider the linear stochastic differential equation dXt = cXt dt + σXt dBt , t ∈ [0, T ]. With a linear drift term, X is not a martingale under P. ˜ t = Bt + c t, and we have Define B σ ˜ t , t ∈ [0, T ]. dXt = σXt d(Bt + σc t) = σXt d B ˜ is a standard Brownian motion under the equivalent martingale B measure Q, and thus X is a martingale under Q. Haijun Li


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Significance of the Change-of-Measure Trick

If we had known the solution only for the case without a linear drift, we could have derived the solution for the case with a linear drift via the change of measure.

Haijun Li


Week 11

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If we had known the solution only for the case without a linear drift, we could have derived the solution for the case with a linear drift via the change of measure. More significantly, X is a martingale under the equivalent martingale measure Q, and one can make use of the martingale property for proving various results about X .

Haijun Li


Week 11

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If we had known the solution only for the case without a linear drift, we could have derived the solution for the case with a linear drift via the change of measure. More significantly, X is a martingale under the equivalent martingale measure Q, and one can make use of the martingale property for proving various results about X . In fact, this is not just a technical trick, and as we demonstrate below, the change of measure provides an effective method to incorporate uncertainty and to hedge against contingent claims.

Haijun Li


Week 11

14 / 20

Recap: The Black-Scholes Model The price of one share of the risky asset (stock) is described by dXt = cXt dt + σXt dBt , t ∈ [0, T ].

Haijun Li


Week 11

15 / 20

Recap: The Black-Scholes Model The price of one share of the risky asset (stock) is described by dXt = cXt dt + σXt dBt , t ∈ [0, T ]. The price of the riskless asset (bond) is described by dβt = r βdβt , t ∈ [0, T ].

Haijun Li


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Portfolio = (at , bt ), with value Vt = at Xt + bt βt at time t.

Haijun Li


Week 11

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Portfolio = (at , bt ), with value Vt = at Xt + bt βt at time t. The portfolio is self-financing: dVt = at dXt + bt dβt , t ∈ [0, T ].

Haijun Li


Week 11

15 / 20


Portfolio = (at , bt ), with value Vt = at Xt + bt βt at time t. The portfolio is self-financing: dVt = at dXt + bt dβt , t ∈ [0, T ]. At time of maturity, VT = h(XT ), where h(Xt ) is the contingent claim at time t. For a European call option, h(x) = (x − K )+ , and for a European put option, h(x) = (K − x)+ .

Haijun Li


Week 11

15 / 20

Pricing via the Change-of-Measure Your gain from the option at time of maturity is h(XT ). To determine the value of this amount of money at t = 0, you have to discount it with given interest rate r : e−rT h(XT ), and take the expectation of it as the price for the option at t = 0.

Haijun Li


Week 11

16 / 20

Pricing via the Change-of-Measure Your gain from the option at time of maturity is h(XT ). To determine the value of this amount of money at t = 0, you have to discount it with given interest rate r : e−rT h(XT ), and take the expectation of it as the price for the option at t = 0. You have to also discount price of one share of stock: ˜t = e−rt Xt , t ∈ [0, T ], and the Itô lemma leads to d X ˜t = σ X ˜t d B ˜t , X c−r ˜ where Bt = Bt + σ t.

Haijun Li


Week 11

16 / 20

Pricing via the Change-of-Measure Your gain from the option at time of maturity is h(XT ). To determine the value of this amount of money at t = 0, you have to discount it with given interest rate r : e−rT h(XT ), and take the expectation of it as the price for the option at t = 0. You have to also discount price of one share of stock: ˜t = e−rt Xt , t ∈ [0, T ], and the Itô lemma leads to d X ˜t = σ X ˜t d B ˜t , X c−r ˜ where Bt = Bt + σ t. ˜ There exists an equivalent martingale measure Q which turns B into a standard Brownian motion, and ˜t = X ˜0 e−0.5σ2 t+σB˜ t X becomes a martingale with respect to the natural Brownian filtration under Q.

Haijun Li


Week 11

16 / 20

Pricing via the Change-of-Measure Your gain from the option at time of maturity is h(XT ). To determine the value of this amount of money at t = 0, you have to discount it with given interest rate r : e−rT h(XT ), and take the expectation of it as the price for the option at t = 0. You have to also discount price of one share of stock: ˜t = e−rt Xt , t ∈ [0, T ], and the Itô lemma leads to d X ˜t = σ X ˜t d B ˜t , X c−r ˜ where Bt = Bt + σ t. ˜ There exists an equivalent martingale measure Q which turns B into a standard Brownian motion, and ˜t = X ˜0 e−0.5σ2 t+σB˜ t X becomes a martingale with respect to the natural Brownian filtration under Q. The value of the portfolio at time t is given by Vt = EQ [e−r (T −t) h(XT )|Ft ], t ∈ [0, T ]. Haijun Li


