Bm Fi6051 Wk7 Lecturer Notes

Financial Derivatives FI6051 Finbarr Murphy Dept. Accounting & Finance University of Limerick Autumn 2009

Week 7 – Relative Value Trading

Arbitrage & Risk Neutral Valuation 

Option Pricing Theory allows us to compute the Fair Value† of an asset.



This lecture examines how we can expolit differences between the fair value and the market price to generate a riskless profit †

Definition : Fair Value is the theoretically correct relative value of the call option such that options traders cannot generate a riskless profit by selling (if “overpriced”) or buying (if “underpriced”) the Call and then “covering” their risk exposures by trading the underlying asset (augmented by borrowing / lending).

Arbitrage & Risk Neutral Valuation 

Lecture Objectives



We will identify and exploit arbitrage opportunities



We will cover the theoritical basis of risk-neutral valuation

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Both of these methods are used to correctly price derivative securities, where the market price differs from the fair-value price, a potential arbitrage opportunity exists

Arbitrage 

Firstly some preliminary comment on the Principles Behind Riskless Portfolio construction Cτ Cτ S-X Ct

• ∆ Ct ∆ St St

∆≈ 

Sτ

∂Ct > 0but < 1 ∂St

We know that as the stock price moves ↑ ,the call price moves ↑.This can be represented by the option delta.

Arbitrage 







In effect, the call price does not move one-forone with the underlyiing equity price Now, assume you hold a long position in the Call option (I.e. +Ct) Construct a portfolio with this Ct position and a “certain amount” of shares such that that portfolio remains riskless at all times. I.e., any unexpected small change in the call price (ΔCt) will be exactly offset by the corresponding change in the share holding

Arbitrage 

If we hold Δ many shares in the opposite direction to the call position, we will be hedged and the portfolio will be riskless



I.e. if we hold –Δ.St shares (minus denotes a short position) against one long call option.

Arbitrage 

Long Put Positions



We know that as the stock price moves ↑ ,the put price moves ↓. Therefore, we need a long position (+Δ.St) shares to offset the change in the put price.



Arbitrage 

Data: S0 = 20, X=21, T=3 months, rF = 12% p.a. SUT = 22 ⇒ CUT = 1

S0 = 20

SDT = 18 ⇒ CDT = 0





Value the Call Option C0 by constructing a riskless portfolio Use a short position in the call security and a long position of Δ shares in the underlying equity

Arbitrage 

Now consider the possible Cash Flow (CF) outcomes at t = T (option expiry/maturity) CFs at T=0.25yrs

Portfolio Value

UP

+22.Δ - 1

DOWN

+18.Δ - 0



N.B. Short Call Payoff at maturity = -Max(St-X, 0)



Financial Engineering sign convention  

+ for Long Positions - for short Positions

Arbitrage 



If our riskless portfolio is truly riskless, then in either state (up or down), the value of the portfolio is the same. Solve the equation: +22.Δ – 1 => Δ = 0.25



= +18.Δ – 0

Therefore PFUT

= 22(0.25) – 1 = +4.5

PFDT

= 18(0.25) – 0 = +4.5

Arbitrage 

The value of the future (t=T) portfolio today = PFt=0

= e-R F.T .PFt=T = e-0.12.0.25 .4.5 = 4.367



But PFt=0



= S0.Δ – C0

=> 4.367

= 20x0.25 – C0

=> C0

= 0.633

C0 = 0.633 represents the fair value of the option

Arbitrage 

There are two types of Arbitrage



Type 1  A riskless investment for which the rate of return r > rF



Type 2  No initial investment, guaranteed positive payoff at expiry  A free-lunch

Arbitrage 

Arbitrage Type 1 Example



Assume that the call option encountered previously was trading at 0.70 rather than 0.633 Step 1: Identify the Mispricing





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Step 2: Implement appropriate Arbitrage Strategy and Hedge any Risk Exposure  



Call is overpriced by €0.067

Sell Call Buy ∆ many share of underlying equity – i.e. a riskless portfolio by construction !

Step 3: Liquidate portfolio on option expiry and see what % return is implied by this strategy.