Week 11

16 / 20

Pricing via the Change-of-Measure Your gain from the option at time of maturity is h(XT ). To determine the value of this amount of money at t = 0, you have to discount it with given interest rate r : e−rT h(XT ), and take the expectation of it as the price for the option at t = 0. You have to also discount price of one share of stock: ˜t = e−rt Xt , t ∈ [0, T ], and the Itô lemma leads to d X ˜t = σ X ˜t d B ˜t , X c−r ˜ where Bt = Bt + σ t. ˜ There exists an equivalent martingale measure Q which turns B into a standard Brownian motion, and ˜t = X ˜0 e−0.5σ2 t+σB˜ t X becomes a martingale with respect to the natural Brownian filtration under Q. The value of the portfolio at time t is given by Vt = EQ [e−r (T −t) h(XT )|Ft ], t ∈ [0, T ]. At time t = 0, V0 = EQ [e−rT h(XT )] is a rational price of the option. Haijun Li


Week 11

16 / 20

The Value of a European Option Write θ = T − t for t ∈ [0, T ].

Haijun Li


Week 11

17 / 20

The Value of a European Option Write θ = T − t for t ∈ [0, T ]. 2

˜

Since Xt = X0 e(r −0.5σ )t+σBt , we have 2 ˜ ˜ Vt = EQ e−r θ h(Xt e(r −0.5σ )θ+σ(BT −Bt ) )|Ft .

Haijun Li


Week 11

17 / 20


˜

Since Xt = X0 e(r −0.5σ )t+σBt , we have 2 ˜ ˜ Vt = EQ e−r θ h(Xt e(r −0.5σ )θ+σ(BT −Bt ) )|Ft . Since σ(Xt ) ⊆ Ft , Xt can be treated as a constant under Ft .

Haijun Li


Week 11

17 / 20


˜

Since Xt = X0 e(r −0.5σ )t+σBt , we have 2 ˜ ˜ Vt = EQ e−r θ h(Xt e(r −0.5σ )θ+σ(BT −Bt ) )|Ft . Since σ(Xt ) ⊆ Ft , Xt can be treated as a constant under Ft . ˜T − B ˜ t ∼ N(0, θ), and is independent of Ft . Under Q, B

Haijun Li


Week 11

17 / 20


˜

Since Xt = X0 e(r −0.5σ )t+σBt , we have 2 ˜ ˜ Vt = EQ e−r θ h(Xt e(r −0.5σ )θ+σ(BT −Bt ) )|Ft . Since σ(Xt ) ⊆ Ft , Xt can be treated as a constant under Ft . ˜T − B ˜ t ∼ N(0, θ), and is independent of Ft . Under Q, B Thus, Vt = f (t, Xt ), where R∞ 2 1/2 f (t, x) = e−r θ −∞ h(xe(r −0.5σ )θ+σy θ )dΦ(y ).

Haijun Li


Week 11

17 / 20


˜

Since Xt = X0 e(r −0.5σ )t+σBt , we have 2 ˜ ˜ Vt = EQ e−r θ h(Xt e(r −0.5σ )θ+σ(BT −Bt ) )|Ft . Since σ(Xt ) ⊆ Ft , Xt can be treated as a constant under Ft . ˜T − B ˜ t ∼ N(0, θ), and is independent of Ft . Under Q, B Thus, Vt = f (t, Xt ), where R∞ 2 1/2 f (t, x) = e−r θ −∞ h(xe(r −0.5σ )θ+σy θ )dΦ(y ). For a European call option, h(x) = (x − K )+ , and thus f (t, x) = xΦ(z1 ) − Ke−r θ Φ(z2 ), z1 =

Haijun Li

ln(x/K ) + (r + 0.5σ 2 )θ , z2 = z1 − σθ1/2 . σθ1/2


Week 11

17 / 20


˜

Since Xt = X0 e(r −0.5σ )t+σBt , we have 2 ˜ ˜ Vt = EQ e−r θ h(Xt e(r −0.5σ )θ+σ(BT −Bt ) )|Ft . Since σ(Xt ) ⊆ Ft , Xt can be treated as a constant under Ft . ˜T − B ˜ t ∼ N(0, θ), and is independent of Ft . Under Q, B Thus, Vt = f (t, Xt ), where R∞ 2 1/2 f (t, x) = e−r θ −∞ h(xe(r −0.5σ )θ+σy θ )dΦ(y ). For a European call option, h(x) = (x − K )+ , and thus f (t, x) = xΦ(z1 ) − Ke−r θ Φ(z2 ), z1 =

ln(x/K ) + (r + 0.5σ 2 )θ , z2 = z1 − σθ1/2 . σθ1/2

For a European put option, f (t, x) = Ke−r θ Φ(−z2 ) − xΦ(−z1 ). Haijun Li


Week 11

17 / 20

Extensions and Limitations of the Model The Black-Scholes model can be extended for variable (but deterministic) rates and volatilities.

Haijun Li


Week 11

18 / 20

Extensions and Limitations of the Model The Black-Scholes model can be extended for variable (but deterministic) rates and volatilities. The model may be also used to value European style options on instruments paying dividends, and closed-form solutions available if the dividend is a known proportion of the stock price.