Arbitrage 

Initial Outlay (Cash Flow) At t = 0 Sell 1 Call @ 0.7

= 0.7

Buy 0.25 shares @ 20

= -5.0

Total 

CF

= -4.30

And recall CFs at T=0.25yrs

Portfolio Value

UP

+22x0.25 – 1 = 4.50

DOWN

+18x0.25 – 0 = 4.50

Arbitrage  

 

So at t=0, we spent 4.30 to own the portfolio At t=T, we unwind (sell the call and buy the stock back) to realise 4.50 This gives us a net profit of 0.20 The net return on this strategy is given by

4.3 = 4.5e 

− r ( 0.25 )

Taking the natural log of both sides

ln(4.3) = −0.25.r. ln(4.5) ⇒ ln = 18.185% 

I.e. r > rF!

Arbitrage 

With r > rF, contradicting the CAPM risk-return trade-off formula!



Given the opportunity described above, many in the market will sell the calls and buy shares This activity will drive the option price towards the fair-value, I.e. 6.33 In an efficient market, this returning to equilibrium will happen quickly

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Exercise, what is the implied rate of return on our strategy if the option is priced at 6.333?

Arbitrage 

Arbitrage Type 2 Example



Lets assume that the call option encountered previously was trading at 0.60 rather than 0.633 Step 1: Identify the Mispricing





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Call is underpriced by €0.033

Step 2: Implement appropriate Arbitrage Strategy and Hedge any Risk Exposure  

Buy Call (since it is underpriced) Sell ∆ many share of underlying equity (you will need to borrow these first to sell them).

Arbitrage 

Step 3: Loan out the surplus cash (invest in riskless government bonds) and receive rF

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Step 4: Liquidate portfolio on option expiry and see what % return is implied by this strategy.

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Note that this strategy does not involve an initial outlay of cash.

Arbitrage 

Initial Outlay (Cash Flow) At t = 0

CF

Buy 1 Call @ 0.6

= -0.6

Sell 0.25 shares @ 20

= +5.0

Sub-Total Lend

= +4.40 4.4 at rF

Net-Total

= zero outlay

Arbitrage 

At t=T (Cash Flow at Maturity) At t = 0

PFUT or PFDT

CF = -4.5

Loan Recuperate

= +4.534

Total

= +0.034



I.e. We have made a riskless profit without spending any money



Exercise, what is the implied rate of return on our strategy if the option is priced at 6.333?

Risk Neutral Valuation 

Consider again SUT = 22 ⇒ CUT = 1

S0 = 20

SDT = 18 ⇒ CDT = 0

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Previously, we used the no-arbitrage approach to value the option. We will now value the option by calculating the expected payoff at expiry and discount this back to today.

Risk Neutral Valuation 

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Previously, we used the no-arbitrage approach to value the option. Construct a riskless portfolio in which you hold the Call security short Therefore ∆ many of the underlying shares in an offsetting long position.  Remember, we don’t know what ∆ is! Let u denote the multiplicative UP factor and d=1/u the DOWN factor – see binomial stock price tree.

Risk Neutral Valuation + SUT = (S0.u).∆ - CUT ≡ - CU = - Max[SUT –X ] S0.∆ - C0

+ SDT = (S0.d).∆ - CDT ≡ - CD = - Max[SDT –X ]



Since the portfolio is riskless, we have S 0u.∆ − C U = S 0 d .∆ − C D

(

)

⇒ ∆ = C U − C D / ( S 0u − S 0 d )

…… (i)

Risk Neutral Valuation 

Initally, the portfolio set up cost was: S 0 ∆ − C0



This must be equal to the discounted payoff at maturity (at either PFU or PFD state) S 0 ∆ − C0 = ( S 0u.∆ − C U )e − RF T

(

)

⇒ C0 = S 0 ∆ − S 0u.∆ − C U e − RF T 



Substituting in ∆ from equation (i) ⇒ C0 = e − RF T ( p.C U + (1 − p ) C D ) where

(

p = e RF T − d

) (u − d )

Risk Neutral Valuation 

Consider again: p

SUT

Note : u and d must imply the same but opposite sign proportional change in price – i.e. returns +/- 10%

E0[ST] = ? 1-p

(

p = e RF T − d

SDT

) (u − d )



where



Let u = 1.10, d = 0.90, rF = 12%, T = 0.25



We can calculate p = 0.6523

Risk Neutral Valuation 

Interpreting p and (1-p) as probabilities, we can calculate the expected future stock price ET[ST] = SU.p + SD.(1-p) = 20x1.1x0.6523 + 20x0.9x(1-0.6523) = 20.61

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BUT note that the expected future stock price: ET[S0] = S0.erFT = 20.e0.12(0.25) = 20.61

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In a risk-neutral world, we expect the stock price to grow at the risk free rate

Risk Neutral Valuation 

This means that the probabilities we have used p and (1-p) are the risk-neutral probabilities

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Confirm this by using them to prove that expectations using them can be discounted to the resent value of the asset, I.e. S0 = e-r FT[SU.p + SD.(1-p)] = 20

Risk Neutral Valuation 

Using the Risk-Neutral Probabilities, we can calculate the present value of the Call Option; C0 = e-r FT[CU.p + CD.(1-p)] C0 = e-0.12(0.25)

[1x0.6523 + 0x(1-0.6523)]

= 0.633 

Now we have revalued the call option using the Risk Neutral Valuation

Risk Neutral Valuation 

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Reason we were able to use RN valuation is because we were able to construct a truly riskless PF comprising the Call and Equity securities. The risk-exposure facing the Call holder is unanticipated changes in the underlying equity price which can induce adverse changes in the market value Call security. If this risk can be hedged by constructing a riskless portfolio then we can equivalently value the Call security using either No-Arbitrage (or relative-value) valuation approach or the equivalent RN methodology.

Risk Neutral Binomial Trees 

In practice, we do not know what u and d are but we can estimate their values as a function of time and volatility u = eσ δt d = e −σ δt 



See Hull page 249 for further information

Further, recall that:

(

p = e − RF T − d

) (u − d )

Risk Neutral Binomial Trees 

Assume the following inputs:

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S0 = 100, X = 98, σ = 20% pa, rF = 5% pa, T = 1.0 years Distinct "States Node (2,2) 132.69 Node (1,1)

34.69

of the World" w=2

115.19 Node (0,1)

19.61062

100 11.064

Node (2,1) 100.00

Node (1,0)

2.00

w=1

86.81 1.080

Node (2,0) 75.36 0.00

Node Time: 0.0

Node Upper value = Stock Price Node Lower value = Option Price

0.5

1.0

w=0

Risk Neutral Binomial Trees 

Firstly, calculate the value for u and d u = 1.1519, d = 1/u = 0.8681

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Next, populate the stock price values, going from left to right: 

Node(1,1) Stock price = S0u = 115.19

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Node(2,1) Stock price = S0ud = S0du = 100

Now calculate the terminal option price values 

Node(2,2) Option Price = max(ST-X,0) = 132.69-98 = 34.69

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Node(2,1) Option Price = max(ST-X,0) = 100.00-98 = 2.00

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Node(2,0) Option Price = max(ST-X,0) = 75.36-98 = zero

Risk Neutral Binomial Trees 

Now calculate the intervening option prices going from right to left  



Node(1,1) OP = e-0.05(0.5) Node(1,1) OP = e-0.05(0.5)

[34.68x0.5539+2x0.4461] = 19.6106 [2x0.5539-0x0.4461] = 1.080

And finally the current (t=0) option price  

Node(0,0) OP = e-0.05(0.5) = 11.064

[19.6106x0.5539- 1.080 x0.4461]

Risk Neutral Binomial Trees 

Notice the sum of the risk neutral probabilities

Probability state

Description

w=2

Two up probabilities in succession

= p.p

= 0.3068

w=1

Up followed by down or down followed by up

= p.u or u.p

= 0.4942

w=0

Two down probabilities in succession

= d.d

= 0.1990

Total

= 1.000

Further reading 

Hull, J.C, “Options, Futures & Other Derivatives”, 2009, 7th Ed. 

Chapter 17