Haijun Li


Week 11

18 / 20

Extensions and Limitations of the Model The Black-Scholes model can be extended for variable (but deterministic) rates and volatilities. The model may be also used to value European style options on instruments paying dividends, and closed-form solutions available if the dividend is a known proportion of the stock price. The model underestimates extreme moves that yields tail risk.

Haijun Li


Week 11

18 / 20

Extensions and Limitations of the Model The Black-Scholes model can be extended for variable (but deterministic) rates and volatilities. The model may be also used to value European style options on instruments paying dividends, and closed-form solutions available if the dividend is a known proportion of the stock price. The model underestimates extreme moves that yields tail risk. In reality security prices do not follow a strict stationary log-normal process, nor is the risk-free interest actually known (and is not constant over time).

Haijun Li


Week 11

18 / 20

Extensions and Limitations of the Model The Black-Scholes model can be extended for variable (but deterministic) rates and volatilities. The model may be also used to value European style options on instruments paying dividends, and closed-form solutions available if the dividend is a known proportion of the stock price. The model underestimates extreme moves that yields tail risk. In reality security prices do not follow a strict stationary log-normal process, nor is the risk-free interest actually known (and is not constant over time). The variance has been observed to be non-constant leading to models such as GARCH to model volatility changes.

Haijun Li


Week 11

18 / 20

Extensions and Limitations of the Model The Black-Scholes model can be extended for variable (but deterministic) rates and volatilities. The model may be also used to value European style options on instruments paying dividends, and closed-form solutions available if the dividend is a known proportion of the stock price. The model underestimates extreme moves that yields tail risk. In reality security prices do not follow a strict stationary log-normal process, nor is the risk-free interest actually known (and is not constant over time). The variance has been observed to be non-constant leading to models such as GARCH to model volatility changes. Pricing discrepancies between empirical and the Black-Scholes model have long been observed in options corresponding to extreme price changes; such events would be very rare if returns were log-normally distributed, but are observed much more often in practice. Haijun Li


Week 11

18 / 20

Historical Notes

A sociologist investigating the behavior of the probability community during the early 1990s would surely report an interesting phenomenon. Many of the best minds of this (or any other) generation began concentrating their research in the area of mathematical finance. The main reason for this can be summed up in two words: option pricing (D. Applebaum, 2004)

Haijun Li


Week 11

19 / 20

Historical Notes

A sociologist investigating the behavior of the probability community during the early 1990s would surely report an interesting phenomenon. Many of the best minds of this (or any other) generation began concentrating their research in the area of mathematical finance. The main reason for this can be summed up in two words: option pricing (D. Applebaum, 2004) The Black-Scholes model is widely employed as a useful approximation, but proper application requires understanding its limitations.

Haijun Li


Week 11

19 / 20

Historical Notes

A sociologist investigating the behavior of the probability community during the early 1990s would surely report an interesting phenomenon. Many of the best minds of this (or any other) generation began concentrating their research in the area of mathematical finance. The main reason for this can be summed up in two words: option pricing (D. Applebaum, 2004) The Black-Scholes model is widely employed as a useful approximation, but proper application requires understanding its limitations. The limitations and defects of the model have led many probabilists to query it.

Haijun Li


Week 11

19 / 20

Lévy Matters Heavy tails of stock prices, which is incompatible with a Gaussian model, suggests that it might be fruitful to replace Brownian motion with a more general Lévy process.

Haijun Li


Week 11

20 / 20

Lévy Matters Heavy tails of stock prices, which is incompatible with a Gaussian model, suggests that it might be fruitful to replace Brownian motion with a more general Lévy process. A Lévy process L = (Lt , t ≥ 0) has independent and stationary increments and is stochastically continuous, i.e., limt→s P(|Lt − Ls | > ) = 0 for any > 0.

Haijun Li


Week 11

20 / 20

Lévy Matters Heavy tails of stock prices, which is incompatible with a Gaussian model, suggests that it might be fruitful to replace Brownian motion with a more general Lévy process. A Lévy process L = (Lt , t ≥ 0) has independent and stationary increments and is stochastically continuous, i.e., limt→s P(|Lt − Ls | > ) = 0 for any > 0. Example: Brownian motion, the Poisson process, compound Poisson processes and their “combinations”.

Haijun Li


Week 11

20 / 20

Lévy Matters Heavy tails of stock prices, which is incompatible with a Gaussian model, suggests that it might be fruitful to replace Brownian motion with a more general Lévy process. A Lévy process L = (Lt , t ≥ 0) has independent and stationary increments and is stochastically continuous, i.e., limt→s P(|Lt − Ls | > ) = 0 for any > 0. Example: Brownian motion, the Poisson process, compound Poisson processes and their “combinations”. The Lévy-Itô decomposition for a one-dimensional Lévy process: R R Lt = bt + Bt + |x| ) = 0 for any > 0. Example: Brownian motion, the Poisson process, compound Poisson processes and their “combinations”. The Lévy-Itô decomposition for a one-dimensional Lévy process: R R Lt = bt + Bt + |x|

An Introduction to Stochastic Calculus - Department of Mathematics

An Introduction to Stochastic Calculus - Department of Mathematics

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